STUDIES IN STATISTICAL INFERENCE, SAMPLING TECHNIQUES AND DEMOGRAPHY
Rajesh Singh, Jayant Singh, Florentin Smarandache
ILQ
2009
1
This book can be ordered in a paper bound reprint from:
Books on Demand ProQuest Information & Learning (University of Microfilm International) 300 N. Zeeb Road P.O. Box 1346, Ann Arbor MI 48106-1346, USA
Copyright 2009 by InfoLearnQuest (Ann Arbor) and the Authors
Many books can be downloaded from the following Digital Library of Science: http://www.gallup.unm.edu/~smarandache/eBooks otherformats.htm
Peer Reviewers: Prof. Mihaly Bencze, Department of Mathematics, Ă prily Lajos College, Bra ov, Romania. Dr. Sukanto Bhattacharya, Department of Business Administration, Alaska Pacific University, U.S.A. (ISBN-10): 1-59973-087-1 (ISBN-13): 978-1-59973-087-5 (EAN): 9781599730875 Printed in the United States of America
2
Contents
Preface: 4 1 Optimum Statistical Test Procedure: 5 2 A Note on Testing of Hypothesis: 21 3 Improvement in Estimating Population Mean using Two Auxiliary Variables in Two-Phase Sampling: 26 4 Improved Exponential Estimator for Population Variance Using Two Auxiliary Variables: 36 5 Structural Dynamics Of Various Causes Of Migration In Jaipur: 45
3
Preface
This volume is a collection of five papers. Two chapters deal with problems in statistical inference, two with inferences in finite population, and one deals with demographic problem. The ideas included here will be useful for researchers doing works in these fields. The following problems have been discussed in the book: Chapter 1. In this chapter optimum statistical test procedure is discussed. The test procedures are optimum in the sense that they minimize the sum of the two error probabilities as compared to any other test. Several examples are included to illustrate the theory. Chapter 2. In testing of hypothesis situation if the null hypothesis is rejected will it automatically imply alternative hypothesis will be accepted? This problem has been discussed by taking examples from normal distribution. Chapter 3. In this section improved chain-ratio type estimator for estimating population mean using some known values of population parameter(s) has been discussed. The proposed estimators have been compared with two-phase ratio estimator and some other chain ratio type estimators. Chapter 4. In this section we have analysed exponential ratio and exponential product type estimators using two auxiliary variables are proposed for estimating unknown population variance S 2y . Problem is extended to the case of two-phase sampling. Chapter 5. In this section structural dynamics of various causes of migration in Jaipur was analysed. Reasons of migration from rural to urban areas and that of males and females are studied.
The Authors
4
Optimum Statistical Test Procedure Rajesh Singh Department of Statistics, BHU, Varanasi (U.P.), India rsinghstat@yahoo.com Jayant Singh Department of Statistics Rajasthan University, Jaipur, India Jayantsingh47@rediffmail.com Florentin Smarandache Chair of Department of Mathematics, University of New Mexico, Gallup, USA smarand@unm.edu
Introduction Let X be a random variable having probability distribution P( X / T), T 4 . It is desired to test H 0 : T 4 0 against H1 : T 41 outcomes of an experiment and x
4 4 0 . Let S denote the sample space of
( x 1 , x 2 , , x n ) denote an arbitrary element of S. A test
procedure consists in diving the sample space into two regions W and S – W and deciding to reject H0 if the observed
x W . The region W is called the critical region. The function
J (T) P T ( x W ) = PT ( W) , say, is called the power function of the test.
5
We consider first the case where 40 consists of a single element, T0 and its complement 41 also has a single element T1 . We want to test the simple hypothesis H0 : T T0 against the simple alternative hypothesis H1 : T T1 . Let L0 = L(X/H0) and L1 = L(X/H1) be the likelihood functions under H0 and H1 respectively. In the Neyman – Pearson set up the problem is to determine a critical region W such that
J(T0 )
PT0 (W)
³ L dx
D, an assigned value
0
(1)
w
and J(T1 )
PT1 (W)
³ L dx 1
is maximum
(2)
w
compared to all other critical regions satisfying (1). If such a critical region exists it is called the most powerful critical region of level . By the Neyman-Pearson lemma the most powerful critical region W0 for testing ܪ ǣ ߠ ൌ ߠ against ܪଵ ǣ ߠ ൌ ߠଵ is given by
W0
{x :L1 t kL0 }
provided there exists a k such that (1) is satisfied. For this test J(T 0 )
D and J (T1 ) o 1 as n o D .
But for any good test we must have J(T 0 ) o 0 and J (T1 ) o 1 as n o f because complete discrimination between the hypotheses H0 and H1 should be possible as the sample size becomes indefinitely large.
6
Thus for a good test it is required that the two error probabilities D and E should depend on the sample size n and both should tend to zero as n o f . We describe below test procedures which are optimum in the sense that they minimize the sum of the two error probabilities as compared to any other test. Note that minimizing ( D E ) is equivalent to maximising 1 - ( D E ) = (1 E) D = Power – Size. Thus an optimum test maximises the difference of power and size as compared to any other test. Definition 1 : A critical region W0 will be called optimum if
Âł L dx Âł L dx t Âł L dx Âł L dx 1
w0
0
w0
1
0
w
(3)
w
for every other critical region W. Lemma 1: For testing H0: T
T0 against H1 : T T1 the region
W 0 {x : L1 t L 0 } is optimum. Proof : W0 is such that inside W0, L1 t L 0 and outside W0 , L1 L 0 . Let W be any other critical region.
Âł (L
W0
1
L 0 )dx Âł (L1 L 0 )dx W
=
Âł
W0 W c
(L1 L 0 )dx
Âł (L
1
L 0 )dx ,
W W0c
by subtracting the integrals over the common region W0 W.
7
t0 since the integrand of first integral is positive and the integrand under second integral is negative. Hence (3) is satisfied and W0 is an optimum critical region. Example 1 : Consider a normal population N(T, V 2 ) where V 2 is known. It is desired to test H0: T
n
T0 against H1 : T T1 , T1 ! T 0 . ( x i T) 2
§ 1 ¡ iŒ1 2V 2 L( x T )= ¨¨ ¸¸ e Š V 2S š n
L1 L0
ÂŚ ( x i T1 ) 2
ÂŚ (x i T0 ) 2
e e
2V 2 2V 2
The optimum test rejects Ho
if
L1 t1 L0
i.e. if
L log 1 t 0 L0
i.e. if
i.e. if
ÂŚ ( x i T1 ) 2 ÂŚ ( x i T 0 ) 2 2V 2
2V 2
t0
2T1 ÂŚ x i nT12 2P 0 ÂŚ x i nT 02 t 0
8
i.e. if
2T1 2T 0 ÂŚ x i t n T12 T02
ÂŚ xi
i.e. if
i.e. if
n
T T t 1 0 2
T T0 xt 1 2
Thus the optimum test rejects H0
T T0 if x t 1 2 We have
D
T T0 º ª PH 0  x t 1 2 Ÿ  = PH 0 ª x T 0
t ÂŤ ÂŤÂŹ V / n
Under Ho,
n T1 T 0 º  2V Ÿ
x T 0 follows N(0,1) distribution. §V ¡ ¨ ¸ nš Š
§ n T1 T 0 ¡ ¸ ‍ ׾‏D 1 )¨ ¸ ¨ 2V š Š
where ) is the c.d.f. of a N(0,1) distribution.
E
ª º T T0 º ª  x T1 n T1 T 0  P PH 1  x 1 = H1   V 2 Ÿ 2V   Ÿ n
9
Under H1,
x T1 follows N(0,1) distribution. §V ¡ ¨ ¸ nš Š
§ n T1 T 0 ¡ ¸ E 1 )¨ ¨ ¸ 2V Š š
Here D
E.
It can be seen that D
E o 0 as n o f .
Example 2 : For testing H0: T
T0 against H1 : T T1 , T1 T0 , the optimum test rejects
T T0 H0 when x d 1 . 2
Example 3 : Consider a normal distribution N T, V 2 , T known.
It is desired to test H0 : V 2
V 02 against H1 : V 2
V12 , V12 ! V 02 .
We have
Lx V
2
n
L1 L0
§ V2 ¡ 2 ¨ 0¸ e ¨ V2 ¸ Š 1š
ÂŚ x i T
2
2V12
ÂŚ x i T
2
2V 02
e
L log 1 L0
x i T 2
n
§ 1 ¡ 2 Œ 2V 2 ¨¨ ¸¸ e Š 2SV 2 š
n ÂŚ x i T 2 ÂŚ x i T 2 log V12 log V 02 2 2V12 2V 02
10
=
n ÂŚ x i T V12 V 02 log V12 log V 02 2 2 V12 V 02
2
The optimum test rejects H0
Ý…Ý‚
â€ŤÜŽâ€ŹŕŹľ ŕľ’Íł â€ŤÜŽâ€ŹŕŹ´
V12 V 02 2V12 V 02
i.e. if
ÂŚ x i T 2 t 2 log V12 log V 02
n
ÂŚ x i T 2
i.e. if
V 02
t
nV12 V12
V 02
log V12 log V 02
2
§ x i T ¡ ¸¸ t nc Š V0 š
Œ ¨¨
i.e. if
where c
V12 V12 V 02
log V12 log V02
்೔ ିŕ°? ଶ
Thus the optimum test rejects H0 if Ďƒ ቀ
ఙఏ
በྒ ݊ܿ.
x T Note that under H 0 : i follows N(0,1) distribution. Hence V0
2
§ xi T ¡ ¸¸ follows, under Š V0 š
Œ ¨¨
H0, a chi-square distribution with n degrees of freedom (d. f.).
Here
D
ª §x T 0 PH 0 Œ ¨¨ i  Š V0 
2 º ¡ ¸¸ t nc  š Ÿ
>
P F (2n ) t nc
@
ª § x T ¡2 º ¸¸ t nc and 1 – = PH 1 Œ ¨¨ i  Š V0 š   Ÿ
11
ª § x T · 2 ncV 2 º 0» ¸ t = PH 1 «¦ ¨¨ i 2 » « © V1 ¸¹ V 1 ¬ ¼
ª ncV 02 º = PH1 «F (2n ) t » V12 ¼» ¬« ௫ ିఏ ଶ
ͳǡ σ ቀ
ఙభ
ቁ
follows a chi-square distribution with n d.f.
It can be seen that D o 0 and E o 0 as n o f. Example 4 : Let X follow the exponential family distributions f x T c T e Q T T ( x ) h ( x )
It is desired to test H0: T
L( x T)
T0 against H1 : T T1
>c(T)@n e Q(T)¦ T( x i ) h ( x i ) i
The optimum test rejects Ho when
݈݃
ܮଵ Ͳ ܮ
i.e. when >Q(T1 ) Q(T 0 )@¦ T( x i ) t n log
c( T 0 ) c(T1 ) i.e. when ¦ T( x i ) t >Q(T1 ) Q(T 0 )@
c(T 0 ) c(T1 )
n log
if Q(T1 ) Q(T0 ) ! 0
and rejects Ho,
12
c(T 0 ) c(T1 ) when ¦ T( x i ) d >Q(T1 ) Q(T 0 )@ n log
if Q(T1 ) Q(T0 ) 0
Locally Optimum Tests:
Let the random variable X have probability distribution ܲሺ ݔΤߠሻ. We are interested in testing H0: T
T0 against H1 : T !T0 . If W is any critical region then the power of the test
as a function of ߠ is ߛሺߠሻ ൌ ܲఏ ሺܹሻ ൌ ௐ ܮሺܺΤߠ ሻ݀ݔ
We want to determine a region W for which ߛሺߠሻ െ ߛሺߠ ሻ ൌ ௐ ܮሺ ݔΤߠ ሻ݀ ݔെ ௐ ܮሺ ݔΤߠሻ
is a maximum. When a uniformly optimum region does not exist, there is not a single region which is best for all alternatives. We may, however, find regions which are best for alternatives close to the null hypothesis and hope that such regions will also do well for distant alternatives. We shall call such regions locally optimum regions. Let ߛሺߠሻ admit Taylor expansion about the point ߠ ൌ ߠ . Then ߛሺߠሻ ൌ ߛሺߠ ሻ ሺߠ െ ߠ ሻߛƴ ሺߠ ሻ ߜ
where ߜ ՜ Ͳ as ߠ ՜ ߠ
ߛ ሺߠሻ െ ߛሺߠ ሻ ൌ ሺߠ െ ߠ ሻߛƴ ሺߠ ሻ ߜ
If ȁߠ െ ߠ ȁ is small ߛሺߠሻ െ ߛሺߠ ሻ is maximised when ߛƴ ሺߠ ሻis maximised.
13
Definition2 : A region W0 will be called a locally optimum critical region if ௐ ܮሖሺ ݔΤߠ ሻ݀ ݔ ௐ ܮሖሺ ݔΤߠ ሻ݀ݔ
(4)
బ
For every other critical region W. Lemma 2 : Let W0 be the region ൛ݔǣ ܮሖሺ ݔΤߠ ሻ ܮሺ ݔΤߠ ሻൟ. Then W0 is locally optimum. Proof: Let W0 be the region such that inside it ܮሖሺ ݔΤߠ ሻ ܮሺ ݔΤߠ ሻ and outside it ܮሖሺ ݔΤߠ ሻ ൏ ܮሺ ݔΤߠ ሻ. Let W be any other region. ௐ ܮሖሺ ݔΤߠ ሻ݀ ݔെ ௐ ܮሖሺ ݔΤߠ ሻ݀ݔ బ
=
ௐ
బתೈ
ܮሖሺ ݔΤߠ ሻ݀ ݔെ ௐ תௐ ܮሖሺ ݔΤߠ ሻ݀ݔ
=
ௐ
బתೈ
ܮሖሺ ݔΤߠ ሻ݀ ݔ ௐ ௐ ܮሖሺ ݔΤߠ ሻ݀ݔ
ௐ
బתೈ
ܮሺ ݔΤߠ ሻ݀ ݔ ௐ ௐ ܮሺ ݔΤߠ ሻ݀ݔ
బ
బ
(*)
బ
since ܮሖሺ ݔΤߠ ሻ ܮሺ ݔΤߠ ሻ inside both the regions of the integrals. 0,
Hence
since ܮሺ ݔΤߠ ሻ Ͳ
in all the regions.
ௐ ܮሖሺ ݔΤߠ ሻ݀ ݔ ௐ ܮሖሺ ݔΤߠ ሻ݀ ݔfor every other region W. బ
To prove (*):
We have ோ ܮሺ ݔΤߠ ሻ݀ ݔ ோ ܮሺ ݔΤߠ ሻ݀ ݔൌ ͳ for every region R. Differentiating we have
14
න ܮሖሺ ݔΤߠ ሻ݀ ݔ න ܮሖሺ ݔΤߠ ሻ݀ ݔൌ Ͳ ோ
ோ
න ܮሖሺ ݔΤߠ ሻ݀ ݔൌ െ න ܮሖሺ ݔΤߠ ሻ݀ ݔൌ Ͳ ோ
ோ
In (*), take ܴ =ܹ ܹ תand the relation is proved. Similarly if the alternatives are H1 : T T0 , the locally optimum critical region is ൛ݔǣ ܮሖሺ ݔΤߠ ሻ ܮሺ ݔΤߠ ሻൟ. Example 5: Consider ܰሺߠǡ ߪ ଶ ሻ distribution, ߪ ଶ known.
It is desired to test H0: T
ܮሺ ݔΤߠ ሻ ൌ ൬
ͳ ߪξʹߨ
൰ ݁
T0 against H1 : T !T0
ି σሺ௫ ିఏሻమ ଶఙ మ
ͳ σሺݔ െ ߠሻଶ ݈ܮ݃ሺ ݔΤߠሻ ൌ ݈݊ ݃൬ ൰െ ʹߪ ଶ ߪξʹߨ ܮሖሺ ݔΤߠ ሻ ߜ݈ܮ݃ሺ ݔΤߠሻ σሺݔ െ ߠሻ ݊ሺݔҧ െ ߠሻ ൌ ൌ ൌ ߜߠ ߪଶ ܮሺ ݔΤߠ ሻ ߪଶ
ܮሖሺ ݔΤߠ ሻ ݊ሺݔҧ െ ߠ ሻ ൌ ܮሺ ݔΤߠ ሻ ߪଶ
The locally optimum test rejects H0 ,if ݊ሺݔҧ െ ߠ ሻ ͳ ߪଶ
i.e. ݔҧ ߠ
ఙమ
Now,
15
ߙ ൌ ܲுబ ቈݔҧ ߠ
ߪଶ ݊
ݔҧ െ ߠ ߪ ൌ ܲுబ ߪ ൗ ξ݊ ൗ ݊ ξ ൌ ͳ െ Ȱ ൬ߪൗ ൰, since under H0, ξ݊
௫ҧ ିఏబ ఙ ൗ ξ
follows N(0,1) distribution.
ଶ
ͳ െ ߚ ൌ ܲுభ ݔҧ ߠ ߪ ൗ݊
ݔҧ െ ߠଵ ߠଵ െ ߠ ߪ ൌ ܲுభ ߪ െ ߪ ξ݊ ൗ ݊ ൗ ݊ ξ ξ െ൫ߠଵ െ ߠ ξ݊൯ ߪ ൌ ͳ െ Ȱቈ ߪ ξ݊
since under H1,
௫ҧ ିఏభ ఙ ൗ ξ
follows N(0,1) distribution.
Exercise : If ߠ ൌ ͳͲǡ ߠଵ ൌ ͳͳǡ ߪ ൌ ʹǡ ݊ ൌ ͳǡ then ߙ = 0.3085, 1-ߚ = 0.9337
Power - Size = 0.6252 ௫ҧ ିఏబ
If we reject H0 when ఙ
ൗ ξ
> 1.64, then ߙ = 0.05, 1-ߚ = 0.6406
Power – Size = 0.5906 Hence Power – Size of locally optimum test is greater than Power – size of the usual test. Locally Optimum Unbiased Test:
16
Let the random variable X follows the probability distribution ܲሺ ݔΤߠ ሻ. Suppose it is desired to test H0: T
T0 against H1 : T zT0 . We impose the unbiasedness restriction
ߛሺߠሻ ߛሺߠ ሻǡ ߠ ് ߠ
and ߛሺߠሻ െ ߛሺߠ ሻ is a maximum as compared to all other regions. If such a region does not exist we impose the unbiasedness restriction ߛƴ ሺߠ ሻ ൌ ͲǤ Let ߛሺߠሻ admit Taylor expansion about the point ߠ ൌ ߠ Ǥ Then ߛሺߠሻ ൌ ߛሺߠ ሻ ሺߠ െ ߠ ሻߛƴ ሺߠ ሻ
ሺఏିఏబ ሻమ ଶ
J cc T 0 +ߟ
where ߟ ՜ Ͳ as ߠ ՜ ͲǤ ߛ ሺߠሻ െ ߛሺߠ ሻ ൌ ሺߠ െ ߠ ሻߛƴ ሺߠ ሻ
ሺఏିఏబ ሻమ ଶ
J cc T 0 +ߟ
Under the unbiasedness restriction ߛƴ ሺߠ ሻ = 0, if ȁߠ െ ߠ ȁ is small ߛሺߠሻ െ ߛሺߠ ሻ is maximised when J cc T 0 is maximised. Definition 3 : A region W0 will be called a locally optimum unbiased region if ߛƴ ሺߠ ሻ ൌ ௐ ܮሖሺ ݔΤߠ ሻ݀ ݔൌ Ͳ
(5)
బ
and
J cc T 0 =
³ Lcc X T 0 dx t ³ Lcc X T 0 dx
W0
(6)
W
for all other regions W satisfying (5). Lemma 3 : Let W0 be the region
^x : Lcc x T0 t L x T0 `
Then W0 is locally optimum unbiased.
17
Proof : Let W be any other region ௐ ܮᇱᇱ ሺ ݔΤߠ ሻ݀ ݔെ ௐ ܮᇱᇱ ሺ ݔΤߠ ሻ݀ݔ బ
න ܮᇱᇱ ሺ ݔΤߠ ሻ݀ ݔ
ൌ
න ܮᇱᇱ ሺ ݔΤߠ ሻ݀ݔ ௐబ תௐ
ௐబ תௐ
by subtracting the common area of W and W0. න ܮᇱᇱ ሺ ݔΤߠ ሻ݀ ݔ
ൌ
ௐబ תௐ
න ௐబ
ܮᇱᇱ ሺ ݔΤߠ ሻ݀ݔǡ
תௐ
since ܮᇱᇱ ሺ ݔΤߠ ሻ ܮሺ ݔΤߠ ሻ inside W0 and outside W. Ͳ
since
ܮሺ ݔΤߠሻ ͲǤ
Example 6:
Consider ܰሺߠǡ ߪ ଶ ሻ distribution, ߪ ଶ known . ܪ ǣ ߠ ൌ ߠ ǡ ܪଵ ǣ ߠ ് ߠ
ܮൌ൬
ͳ ߪξʹߨ
൰ ݁
ି
σሺ௫ ିఏሻమ ଶఙమ
̶ܮሺ ݔΤߠ ሻ ݊ଶ ሺݔҧ െ ߠሻଶ ݊ ൌ െ ଶ ߪ ߪସ ܮሺ ݔΤߠሻ
Locally optimum unbiased test rejects H0 ݂݅
݊ଶ ሺݔҧ െ ߠ ሻଶ ݊ െ ଶͳ ߪ ߪସ
݅Ǥ ݁Ǥ
݊ሺݔҧ െ ߠ ሻଶ ߪଶ ͳ ଶ ߪ ݊
ܷ݊݀݁ܪ ݎǡ
ሺ௫ҧ ିఏబ ሻమ ఙమ
ଶ follows ߯ሺଵሻ distribution.
18
Testing Mean of a normal population when variance is unknown.
Consider ܰሺߠǡ ߪ ଶ ሻ distribution, ߪ ଶ known. For testing H0: T
T0 against H1 : T T1 , the critical function of the optimum test is
given by ߶ ሺݔሻ ൌ ቄ
ͳ ݂݅ ܮሺ ݔΤߠଵ ሻ ܮሺ ݔΤߠ ሻ Ͳ ݁ݏ݅ݓݎ݄݁ݐ
On simplification we get ߠ ߠଵ ݂݅ ߤଵ ߤ ߶ ሺݔሻ ൌ ൝ͳ ݂݅ ݔҧ ʹ Ͳ ݁ݏ݅ݓݎ݄݁ݐ Ʌ Ʌଵ ρଵ ൏ ρ Ԅ୫ ሺ ሻ ൌ ൝ ͳ ത ൏ ʹ Ͳ Consider the case when ࣌ is unknown.
For this case we propose a test which rejects H0 when ܮሺ ݔΤߠଵ ሻ ͳ ܮሺ ݔΤߠ ሻ
where ܮሺ ݔΤߠ ሻ, (i=0,1) is the maximum of the likelihood under Hi obtained from ܮሺ ݔΤߠ ሻ by replacing ߪ ଶ by its maximum likelihood estimate ଶ ଵ ଶ ߪ ప ൌ σୀଵ൫ݔ െ ߠ ൯ ; i=0,1.
Let ߶ ሺݔሻ denote the critical function of the proposed test, then ሺ ݔΤߠଵ ሻ ܮሺ ݔΤߠ ሻ ߶ ሺݔሻ ൌ ൜ͳ ݂݅ ܮ Ͳ ݁ݏ݅ݓݎ݄݁ݐ
19
On simplification we get
߶ ሺݔሻ ൌ ቊ
ͳ ݂݅ ݔҧ
ఏబ ାఏభ ଶ
݂݅ ߠଵ ߠ
݁݅ݏݓݎ݄݁ݐ ఏ ାఏ
and
ͳ ݂݅ ݔҧ ൏ బ భ ݂݅ ߠଵ ൏ ߠ ߶ ሺݔሻ ൌ ቊ ଶ ݁݅ݏݓݎ݄݁ݐ
Thus the proposed test ߶ ሺݔሻ is equivalent to ߶ ሺݔሻ which is the optimum test that minimizes the sum of two error probabilities ሺߙ ߚሻ. Thus we see that one gets the same test which minimises the sum of the two error probabilities irrespective of whether ߪ ଶ is known or unknown. Acknowledgement : Authors are thankful to Prof. Jokhan Singh for his guidance in writing
this article. References
Pandey, M. and Singh,J. (1978) : A note on testing mean of a normal population when variance is unknown. J.I.S.A. 16, pp. 145-146. Pradhan,M. (1968) : On testing a simple hypothesis against a simple alternative making the sum of the probabilities of two errors minimum. J.I.S.A. 6, pp. 149-159. Rao, C.R.(1973) : Linear Statistical Inference and its Applications. John Wiley and Sons Singh,J. (1985): Optimum Statistical Test Procedures. Vikram Mathematical Journal Vol V, pp. 53-56.
20
A Note on Testing of Hypothesis
Rajesh Singh School of Statistics, D. A.V.V., Indore (M.P.), India rsinghstat@yahoo.co.in
Jayant Singh Department of Statistics Rajasthan University, Jaipur, India Jayantsingh47@rediffmail.com
Florentin Smarandache Chair of Department of Mathematics, University of New Mexico, Gallup, USA fsmarandache@yahoo.com
Abstract :
In testing of hypothesis situation if the null hypothesis is rejected will it automatically imply alternative hypothesis will be accepted. This problem has been discussed by taking examples from normal distribution. Keywords : Hypothesis, level of significance, Baye’s rule.
21
1. Introduction
Let the random variable (r.v.) X have a normal distribution N( T , V 2 ), V 2 is assumed to be known. The hypothesis H0 : T = T 0 against H1 : T = T1 , T1 > T 0 is to be tested. Let X1, X2, …, Xn be a random sample from N( T , V 2 ) population. Let X (=
1 n
n
¦X
i
) be the sample
i 1
mean. By Neyman – Pearson lemma the most powerful test rejects H0 at D % level of significance, n X T o
t d D , where d D is such that V
if
f
³
dD
1 2S
e
Z2 2 dZ
=D
If the sample is such that H0 is rejected then will it imply that H1 will be accepted? In general this will not be true for all values of T1 , but will be true for some specific value of T1 i.e., when T1 is at a specific distance from T 0 .
H0 is rejected if
i.e. X t T 0 + d D
n X T o
t dD V
V n
(1)
Similarly the Most Powerful Test will accept H1 against H0
if
n X T1
t - dD V
22
V
X t T1 - d D
i.e.
(2)
n
Rejecting H0 will mean accepting H1 (1) Â&#x; (2)
if
i.e.
X t T0 + d D
i.e.
T1 - d D
V
V n
V n
V
d T0 + d D
n
X t T1 - d D
Â&#x;
(3)
n
Similarly accepting H1 will mean rejecting H0 (2) Â&#x; (1)
if
V
T0 + d D
i.e.
V
T1 - d D
d
n
(4)
n
From (3) and (4) we have
T0 + d D
V n
= T1 - d D
i.e. T1 - T 0 = 2 d D
Thus d D
V n
=
V n
V
(5)
n
V T1 T 0 and T1 = T 0 +2 d D . 2 n
Reject H0 if
X >
T0 +
and from (2) Accept H1 if
X >
T1 -
From (1)
T1 T 0 T T = 0 1 2 2 T1 T 0 = 2
T 0 T1 2
Thus rejecting H0 will mean accepting H1
23
when
T 0 T1 . 2
X >
From (5) this will be true only when T1 = T 0 + 2 d D
T1 z T 0 + 2 d D
V n
V n
. For other values of
rejecting H0 will not mean accepting H1.
It is therefore, recommended that instead of testing H0 : T = T 0 against H1 : T = T1 , T1 > T 0 , it is more appropriate to test H0 : T = T 0 against H1 : T > T 0 . In this situation rejecting H0 will mean T > T 0 and is not equal to some given value T1 . But in Baye’s setup rejecting H0 means accepting H1 whatever may be T 0 and T1 . In this set up the level of significance is not a preassigned constant, but depends on T 0 , T1 , V 2 and n. Consider (0,1) loss function and equal prior probabilities ½ for T 0 and T1 . The Baye’s test rejects H0 (accepts H1)
if
X >
T 0 T1 2
and accepts H0 (rejects H1)
if
X <
T 0 T1 . 2
[See Rohatagi, p.463, Example 2.] The level of significance is given by
PH0 [ X >
(T T ) n (X T0 ) n T 0 T1 ] = PH0 [ > 1 0 ] 2 V 2V
24
§ n (T1 T 0 2V ©
= 1 - ) ¨¨
t
where )(t ) =
³
f
1 2S
e
Z2 2 dZ
· ¸ ¸ ¹
.
Thus the level of significance depends on T 0 , T1 , V 2 and n. Acknowledgement : Author’s are thankful to Prof. Jokhan Singh for suggesting this
problem.
References
Lehmann, E.L. (1976) : Testing Statistical Hypotheses, Wiley Eastern Ltd., New Delhi. Rohatagi, V.K. (1985) : An introduction to probability theory and mathematical statistics, Wiley Eastern Ltd., New Delhi.
25
Improvement in Estimating Population Mean using Two Auxiliary Variables in Two-Phase Sampling
Rajesh Singh Department of Statistics, Banaras Hindu University(U.P.), India (rsinghstat@yahoo.com) Pankaj Chauhan and Nirmala Sawan School of Statistics, DAVV, Indore (M.P.), India Florentin Smarandache Department of Mathematics, University of New Mexico, Gallup, USA (smarand@unm.edu) Abstract
This study proposes improved chain-ratio type estimator for estimating population mean using some known values of population parameter(s) of the second auxiliary character. The proposed estimators have been compared with two-phase ratio estimator and some other chain ratio type estimators. The performances of the proposed estimators have been supported with a numerical illustration. Key words: Auxiliary variables, chain ratio-type estimator, bias, mean squared error. 1. Introduction
The ratio method of estimation is generally used when the study variable Y is positively correlated with an auxiliary variable X whose population mean is known in advance. In the absence of the knowledge on the population mean of the auxiliary character we go for twophase (double) sampling. The two-phase sampling happens to be a powerful and cost effective (economical) procedure for finding the reliable estimate in first phase sample for the unknown parameters of the auxiliary variable x and hence has eminent role to play in survey sampling, for instance, see Hidiroglou and Sarndal (1998).
26
Consider a finite population U
(U1 , U 2 ,......, U N ) . Let y and x be the study and
auxiliary variable, taking values yi and x i respectively for the i th unit U i . Allowing SRSWOR (Simple Random Sampling without Replacement) design in each phase, the two-phase sampling scheme is as follows: the first phase sample s nc (s nc U) of a fixed size n c is drawn to measure only x
(i)
in order to formulate a good estimate of a population mean X , Given s nc , the second phase sample s n (s n s nc ) of a fixed size n is drawn to
(ii)
measure y only. Let x
1 ¦ xi , y n i sn
1 ¦ yi and xc n i s n
1 ¦ xi . n c i s nc
The classical ratio estimator for Y is defined as
yr
y X x
(1.1)
If X is not known, we estimate Y by two-phase ratio estimator
y rd
y xc x
(1.2)
Some times even if X is not known, information on a cheaply ascertainable variable z, closely related to x but compared to x remotely related to y, is available on all units of the population. For instance, while estimating the total yield of wheat in a village, the yield and area under the crop are likely to be unknown, but the total area of each farm may be known from village records or may be obtained at a low cost. Then y, x and z are respectively yield, area under wheat and area under cultivation see Singh et al. (2004). Assuming that the population mean Z of the variable z is known, Chand (1975) proposed a chain type ratio estimator as t1
y § xc · ¨ ¸Z x © zc ¹
(1.3)
Several authors have used prior value of certain population parameter(s) to find more precise estimates. Singh and Upadhyaya (1995) used coefficient of variation of z for defining
27
modified chain type ratio estimator. In many situation the value of the auxiliary variable may be available for each unit in the population, for instance, see Das and Tripathi (1981). In such situations knowledge on Z , C z , E1 (z) (coefficient of skewness), E 2 (z) (coefficient of kurtosis) and possibly on some other parameters may be utilized. Regarding the availability of information on C z , E1 (z) and E 2 (z) , the researchers may be referred to Searls(1964), Sen(1978), Singh et al.(1973), Searls and Intarapanich(1990) and Singh et.al.(2007). Using the known coefficient of variation C z and known coefficient of kurtosis E 2 (z) of the second auxiliary character z Upadhyaya and Singh (2001) proposed some estimators for Y . If the population mean and coefficient of variation of the second auxiliary character is known, the standard deviation V z is automatically known and it is more meaningful to use the V z in addition to C z , see Srivastava and Jhajj (1980). Further, C z , E1 (z) and E 2 (z) are the unit free constants, their use in additive form is not much justified. Motivated with the above justifications and utilizing the known values of V z , E1 (z) and E 2 (z) , Singh (2001) suggested some modified estimators for Y . In this paper, under simple random sampling without replacement (SRSWOR), we have suggested improved chain ratio type estimator for estimating population mean using some known values of population parameter(s). 2. The suggested estimator
The work of authors discussed in section 1 can be summarized by using following estimator t
§ x c ¡§ aZ b ¡ y¨ ¸¨¨ ¸¸ Š x šŠ az c b š
(2.1)
where a (z 0) , b are either real numbers or the functions of the known parameters of the second auxiliary variable z such as standard deviation ( V z ), coefficient of variation ( C z ), skewness ( E1 (z) ) and kurtosis ( E 2 (z) ). The following scheme presents some of the important known estimators of the population mean which can be obtained by suitable choice of constants a and b.
28
Estimator
t1
§ x c ¡§ Z ¡ y¨ ¸¨¨ ¸¸ Š x šŠ z c š
Values of a
b
1
0
1
Cz
E 2 ( z)
Cz
Cz
E 2 ( z)
1
Vz
E1 (z)
Vz
E 2 ( z)
Vz
Chand (1975) chain ratio type estimator
t2
§ x c ¡§ Z C z ¡ ¸¸ y¨ ¸¨¨ Š x šŠ zc C z š
Singh and Upadhyaya (1995) estimator
§ x c ¡§ E (z) Z C z ¡ ¸¸ y¨ ¸¨¨ 2 Š x šŠ E 2 (z) zc C z š
t3
Upadhyaya and Singh (2001) estimator
§ x c ¡§ C Z E 2 (z) ¡ ¸¸ y¨ ¸¨¨ z Š x šŠ C z zc E 2 (z) š
t4
Upadhyaya and Singh (2001) estimator
t5
§ x c ¡§ Z V z ¡ ¸ y¨ ¸¨¨ Š x šŠ zc V z ¸š
Singh (2001) estimator
t6
§ x c ¡§ E (z) Z V z ¡ ¸¸ y¨ ¸¨¨ 1 Š x šŠ E1 (z) zc V z š
Singh (2001) estimator
t7
§ x c ¡§ E (z) Z V z ¡ ¸¸ y¨ ¸¨¨ 2 Š x šŠ E 2 (z) zc V z š
In addition to these estimators a large number of estimators can also be generated from the estimator t at (2.1) by putting suitable values of a and b. Following Kadilar and Cingi (2006), we propose modified estimator combining t 1 and t i
i
2,3,....,7 as follows
29
Dt1 1 D t i , i
t *i
2,3,....,7
(2.2)
where D is a real constant to be determined such that MSE of t *i is minimum and t i
i
2,3,....,7 are estimators listed above.
To obtain the bias and MSE of t *i , we write
X 1 e1 , xc
Y 1 e 0 , x
y
X 1 e1c , zc
Z 1 ec2
such that
E e 0 = E e1 = E e1c = E ec2 = 0 and
f1C 2y ,
E e12
f 2 C 2z ,
E ( e 0 e1 )
E e 02
E ec22
E (e 0 ec2 )
f 2 U yz C y C z , E(e1e1c )
E(e1cec2 )
f 2U xz C x C z
E e1c 2
f1C2x ,
f 2 C 2x
f1U xy C x C y , E (e 0 e1c )
f 2 U xy C x C y
E(e1ec2 )
f 2U xz C x C z
f 2 C2x ,
where f1
C 2y
U xy
§1 1 ¡ ¨ ¸, Šn Nš S 2y Y
2
,
S xy Sx Sy
,
f2
§1 1¡ ¨ ¸, Š nc N š
C 2x
S 2x , X2
C 2z
U xz
S xz , S x Sz
U yz
S yz S ySz
S 2y
N 1 ÂŚ iÂ?U
Y ,
S 2x
N 1 ÂŚ iÂ?U
S2z
1 z i Z 2 , ÂŚ N 1 iÂ?U
S xy
1 x i X y i Y
N 1 ÂŚ iÂ?U
1
y
S 2z Z2
2
i
1
x
X
2
i
30
1
N 1 ÂŚ iÂ?U
S xz
x
i
X z i Z ,
S yz
1
N 1 ÂŚ iÂ?U
y
i
Y z i Z .
Expressing t *i in terms of eâ&#x20AC;&#x2122;s, we have
>
where T
@
1 1 1 1 (2.3) Y 1 e 0 D 1 e1c 1 e1 1 ec2 1 D 1 e1c 1 e1 1 Tec2
t *i
aZ aZ b
(2.4)
Expanding the right hand side of (2.3) and retaining terms up to second power of eâ&#x20AC;&#x2122;s, we have t *i # Y>1 e 0 e1 e1c ec2 (D T DT)@
(2.5)
t *i Y # Y >e 0 e1 e1c ec2 (D T DT)@
(2.6)
or
Squaring both sides of (2.6) and then taking expectation, we get the MSE of the estimator t *i , up to the first order of approximation, as
MSE ( t *i )
>
Y 2 f1C 2y f 3C 2x (D T DT) 2 f 2 C 2z 2f 3UC y C x 2(D T DT)f 2 UC y C z
@
(2.7) where f3
§1 1 ¡ ¨ ¸. Š n nc š
Minimization of (2.7) with respect to D yield its optimum value as D opt
K yz T 1 T
(2.8)
where K yz
U yz
Cy Cz
.
31
Substitution of (2.8) in (2.7) yields the minimum value of MSE ( t *i ) as â&#x20AC;&#x201C;
min .MSE ( t *i )
>
Y 2 f1C 2y f 3 (C 2x 2U yx C y C x ) f 2U 2yz C 2y
Mo
@
(2.9)
3. Efficiency comparisons
In this section, the conditions for which the proposed estimator is better than t i (i 1,2,....7) have been obtained. The MSEâ&#x20AC;&#x2122;s of these estimators up to the order o(n ) 1 are derived as â&#x20AC;&#x201C; MSE ( y rd )
>
Y 2 f1C 2y f 3 (C 2x 2U yx C y C x )
@
>
(3.1)
@
MSE ( t 1 )
Y 2 f1C 2y f 2 (C 2z 2U yz C y C z ) f 3 (C 2x 2U yx C y C x )
MSE ( t 2 )
Y 2 f1C 2y f 2 (T 22 C 2z 2T 2U yz C y C z ) f 3 (C 2x 2U yx C y Cx )
MSE ( t 3 )
Y 2 f1C 2y f 2 (T32 C 2z 2T3U yz C y C z ) f 3 (C 2x 2U yx C y Cx )
MSE ( t 4 )
Y 2 f1C 2y f 2 (T 24 C 2z 2T 4U yz C y C z ) f 3 (C 2x 2U yx C y Cx )
MSE ( t 5 )
Y 2 f1C 2y f 2 (T52 C 2z 2T5U yz C y C z ) f 3 (C 2x 2U yx C y Cx )
MSE ( t 6 )
Y 2 f1C 2y f 2 (T 62 C 2z 2T 6U yz C y C z ) f 3 (C 2x 2U yx C y Cx )
MSE ( t 7 )
Y 2 f1C 2y f 2 (T 72 C 2z 2T 7 U yz C y C z ) f 3 (C 2x 2U yx C y Cx )
(3.2)
>
@
(3.3)
>
@
(3.4)
>
@
(3.5)
>
@
(3.6)
>
@
(3.7)
>
@
(3.8)
and
where
E 2 ( z) Z , T4 E 2 ( z) Z C z
T2
Z , T3 Z Cz
T6
E1 (z) Z , T7 E1 (z) Z V z
Cz Z , T5 C 2 Z E 2 (z)
Z , Z Vz
E 2 ( z) Z . E 2 (z) Z V z
From (2.9) and (3.1), we have MSE( y rd ) M o
f 2U 2yz C 2y t 0
(3.9)
Also from (2.9) and (3.2)-(3.8), we have
32
2 f 2 Ti C z U yz C y t 0 , i
MSE ( t i ) M o
2,3,....,7
(3.10)
Thus it follows from (3.9) and (3.10) that the suggested estimator under optimum condition is always better than the estimator t i (i 1,2,....7) . 4. Empirical study
To illustrate the performance of various estimators of Y , we consider the data used by Anderson (1958). The variates are y : Head length of second son x : Head length of first son z : Head breadth of first son N
25 ,
Y 183.84 =,
Cx
0.0526 , C z
X 185.72 ,
0.0488 , U yx
Z 151.12 ,
0.7108 , U yz
Vz
7.224 ,
0.6932 , U xz
Cy
0.0546 ,
0.7346 , E1 z 0.002 ,
E 2 z 2.6519 . Consider n c 10 and n = 7. We have computed the percent relative efficiency (PRE) of different estimators of Y with respect to usual estimator y and compiled in the table 4.1: Table 4.1: PRE of different estimators of Y with respect to y
estimator
PRE
y
100
y rd
122.5393
t1
178.8189
t2
178.8405
t3
178.8277
t4
186.3912
t5
181.6025
t6
122.5473
33
t7
179.9636
t *i
186.6515
5. Conclusion
We have suggested modified estimators t *i (i
2,3,....,7) . From table 4.1, we conclude that the
proposed estimators are better than usual two-phase ratio estimator y rd , Chand (1975) chain type ratio estimator t 1 , estimator t 2 proposed by Singh and Upadhyaya (1995), estimators
t i i
3,4 and than that of Singh (2001) estimators t i i
5,6,7 . For practical purposes the
choice of the estimator depends upon the availability of the population parameter(s).
References
Anderson, T. W. (1958), An Introduction to Multivariate Statistical Analysis. John Wiley & Sons, Inc., New York. Chand, L. (1975): Some ratio type estimators based on two or more auxiliary variables. Unpublished Ph. D. thesis, Iowa State University, Ames, Iowa (USA). Das, A. K. and Tripathi, T. P. (1981): a class of sampling strategies for population mean using information on mean and variance of an auxiliary character. Proc. of the Indian Statistical Institute Golden Jubilee International Conference on Statistics. Applications and New Directions, Calcutta, 16-19, December 1981, 174-181. Hidiroglou, M. A. and Sarndal, C.E. (1998): Use of auxiliary information for two-phase sampling. Survey Methodology, 24(1), 11-20. Searls, D.T. (1964): The utilization of known coefficient of variation in the estimation procedure. Journal of American Statistical Association, 59, 1125-1126. Searls, D.T. and Intarapanich, R. (1990): A note on an estimator for variance that utilizes the kurtosis. Amer. Stat., 44(4), 295-296. Sen, A.R. (1978): Estimation of the population mean when the coefficient of variation is known. Comm. Stat.-Theory Methods, A7, 657-672.
34
Singh, G. N. (2001): On the use of transformed auxiliary variable in the estimation of population mean in two-phase sampling. Statistics in Transition, 5(3), 405416. Singh, G. N. and Upadhyaya, L. N. (1995): A class of modified chain type estimators using two auxiliary variables in two-phase sampling. Metron, LIII,
117-125.
Singh, H. P, Upadhyaya, L. N. and Chandra, P. (2004): A general family of estimators for estimating population mean using two auxiliary variables in two-phase sampling. Statistics in transition, 6(7), 1055-1077. Singh, J., Pandey, B. N. and Hirano, K. (1973): On the utilization of a known coefficient of kurtosis in the estimation procedure of variance. Ann. Inst. Statist. Math., 25, 51-55. Singh, R. Chauhan, P. Sawan, N. and Smarandache, F. (2007): Auxiliary information and a priori values in construction of improved estimators. Renaissance high press, USA. Srivastava, S.K. and Jhajj, H.S. (1980): A class of estimators using auxiliary information for estimating finite population variance. Sankhya, C, 42, 87-96. Upadhyaya, L. N. and Singh, G. N. (2001): Chain type estimators using transformed auxiliary variable in two-phase sampling. Advances in Modeling and Analysis, 38, (12), 1-10.
35
Improved Exponential Estimator for Population Variance Using Two Auxiliary Variables
Rajesh Singh Department of Statistics,Banaras Hindu University(U.P.), India (rsinghstat@yahoo.com) Pankaj Chauhan and Nirmala Sawan School of Statistics, DAVV, Indore (M.P.), India Florentin Smarandache Department of Mathematics, University of New Mexico, Gallup, USA (smarand@unm.edu)
Abstract
In this paper exponential ratio and exponential product type estimators using two auxiliary variables are proposed for estimating unknown population variance S 2y . Problem is extended to the case of two-phase sampling. Theoretical results are supported by an empirical study. Key words: Auxiliary information, exponential estimator, mean squared error. 1.
Introduction
It is common practice to use the auxiliary variable for improving the precision of the estimate of a parameter. Out of many ratio and product methods of estimation are good examples in this context. When the correlation between the study variate and the auxiliary variate is positive (high) ratio method of estimation is quite effective. On the other hand, when this correlation is negative (high) product method of estimation can be employed effectively. Let y and (x,z) denotes the study variate and auxiliary variates taking the values yi and (xi,zi) respectively, on the unit Ui (i=1,2,â&#x20AC;Śâ&#x20AC;Ś,N), where x is positively correlated with
36
y and z is negatively correlated with y. To estimate S 2y that S 2x
N 1 (x i X ) 2 ÂŚ ( N 1) i 1
and S 2z
N 1 ( y i y) 2 , it is assumed ÂŚ ( N 1) i 1
N 1 ( z i Z) 2 are known. Assume that ÂŚ ( N 1) i 1
population size N is large so that the finite population correction terms are ignored. Assume that a simple random sample of size n is drawn without replacement (SRSWOR) from U. The usual unbiased estimator of S 2y is
s 2y
where y
n 1 ( y i y) 2 ÂŚ ( n 1) i 1
(1.1)
1 n ÂŚ y i is the sample mean of y. ni1
When the population variance S 2x
N 1 ( x i X ) 2 is known, Isaki (1983) proposed a ÂŚ ( N 1) i 1
ratio estimator for S 2y as
tk
where s 2x
s 2y
S 2x s 2x
(1.2)
n 1 2 ( x i X ) 2 is an unbiased estimator of S x . ÂŚ ( n 1) i 1
Up to the first order of approximation, the variance of S 2y and MSE of tk (ignoring the finite population correction (fpc) term) are respectively given by
var s 2y
§ S 4y ¨ ¨ n Š
¡ ¸>w 400 1@ ¸ š
§ S 4y MSE t k ¨ ¨ n Š
¡ ¸>w 400 w 040 2w 220 @ ¸ š
(1.3)
(1.4)
37
where G pqr
P pqr
P
P pqr p/2 200
r/2 P q020/ 2 P 002
,
1 N p q r ÂŚ y i Y x i X z i Z ; p, q, r being the non-negative integers. Ni1
Following Bahl and Tuteja (1991), we propose exponential ratio type and exponential product type estimators for estimating population variance S 2y as â&#x20AC;&#x201C;
t1
ÂŞ S 2 s 2x Âş s 2y expÂŤ 2x 2 Âť ÂŹSx s x Âź
(1.5)
t2
ÂŞ s 2 S 2z Âş s 2y expÂŤ 2z 2 Âť ÂŹ s z Sz Âź
(1.6)
2. Bias and MSE of proposed estimators To obtain the bias and MSE of t1, we write s 2y
S 2y 1 e 0 , s 2x
S2x 1 e1
Such that E(e0) = E(e1) = 0 and E(e 02 )
1 w 400 1 , E(e12 ) n
1 w 040 1 , E(e 0 e1 ) n
1 w 220 1 . n
After simplification we get the bias and MSE of t1 as
B( t 1 ) #
S 2y ÂŞ w 040 w 220 3 Âş Âť n ÂŤÂŹ 8 2 8Âź
MSE( t 1 ) #
S 2y ÂŞ w 1Âş w 400 040 w 220 Âť ÂŤ n ÂŹ 4 4Âź
(2.1)
(2.2)
To obtain the bias and MSE of t2, we write s 2y
S 2y 1 e 0 , s 2z
S2z 1 e 2
Such that E(e0) = E(e2) = 0
38
E(e 22 )
1 w 004 1 , E(e 0 e 2 ) n
1 w 202 1
n
After simplification we get the bias and MSE of t2 as
B( t 2 ) #
S 2y ÂŞ w 004 w 202 5 Âş Âť n ÂŤÂŹ 8 2 8Âź
MSE( t 2 ) #
3.
(2.3)
S 2y ÂŞ w 9Âş w 400 004 w 202 Âť n ÂŤÂŹ 4 4Âź
(2.4)
Improved Estimator
Following Kadilar and Cingi (2006) and Singh et. al. (2007), we propose an improved estimator for estimating population variance S 2y as-
t
ª  S 2 s 2x ½  s 2z S 2z ½º s 2y D exp Ž 2x ( 1 ) exp D ž Ž 2 2 ž 2 ¯S x s x ¿ ¯ s x S z ¿Ÿ 
(3.1)
where D is a real constant to be determined such that the MSE of t is minimum. Expressing t in terms of eâ&#x20AC;&#x2122;s, we have
t
ÂŞ ° e § e ¡ 1 ½° ° e § e ¡ 1 ½°º S 2y 1 e 0 ÂŤD expÂŽ 1 ¨1 1 ¸ ž (1 D) expÂŽ 2 ¨1 2 ¸ ž 2 š °¿ 2 š °¿ °¯ 2 Š °¯ 2 Š ÂŤÂŹ Âź
(3.2)
Expanding the right hand side of (3.2) and retaining terms up to second power of eâ&#x20AC;&#x2122;s, we have ÂŞ § e e e2 e e e2 t # S 2y ÂŤ1 e 0 2 2 0 2 D¨¨ 1 1 2 8 2 8 ÂŤÂŹ Š 2
¡ §e e2 ¸¸ D¨¨ 2 2 8 š Š 2
§ e §e e 2 ¡º e2 ¡ e 0 D¨¨ 1 1 ¸¸ De 0 ¨¨ 2 2 ¸¸  8 š Ÿ 8 š Š 2 Š 2
¡ ¸¸ š
(3.3)
Taking expectations of both sides of (3.3) and then subtracting S 2y from both sides, we get the bias of the estimator t, up to the first order of approximation, as
39
S 2y ÂŞ D w 040 1 1 D w 004 1 1 D w 202 1
n ÂŤÂŹ 8 8 2
B( t )
D w 220 1 º 2 Ÿ
(3.4)
From (3.4), we have
t S # S 2 y
2 y
De1 1 D Âş ÂŞ ÂŤe 0 2 2 e 2 Âť Âź ÂŹ
(3.5)
Squaring both the sides of (3.5) and then taking expectation, we get MSE of the estimator t, up to the first order of approximation, as
MSE( t ) #
2 S 4y ÂŞ D2 w 040 1 1 D w 004 1
ÂŤ w 400 1 n ÂŹ 4 4
D w 220 1 1 D w 202 1
D 1 D
w 022 1 º 2 Ÿ
(3.6)
Minimization of (3.6) with respect to D yields its optimum value as
D
^w 004 2(w 220 w 202 ) w 022 6` w 040 w 004 2w 022 4
D 0 (say)
(3.7)
Substitution of D 0 from (3.7) into (3.6) gives minimum value of MSE of t. 4.
Proposed estimators in two-phase sampling
In certain practical situations when S 2x is not known a priori, the technique of twophase or double sampling is used. This scheme requires collection of information on x and z the first phase sample sâ&#x20AC;&#x2122; of size nâ&#x20AC;&#x2122; (nâ&#x20AC;&#x2122;<N) and on y for the second phase sample s of size n (n<nâ&#x20AC;&#x2122;) from the first phase sample. The estimators t1, t2 and t in two-phase sampling will take the following form, respectively
t 1d
ª s' 2x s 2x º s 2y exp  2 2   s' x s x Ÿ
(4.1)
40
t 2d
ÂŞ s' 2 s 2 Âş s 2z expÂŤ 2z z2 Âť ÂŹ s ' z s z Âź
(4.2)
ª  s' 2 s 2 ½  s ' 2 s 2 ½ º s 2y k exp Ž 2x x2 ž (1 k ) exp Ž 2z z2 ž ¯ s' x s x ¿ ¯ s' z s z ¿Ÿ 
td
(4.3)
To obtain the bias and MSE of t1d, t2d, td, we write
S2x 1 e1 , s' 2x
s 2y
S 2y 1 e 0 , s 2x
s 2z
S2z 1 e 2 , s' 2z S2z 1 e' 2
1 n' x i x ' 2 , s 2z n ' 1 ÂŚ i 1
where s' 2x
x'
1 n' ÂŚ x i , zc n' i 1
S2x 1 e'1
1 n' z i z ' 2 n ' 1 ÂŚ i 1
1 n' ÂŚ zi n' i 1
Also, E(eâ&#x20AC;&#x2122;1) = E(eâ&#x20AC;&#x2122;2) = 0,
1 w 040 1 , E(e' 22 ) n'
E(e'12 )
E(e'1 e' 2 )
1 w 004 1 , n
1 w 220 1
n'
Expressing t1d, t2d, and td in terms of eâ&#x20AC;&#x2122;s and following the procedure explained in section 2 and section3 we get the MSE of these estimators, respectively asÂŞ1 1§1 1 ¡ MSE ( t 1d ) # S 4y ÂŤ w 400 1 ¨ ¸ w 040 1
4 Š n n' š n º § 1 1¡ ¨ ¸ w 220 1  n ' n Š š Ÿ
(4.4)
41
ª1 1§1 1 ¡ MSE ( t 2d ) # S 4y  w 400 1 ¨ ¸ w 004 1
n 4 Š n n' š  º § 1 1¡ ¨ ¸ w 202 1  Š n' n š Ÿ
ÂŞ1 k2 MSE( t d ) # S 4y ÂŤ w 400 1 4 ÂŹn
(4.5)
§¨ 1 1 ¡¸ w
k 2 1 §1 1¡ ¨ ¸ w 040 1 4 Š n n' š
Šn
n' š
004
1
§1 1 ¡ § 1 1¡ k ¨ ¸ w 220 1 k 1 ¨ ¸ w 202 1
Š n n' š Š n' n š
k ( k 1) § 1 1 ¡ ¨ ¸ w 022 1
2 Š n' n š
(4.6)
Minimization of (4.6) with respect to k yields its optimum value as
k
^w 004 2(w 220 1) w 022 6` w 040 w 004 2w 022 4
k 0 (say)
(4.7)
Substitution of k0 from (4.7) to (4.6) gives minimum value of MSE of td.
5.
Empirical Study
To illustrate the performance of various estimators of S 2y , we consider the data given in Murthy(1967, p.-226). The variates are: y: output, x: number of workers, z: fixed capital, N=80, nâ&#x20AC;&#x2122;=25, n=10.
w 400
2.2667 , w 040
3.65 , w 004
2.8664 , w 220
2.3377 , w 202
2.2208 , w 400
3.14
The percent relative efficiency (PRE) of various estimators of S 2y with respect to conventional estimator s 2y has been computed and displayed in table 5.1.
42
Table 5.1 : PRE of s 2y , t1, t2 and min. MSE (t) with respect to s 2y Estimator
PRE(., s 2y )
s 2y
100
t1
214.35
t2
42.90
t
215.47
In table 5.2 PRE of various estimators of s 2y in two-phase sampling with respect to S 2y are displayed. Table 5.2 : PRE of s 2y , t1d, t2d and min.MSE (td) with respect to s 2y
6.
Estimator
PRE (., s 2y )
s 2y
100
t1d
1470.76
t2d
513.86
td
1472.77
Conclusion
From table 5.1 and 5.2, we infer that the proposed estimators t performs better than conventional estimator s 2y and other mentioned estimators.
43
References
Isaki, C. T. (1983): Variance estimation using auxiliary information. Journal of American Statistical Association. Bahl, S. and Tuteja, R.K. (1991): Ratio and Product type exponential estimator, Information and Optimization sciences, Vol. XII, I, 159-163. Kadilar,C. and Cingi,H. (2006) : Improvement in estimating the population mean in simple random sampling. Applied Mathematics Letters 19 (2006), 75â&#x20AC;&#x201C;79. Singh, R. Chauhan, P. Sawan, N. and Smarandache, F. (2007): Auxiliary information and a priori values in construction of improved estimators. Renaissance high press, USA. Murthy, M.N.(1967): Sampling Theory and Methods. Statistical Publishing Society, Calcutta, India.
44
STRUCTURAL DYNAMICS OF VARIOUS CAUSES OF MIGRATION IN JAIPUR
Jayant Singh and Hansraj Yadav Department of Statistics, University of Rajasthan, Jaipur, India jayantsingh47@rediffmail.com, hansraj_yadav@rediff.com Rajesh Singh Department of Statistics, BHU, Varanasi (U.P.), India rsinghstat@yahoo.com Florentin Smarandache Chair of Department of Mathematics, University of New Mexico, Gallup, USA fsmarandache@yahoo.com
Abstract
Jaipur urban area has grown tremendously in last three decades. Composition of People migrating due to various reasons has display a meticulous trend. Dominance of people moving due to marriages is getting sturdy whereas Jaipur city is losing its lustre in attracting persons for education and business. Short duration migration from Jaipur district to urban area has gone down to a very low level. Flow of migrants from Rural areas to Jaipur outpaced the migrants from urban areas and its composition from various in terms long and short distances migration has substantially changed over two consecutive decades. Movements of males and females were differ on many criterion as male moving faster than females for employment & education and females move faster than male for marriages and moving along family was found evident in short, medium and long distances migration. Gender gap in
45
people migration from different reasons was observed and a gender specific trend was seen favour. Short duration migration and migration due to education & employment is not as prominence as it was two decade back. Introduction:
Migration from one area to another in search of improved livelihoods is a key feature of human history. While some regions and sectors fall behind in their capacity to support populations, others move ahead and people migrate to access these emerging opportunities. There are various causes like political, cultural, social, personal and natural forces but aspire for betterment, higher earning, more employment opportunities receive special attention. There are four type of migration namely i.
Rural-Rural
ii.
Rural-Urban
iii.
Urban-Urban
iv.
Urban-Rural
Though all of these have different implication over the various demographic and socio-economic characteristics of the society but rural-urban & urban-urban migration is a cause of concern in reference to migration process to Jaipur urban agglomeration. The dynamics of migration for three census (1981, 1991, 2001) has been analysed from different angles at destination i.e. Jaipur Urban Agglomeration. The peoples of two places have different socio-economic character like education attainment, availability of land to the rural labour and agriculture production capacity,
46
industrialization etc and the difference of these factors at two places gear the migration process. Distance plays a prominent role in migration of peoples, in general people from nearby area show a faster pace than the distant places due to psychic of being come back or feel like at home or the reason that some acquaintance in nearby area plays a big pull factor. However these assumptions do govern by other consideration of pull and push factor and the prevalent socio-economic aspects of the origin and destination places. Jaipur being the capital of the state and proximity to the national state has been a great potential to draw peoples. It has not been attracting peoples from the nearby areas but it has influence on the persons of entire state and other states of the country. Majority of immigrants to Jaipur belongs to different parts of the states followed by its adjoining states. However it has been able to attract people from all over the country and overseas as well though their contribution in totality is not as significant. Seeing at this scenario it is worthwhile to limit the migrants from the following area to comprehend the migrant process of Jaipur. In-migrants to Jaipur urban area from (a) various parts of Jaipur district (b) other districts of the state (c) adjoining states of the state having fair share in migrants and (d) total migration which is overall migration from all the areas. COMPOSITION OF IN-MIGRANTS TO JAIPUR:
In-migrants to Jaipur has grown by leaps and bounds in the last three decades. The decadal growth of in-migrants to Jaipur in last four decades synchronized with the growth of urban population of the Jaipur. Though the decadal rate of growth of migrants is lagging behind to the growth of the urban population as both has been
47
59.3% & 45.2% in decade 1991-2001, 49.5% & 35.8 % in decade 1991-81 respectively. Short Distance migration is considered, people from the other parts of Jaipur district who are coming to Jaipur urban area, migration from other parts of the state is relatively longer distance migration and put in the moderate (medium) distance migration whereas the people from out side the state are in the category of long distance migration. The contribution of the short distance migration in total migration as per census of 2001, it was 17.1% against the 51% were medium distant migrants as they came from other districts of the state and long distance migration from some most contributing states namely Punjab, U. P., & Delhi have there share as 9.6%, 3.3% & 2.3% in total migrants to Jaipur in this same duration. These three states accounted for half of the long distance migration. These different types of migration spell a meticulous trend over the years. As small distance migration shows a downward trend as its share in total migration which was 28.8% in yr 1981 came to 25.8% in according to census of 1991 and further slipped to 17.1% in census 2001. Medium distance migration exhibited a opposite path to the short distance migration as it advanced to 47% in yr 1991 against 45% in yr 1981 which further ascended to 51% in yr 2001. Contribution of long distance migration in total migration from all states also exhibited rolling down trend. This trend followed suite for the migration from the adjoining states.
COMPOSITIONAL DYNAMICS OF REASON FOR MIGRATION TO JAIPUR URBAN AREA:
Affect of various reasons of migration on peoples of diverse areas is different. Some reasons are more common than others moreover their affect on male and
48
females is also different. Share of Rural and Urban in-migrants population will widely vary for various cause of migration. Distance of place of origin is also a crucial factor in migration process to any area. Dynamics of various reasons for migration will be analysed from four perspectives. 1.
Dominance of various reasons for migration;
2.
Rural-Urban Paradigm and changes taking place;
3.
Gender issues and disparity.
DOMINANCE OF VARIOUS REASONS FOR MIGRATION IN MIGRATION PROCESS:
Person do migrate from a variety of reasons, prominent of them are migration due to 1. Employment 2. Education 3. Marriage 4. Moving with family. Marriage has been the foremost reason for migration as its share in total migrants to Jaipur was 32.1% in yr 2001. People migrating for the employment and/or business with 27.3% contribution in total migration seconded the marriage cause. It was distantly followed by category of persons moving with family with 17.6 % share in total migration. There was a remarkable difference in two dominating categories of people moving due to employment and marriage and it was that the people migrating to Jaipur due to employment is on declining side as it came down to 25% in yr 2001 from 27.3% in yr 1991 and 30.2% in yr 1981 contrary to a gradual increase in people migrating to Jaipur because of marriage as it raised to 32.1% in yr 2001 from 27.8% in yr 1991 and 25.2% in yr 1981. Education as a cause of migration doesnâ&#x20AC;&#x2122;t have significant contribution in total migration to Jaipur and it is getting meagre over the years. As in yr 1981 its share in
49
total migration was 6.1% and the figure came to 4.4% in yr 1991 and further dip to 2.7% in yr 2001. This movement is also followed by migrants for education from all the adjoining state, within state and from Jaipur district to Jaipur urban area. People moving with household also followed
the decline suite though the rate of decline
was steeper than the others as the share of people migrating under this category which was 30.2% in yr 1981 fall to 28.5% in yr 1991 and further it slip to 17.6 % in yr 2001. Composition of various reasons for migration over last three decades is depicted in coming Graph.
Composition of Migrants according to Reason of Migration
450000 400000 350000 300000 250000 200000 150000 100000 50000 0
1981 1991 2001
Employment
Education
Marriage
Moved with hh
Reason for Migration
RURAL-URBAN PARADIGM:
Intensity of migration widely differs for persons migrating form Rural and Urban areas for various reasons for migration. Flow of migrants from Rural areas to Jaipur outpaced the migrants from urban areas. According to data of census in yr 1981, the share of migrants to Jaipur urban area from rural and urban areas was 53%
50
& 47 % respectively and this gap remained intact in the coming decades. The trend in rural, urban and combined for last three decades is depicted in graph on next page.
Share of Rural and Urban Migrants in Total Migration over last three decades 1400000 1200000
M igrants
1000000 800000 600000 400000
Rural and Urban Combined
200000
Rural
0
2001
1991
1881
URBAN
year
The contribution of rural & urban migrants within a category of reason for migration over last two consecutive decades is tested by calculating the z-values for various category of reason for migration for Rural & Urban areas and significance was tested at 5% level of significance. To test the equality of share of Rural/Urban migrants from any reason of migration over a decade period following hypothesis was set up. H0 : Share of Rural (or Urban) migrants due to any reason of migration in a decade is equal. (p1=p2) Against H1 : p1 p2 This is tested for two decadal period 1981-9991 & 1991-2001. where
51
ܲൌ
݊ଵ ଵ ݊ଶ ଶ ǡ ݊ଵ ݊ଶ
ܽ݊݀ ܳ ൌ ͳ െ ܲǤ
p1 is the share of rural/urban migrants due to any reason at a point of time in total migration, p2 is the share of rural/urban migrants due to that reason after a decade in total migration To test this hypothesis, Z-value for equality of proportions of migrants from any reason over a decade is calculated and compared with tabulated value at 5% level of significance for the period 1981-9991 & 1991-2001 for rural and urban migrants separately. Four groups according to share of migrants from any reason of migration over a decade period are formed to analyse the Rural-Urban dynamics of the migrant process. Group1: Share of migrants from any reason of migration from Rural/Urban area over a decade period (in 1981-991 & 1991-2001) is not equal. Means share of peoples migrating from rural & urban areas for a particular reason of migration differ significantly over the period 1981-991 & 1991-2001. Areas falling under this group shows a change in similar direction (i. e. share of urban & rural migrants for that reason of migration has changed considerably over a decade period) for Rural & Urban migrants in terms of their share in total migration for that reason of migration over a decade period. Group 2: Share of migrants from any reason of migration from Rural/Urban area over a decade period (in 1981-991 & 1991-2001) is equal. Means share of peoples migrating from rural & urban areas for a particular reason of migration don’t differ significantly over the period 1981-991 & 1991-2001. Areas falling under this group don’t shows any change (i. e. share of urban & rural migrants for that reason of
52
migration is has not changed over a decade period) for Rural & Urban migrants in terms of their share in total migration for that reason of migration over a decade period. Group 3: Share of migrants from any reason of migration from Rural area is not equal whereas for migrants from urban areas due to this reason is equal over a decade period (in 1981-991 & 1991-2001). Means share of peoples migrating from Rural areas for a reason of migration differ significantly whereas share of peoples migrating from Urban areas for this reason of migration donâ&#x20AC;&#x2122;t differ significantly over the period 1981-991 & 1991-2001.
Areas falling under this group shows
different story as share of Urban migrants for any reason of migration in total migration is not equal though for Rural Migrants it is equal over a decade period. Group 4: Share of migrants from any reason of migration from Urban area is not equal whereas for migrants from Rural areas due to this reason is equal over a decade period (in 1981-991 & 1991-2001). Means share of peoples migrating from Urban areas for a reason of migration differ significantly whereas share of peoples migrating from Rural areas for this reason of migration donâ&#x20AC;&#x2122;t differ significantly over the period 1981-991 & 1991-2001. Areas falling under this group shows different story as share of Rural migrants for any reason of migration in total migration is not equal though for Urban Migrants it is equal over a decade period. In Group 1 & 2, migration due to any reason from rural and urban areas is in agreement i.e. share of migrants due to any reason over a decade either is significant or insignificant for both rural and urban migrants. In contrary to this In Group 3 & 4, migration due to any reason from rural and urban areas is not in agreement i.e. share
53
of migrants due to any reason over a decade is significant for urban migrants than it is insignificant for rural migrants or vice-versa. Z-value for testing hypothesis at 5% level of significance in a group will be as under. Group 1: Zu & Zr>1.96 Group 2: Zu & Zr <1.96 Group 3: Zu >1.96 & Zr <1.96 Group 4: Zu <1.96 & Zr >1.96 Where Zu and Zr is the calculated value of Z for migrants due to a reason from Urban & Rural area. The significance of Null hypothesis for all the groups is summarized in table on ensuing page. Reason for Contribution of Rural & Urban Migrants over a decade period Migration is in agreement for any reason of Migration
Duration 1991-2001
Zu & Zr>1.96
Zu & <1.96
Duration 1981-1991
Zr Zu & Zr>1.96
Employment
Total Migration, Haryana, Elsewhere Jaipur U.P., Delhi District, Gujrat
Total Migration, Elsewhere in Jaipur District, in other Districts, Gujarat, Hrayana, U.P., Punjab, Delhi
Education
Total Migration, Gujarat, Punjab Haryana, U.P., Delhi
-do-
Zu & Zr <1.96
54
Marriage
Moved Family
U.P., Punjab, Haryana, Delhi
Elsewhere in Jaipur District, in other Districts, Gujarat, Hrayana, U.P., Punjab, Delhi
with Total Migration, Elsewhere in Jaipur District, in other Districts, U.P., Punjab, Delhi
Total Migration, Elsewhere in Jaipur District, in other Districts, Gujarat, Hrayana, U.P., Punjab, Delhi
Reason for Contribution of Rural & Urban Migrants over a decade period Migration is not agreement for any reason of Migration
Duration 1991-2001
Duration 1981-1991
Zu >1.96 Zu <1.96 & Zr >1.96 Zr & <1.96
Zu >1.96 & Zr Zu <1.96 <1.96 &Zr >1.96
Employment
in other Districts, Punjab
Education
Elsewhere in Jaipur District, in other Districts,
Marriage Moved Family
Gujarat
Total Migration
with Haryana
It is apparent from this summarization that share of rural & urban migrants in the period 1981 & 1991 differ widely for migrants coming from various places.
55
Especially for migrants coming from other states the share of rural & urban population in yr 1981 & 1991 differ significantly for all the four categories of reason for migration. However this fact was a little bit different in the period of 1991-2001 as migrants coming for education & employment from rural & urban areas of various states donâ&#x20AC;&#x2122;t differ significantly in terms of their share in year 1991 & 2001 in total migration. Migrants from rural & urban areas due to marriage, employment & education were not in agreement as from some of the areas the proportion of rural migrants in year 1991 & 2001 was significant whereas for urban it was not. Therefore for the duration 1991-2001 migrants from some of the places are not making significant difference in terms of their contribution for some of the reasons to migrate or for rural migrants it is not significant whereas for urban migrants it is significant or vice-versa. This situation was missing in the duration 1981-9991. GENDER ISSUES AND DISPARITY:
Flow of male and female migration governed by different reasons differently and exhibit a different trait over the years. Looking at total in-migration in Jaipur it is found that contribution of males were phenomenal high in the category of people migrating due to employment and education as against the share of female was higher than males in category of persons migrating due to marriages and moving with family. Moreover the fact of male moving faster than females for employment & education and females move faster than male for marriages and moving along family was also evident in short, medium and long distances migration and this gap at the segregated levels was much explicit than the aggregated level. Following hypothesis was formulated to test the gender disparity in migration.
56
H0 : Share of males (or females) migrants due to any reason for migration in a decade is equal (i.e. p1=p2) Against H1 : p1 p2 Formula for Z remain same whereas p1 is the share of male/female migrants due to any reason at a point of time in total migration and p2 is the share of male/female migrants due to that reason after a decade in total migration.
To test this hypothesis Z-value for equality of proportions of migrants from any reason over a decade is calculated and compared with tabulated value at 5% level of significance for the period 1981-9991 & 1991-2001 for male and female migrants separately. Four groups according to share of migrants of any reason for migrations over a decade period are formed to analyse the Rural-Urban dynamics of the migrant process. Z-value for testing hypothesis at 5% level of significance for the four groups will be as under. Group 1: Zm & Zf>1.96 Group 2: Zm & Zf <1.96 Group 3: Zm >1.96 & Zf <1.96 Group 4: Zm <1.96 & Zf >1.96
57
Where Zm and Zf is the calculated value of Z for male & female migrants due to a reason. The significance of Null hypothesis for all the groups is summarized in table inserted below.
Reason for Migration
Contribution of Male & Female Migrants over a decade period is in agreement for any reason of Migration Duration 1991-2001
Duration 1981-1991
Zm & Zf>1.96
Zm & Zf>1.96
Zm & Zf <1.96
Employment Total Migration, in other Districts
Gujarat, Punjab, Total Migration, in Haryana, U.P., other Districts, Delhi Gujarat, Punjab,
Education
Gujarat, Punjab, Haryana, U.P., Delhi
Total Migration
Zm & Zf <1.96
Gujarat, Punjab, Haryana, U.P., Delhi
Marriage
Total Migration,
Total Migration, Elsewhere in Jaipur District, in other Districts, Haryana, Punjab,
Moved with Family
Total Migration, Elsewhere in Jaipur District
Total Migration, Elsewhere in Jaipur District, in other Districts, Gujarat, U.P, Delhi
Reason for Migration
Contribution of Male & Female Migrants over a decade period is not in agreement for any reason of Migration Duration 1991-2001 Zm >1.96 & Zf <1.96
Employment Elsewhere in Jaipur District in
Zm <1.96 & Zf >1.96
Duration 1981-1991 Zm >1.96 & Zf <1.96
Zm <1.96 & Zf >1.96
Gujarat, Punjab, Haryana, U.P., Delhi
58
other districts Education
Total Migration, in other Districts
Elsewhere in Jaipur District, in other Districts
Marriage
Elsewhere in Jaipur District, in other Districts Gujarat, Punjab, Haryana, U.P., Delhi
Moved with Family
in other Districts Gujarat, Punjab Haryana, U.P., Delhi
Gujarat
Punjab, Haryana, U.P., Delhi
It is evident from the above results that the contribution of male and females in different categories over two decades (1981-91 & 1991-2001) has changed considerably and the disparity is widened. As most of the categories in duration (1981-1991) fall in the group where both Zm & Zf>1.96 which means proportion of the males & females over a decade was significantly different. In this way male & females for most of the categories were in agreement (Zm & Zf >1.96) as both were significant as far as their contribution in total migration over a decade is concerned. Except for the people moving due to education from other states as Zm & Zf <1.96 for this category. This means that share male & females migrating due to education from other states in total migration in the year 1981 & 1991 was same and this remained stabilized in year 2001. People migrating due to marriages & moving with family also showed a change in this three decade period as migrating from most of the areas in year 1991 over 1981 exhibited that the share was considerably changed (Zm & Zf>1.96) whereas in year 2001 over 1991 it showed that it has not changed for males
59
though for females it has changed. Thus people moving under these categories have shown a shift in term of increasing share toward females. Migrants from different areas exhibit a considerable shift in terms of contribution of males or females in total migration over a period of ten years. However overall migrants say that three categories (employment, marriage & moved with family) followed the same suite as the share of male & female was significant for testing hypothesis for equality of the same over the duration 1981-991 & 1991-2001. Summary:
Contribution of people migrating for education in total migration is on a steep declining as its contribution in total migration has decreased by one third over a two decade period. People migrating due to marriage is showing a phenomenal incremental growth & it is supposed to grow with a faster pace due to decline sex ratio in the city. Migration due to education is having less contribution in total migration and it is going thinner over the years because of education facilities in smaller town and easy accessibility to them in small town. Therefore no longer education is as significant for tempting to migrate as it used to be two decades back. In the coming years this cause of migration will further tend to lose its impact in overall mobility of peoples. People migrating with family is also on a downward trend as people moving with family and due to marriage are together constitute inactive movement as people are not necessarily moving by choice or primarily donâ&#x20AC;&#x2122;t have motive of employment, business or education which itself are related to betterment of life/career. The share of inactive movements in total migration has came down by 5% over a decade. If this trend continues and the economic progress of the Jaipur
60
indicates that it will attract the people for economic reasons than the share of migrants in working population will grow which in turns contribute for the economy of the City as the share of people moving with family is declining sharply. Migration from urban areas due to marriages is also getting bigger and voluminous in coming decades this will in turn affect the cultural & social structure of the society and a cosmopolitan culture will emerged. Analysis of trend of the male & femalesâ&#x20AC;&#x2122; migration it can be interpreted that share of employment & education from other states to Jaipur is leading to stabilization & it was not found significant for testing the hypothesis of equality of their share over decades. Whereas for people moving with family the share of males is getting stabilized though for females it was growing. If this scenario continue than growing migration of females in this category will, to some extent, be beneficial to the decreasing sex ratio the city. Short distance migration which consist the in-migration from various parts of the Jaipur district to Jaipur Urban area is one a sharp decline path in terms of its contribution in total migration. It clearly indicates the tendency of migrating to Jaipur urban area is lower down as periphery of Jaipur urban area is also being developed as its suburb. Better connectivity is raising the number of daily commuter and in near future entire district may be developed as a part of Jaipur urban area and a new Jaipur is shaping up. In such a scenario overall migration to Jaipur urban area from the various parts of Jaipur district will lose its relevance. References:
Bhagat, R. B. (992). Components of Urban Growth in India with Reference to Haryana: Findings from Recent Censuses, Centre for Training and Research in
61
Municipal Administration, University of California, Nagarlok, Vol.24, No. 3, 1992, pp. 10-14
Bhattacharya, P. (1988). The informal sector and rural to-urban migration: Some Indian evidence, Economic and Political Weekly vol.33(21), 12551262. Jain, M.K., Ghosh, M. and Won Bae, K. (1993). Emerging Trends of Urbanization In India: An Analysis Of 1991 Census Results, Office of Registrar General, New Delhi.
Registrar General of India: Census of India (1971, 1981, 1991, 2001) Migration tables Part V A and B, Rajasthan State.
62
This volume is a collection of five papers/chapters. Two chapters deal with problems in statistical inference, two with inferences in finite population, and one deals with demographic problem. The ideas included here will be useful for researchers doing works in these fields.