An alternative estimation procedures for the point estimation of scale parameter of wald distributio

An Alternative Estimation Procedures for the Point Estimation of Scale Parameter of Wald Distribution Sandeep Singh Charak1, Rahul Gupta2 1,2

Department of Statistics,University of Jammu,Jammu-180006

Abstract: Wald distribution also known as Inverse Gaussian appears to have been first derived by SchrĂśdinger in 1915 as the time to first passage of a Brownian motion.The name inverse Gaussian was proposed by Tweedie (1945). Wald re-derived this distribution in 1947 as the limiting form of a sample in a sequential probability ratio test. Tweedie investigated this distribution in 1957 and established some of its statistical properties. In this paper we consider the problem of point estimation of the scale parameter of the Wald distribution. We prove the failure of the fixed sample size procedures to handle the estimation problems. Alternative procedure based on a sequential sample is developed to tackle the situation and the second-order approximations are obtained for the proposed procedure. In this paper we describe the set-up of the estimation problem and prove the failure of the fixed sample size procedures to deal with them and also develop an alternative procedures for point estimation of scale parameter of the Wald distribution.

Key words: Sequential procedure, Point estimation, Second-order approximations, Stopping rule .

I. INTRODUCTION Wald or Inverse Gaussian distributions have attracted considerable attention during the last 20 years and are being widely used to explain the motion of particles influenced by Brownian motion and is also applied to study the 27 motion of particles in a colloidal suspension under an electric field. In Russian literature on electronics and radio technique, the Wald distribution is often used. In recent years the Wald distribution has played versatile roles in models of stochastic processes including the theory of generalized linear models, reliability, lifetime data analysis and repair time distributions, especially in cases of preponderance of early failures. Pioneering work related to estimation of this distribution was done by Chhikara and Folks(1974) and recently Lu(2016) proposed certain approximations to achieve sharp lower and upper bounds for the Mills’ ratio of the Wald distribution.

Gupta, Bhougal and Joorel(2003) proposed a procedure based on maximum likelihood estimator for fixed width confidence interval of mean of an Wald distribution when dispersion parameter is known and showed the procedure to be efficient and consistent and also obtained distribution of the stopping time. In this paper the same distribution is revisited but the problem of point estimation of the scale parameter is considered under squared error loss function with linear cost of sampling in subsequent sections besides providing an alternative procedure to obtain estimates of scale parameter along with some related optimal properties.

II.

The Set Up of the Estimation Problems and the Failure of the Fixed Sample size Procedures {đ?‘‹ } Let us consider a sequence đ?‘– , đ?‘– = 1,2, ‌ of i.i.d random variables from a Wald distribution đ?‘“(đ?‘‹; đ?œ‡, đ?œ†) = [

đ?œ†

] 2đ?œ‹đ?‘Ľ 3

1â „ 2

đ?‘’đ?‘Ľđ?‘? [−

đ?œ† (đ?‘‹âˆ’đ?œ‡)2 2đ?œ‡2

đ?‘‹

] ; đ?‘‹ ≼ 0, where đ?œ‡ đ?œ–(0, ∞) and đ?œ†âˆ’1 đ?œ– (0, ∞) are the

unknown mean and scale parameters, respectively. Let (đ?œ‡ , đ?œ†âˆ’1 ) đ?œ– â„œ1 Ă— â„œ+ , where â„œ1 and â„œ+ denote, respectively, the one dimensional Eucliden space and the positive half of the real line.

Having recorded a random sample �1 , ‌ . . , �� of size n(≼2), Let us define @IJRTER-2016, All Rights Reserved

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International Journal of Recent Trends in Engineering & Research (IJRTER) Volume 02, Issue 08; August - 2016 [ISSN: 2455-1457] đ?‘›

đ?‘‹Ě…đ?‘› = đ?‘›âˆ’1 ∑ đ?‘‹đ?‘– đ?‘–=1

and

đ?‘›

đ?œ†Ě‚đ?‘› = (đ?‘› − 1)−1 ∑(đ?‘‹đ?‘– −1 − đ?‘‹đ?‘› −1 ) đ?‘–=1

be the estimators of đ?œ‡ and đ?œ†âˆ’1 , respectively, satisfying the following assumption:2 (A1): đ?‘› đ?œ†âˆ’1 (đ?‘‹Ě…đ?‘› − đ?œ‡)2 ~ đ?œ’(1) 2 where đ?œ’(1) denotes the Chi- square random variable with one degree of freedom.

(A2): for all đ?‘› ≼ 2, đ?‘‹Ě…đ?‘› and đ?œ†Ě‚đ?‘› are stochastically independent.

(A3):

(đ?‘› − 1)

Ě‚đ?‘› đ?œ†

(1) = ∑đ?‘›âˆ’1 đ?‘—=1 đ?‘?đ?‘— ,

đ?œ†âˆ’1 (1) đ?‘?đ?‘— ~

2 where đ?œ’(1) Our problem is the point estimation of Scale parameter (đ?œ†âˆ’1 ) of Wald distribution by đ?œ†Ě‚đ?‘› .

Let the loss incurred in estimating đ?œ†âˆ’1 by đ?œ†Ě‚đ?‘› by squared-error plus linear cost of sampling, .i.e., 2 đ??ż(đ?œ†âˆ’1 , đ?œ†Ě‚đ?‘› ) = đ??´(đ?œ†Ě‚đ?‘› − đ?œ†âˆ’1 ) + đ??śđ?‘›

(1.2.1)

where A(>0) is the Known weight and C(>0) is the known cost per unit sample observation. Using (A3), the risk corresponding to loss function (1.2.1) is đ?œ†Ě‚đ?‘› đ?‘…đ?‘› (đ??ś) = đ??´đ?œ†âˆ’2 đ??¸ ( −1 − 1) + đ??śđ?‘› đ?œ† đ?œ†âˆ’2 = 2đ??´ + đ??śđ?‘›. (đ?‘› − 1)2

(1.2.2)

The value đ?‘› = đ?‘›0 , minimizing the risk (1.2.2), is given by 2đ??´ 1/2 đ?‘›0 = 1 + ( ) đ?œ†âˆ’1 , đ?‘?

And substituting đ?‘› = đ?‘›0 in (1.2.2), the associated minimum risk comes out to be đ?‘…đ?‘›0 (đ??ś) = đ??ś(2đ?‘›0 − 1)

(1.2.3)

(1.2.4)

Once again, since đ?‘›0 depends on đ?œ†âˆ’1 , in the absence of any knowledge about đ?œ†âˆ’1 , no fixed sample size procedure minimizes the risk simultaneously for all values of đ?œ†âˆ’1 . From (1.2.3), we conclude that due to the dependence of ‘optimal’ fixed sample size solutions on đ?œ†âˆ’1 , the fixed sample size procedure fail to gives solution to the estimation problem discussed in this section.

III.

THE SEQUENTIAL PROCEDURE FOR THE POINT ESTIMATION OF đ?œ†âˆ’1

Taking m  2 as the initial sample size. Then, the stopping time N≥N(C) for selecting sequential sample is defined by

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International Journal of Recent Trends in Engineering & Research (IJRTER) Volume 02, Issue 08; August - 2016 [ISSN: 2455-1457] 1   2 2 A   ˆ  N = inf n  m : n 1    n  .   C     1 ˆ After stopping we estimate  by  N , incurring the risk





2 RN (C) = A E ˆ N  1  + C E(N).   We defined the ‘regret’ of the sequential procedure as

Rg (C) = RN (C) - R no (C ) .

(1.3.1)

(1.3.2)

(1.3.3)

In what follows, we provide second order approximation for expected sample size and ‘regret’ associated with the proposed sequential procedure.

Theorem 1.

For the sequential procedure defined at (1.3.1) and all m  5 , as C→0, E(N) = no + υ – 3 + o(1),

(1.3.4)

Rg (C) = -2C + o(1).

(1.3.5)

and Proof The stopping time rule (1.3.1) can be re-written as n 1  n  12  .

N = inf n  m :  Zj1  n o  1 j1 

(1.3.6)

Let us define a new stopping variable N* by N*

 n2 = inf n  m  1 : Sn  n o  11 

 . 

(1.3.7)

where, n

Sn =

Z  . j1

1 j

It follows that lemma 1 of Swanepoel and Van Wyk (1982) that the stopping rule N and N* follows the same probability distribution. From (1.3.7) and equation (1.1) of Woodroofe 1 (1977),  =2,  =1,  =1, 2 = 2, C = n o  1 , Lo = 0, λ = no – 1 and a = ½.

It follows from theorem 2.4 of Woodroofe (1977) that, for all m > 3 as C→0, E(N) = λ + β μ-1 υ – β Lo - (1/2)  β2 μ-2 τ2 + o(1) = no – 1 + υ – 2 + o(1) and the result (1.3.4) follows.

Denoting by noo = no – 1 and T = N –1, we can write 2 ˆ   2   1 A ˆ N   = A 2  N1  1     2 A = (ST – T)2 2 T





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 n2  = (C/2) (ST – T)2 + (C/2)  oo2 1 (ST – T)2 . T  Form (1.3.8) and Wald’s Lemma for cumulative sums,  n2  E L 1 , ˆ N = 2 (C/2) E(T) + (C/2) E[  oo2 1 (ST – T)2 ] + C E(T+ 1) T 



(1.3.8)



= 2C E(T) + C + E(  T ), say

(1.3.9)

where, 2  n oo    T = (C/2)  2 1 (ST – T)2 T 

n2 2 = (C/2) (ST – T)  oo2 T 

 T2  1   2 n oo  

2

 T2 T4   2  4  n oo n oo 2

 n2   T2  = (C/2)  oo2  (ST – T)2 1  2  + (C/2) n oo  T   = I + II say, From (1.3.6) in Woodroofe’s (1977) notations, denoting by  T2  RC =  2  – ST , we can write  n oo   T2 1  2  n oo

  T  = 1    n oo

 T 1   n oo

  

      

 T2 1  2  n oo

  (ST – T)2  (1.3.10)

  

 T  1  T ( T – ST – RC ) = 1  n oo    1 1   ( ST – T + RC ) . = –  T  n oo   T a.s Utilizing (1.3.11) and the result that   1 as C→0, we get n oo 2  n oo  I = (C/2)  2 T

= 2C Since ST ≤

2

  2    ST  T Op c (ST – T)2   n oo 

ST  T 4 2 n oo

(1.3.11)

+ Op (c).

(1.3.12)

T2 , we have n oo

ST  T 4 2 n oo

T4 ≤ n oo  T   =   n oo 

 T    1  n oo  4

4

 T  n oo     .

12  n oo 

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(1.3.13)

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 T   and It follows from lemma 2.1 and theorem 2.3 of Woodroofe (1977) that   n oo   T  n oo     are uniformly integrable for all m > 5. Moreover, theorem 1 of Anscombe (1952) 12  n oo  4

1 L leads us to the result that S T  T   n oo2   N(0, 2) as C→0. Using these result and    (1.3.13), we obtain from (1.3.12) that for all m > 5 as C→0,





E(I) = 2C E ST  T noo 1 2 = 24 C. Furthermore, we can write 2  T  T T2      2  2 2  (ST – T)2 II = (C/2) 1   n oo  n oo n oo     2   T    2 ST  T  R C   1    (ST – T)2 = (C/2)  n   n oo    oo  4

(1.3.14)

2

 T   (ST – T)2 = – (C/noo) (ST – T)3 – (C/noo) RC (ST – T)2 + (C/2) 1  n oo   = – II1 – II2 + II3 , say.

(1.3.15) Proceeding as for I, it can be shown that II2 is uniformly integrable for all m> 3.

Form theorem 2.1 of Woodroofe (1977), since RC and (ST – T)2 are asymptotically 12 L independent, the mean of the asymptotic distribution of RC is ‘υ’ and S T  T  / n o    N(0, 2) as C→0, we get for all m > 3, as C→0, E(II2) = C υ.(2) = 2C υ.

(1.3.16) 2 T  n oo  ST  T 2 Once again, using the asymptotic independence of and and the result n oo  n oo 

that they are uniformly integrable for all m > 3 , which each one distributed as 2  21 , we get E(II3) = 4 (C/2) E( 21 )2 = 6 C. It follows from theorem 8 of Chow, Robbins and Teicher (1965) that

(1.3.17)

E[(ST – T)3] = 2 Var Zj(1)  E(T) + 3 Var Zj(1)  E[T(ST – T )] = 8 E(T) + 6 E[T(ST – T )], and hence.

E(II1) = (C/noo) [8 E(T) + 6 E[T(ST – T )] ].

We have,  T   T   T2    (ST – T ) =     R C  T   n oo   n oo   n oo 

(1.3.18)

 





2



 T   T 1 2  1 –  = T  n oo  + noo  T2  noo  n oo   n oo = II11 + II12 – II13 , say.

Form (1.3.4), for all m > 3, as C→0,



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

  RC 

(1.3.19)

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E[II11] = υ – 3 + o(1), and, as C→0, E[II13] = υ.

Moreover, since II12 = n00 

1

(1.3.20) (1.3.21)

T  n oo   T  n oo   , 



2

n oo





 T  a.s   1 as C→0, using the results that   n oo  T  n oo  L N(0, 2) and its square is uniformly integrable for all m > 3, one gets   n oo 1 2 E[II12] = 4.

(1.3.22) Making substitutions from (1.3.20), (1.3.21) and (1.3.22) in (1.3.19), we obtain from (1.3.18), for all m > 3 as C→0, E(II1) = (8C/noo) [ noo + υ – 3 + o(1) ] + 6C [1 + o(1) ].

(1.3.23) From (1.3.15), (1.3.16), (1.3.17) and (1.3.23), for all m > 3, as C→0, E[II] = – 26C + 6C – 2Cυ + o(1).

(1.3.24) We can conclude form (1.3.10), (1.3.14) and (1.3.24) that, for all m > 5, as C→0 E  T  = – 2C + 6C – 2Cυ + o(1).

(1.3.25) Making substitution from (1.3.4) and (1.3.25) in (1.3.9), we get E L 1 , ˆ N = 2 C [no + υ – 3 + o(1)] + C +[– 2C + 6C – 2Cυ + o(1)] = 2Cno – C + o(1).

(1.3.26) Finally substituting (1.2.8) and (1.3.26) in (1.3.3), we get Rg (C) = -2C + o(1), and the theorem follows.





REMARK In order to estimate the nuisance parameter 1 alone, we do not need the assumptions (A1) and (A2) and only (A3) will suffice the purpose. However, since we have considered the joint estimation problems concerning  , as well as 1 , we assume both (A1) and (A2) to hold.

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Gupta, R, Bhougal, S and Joorel, J.P.S (2003): Sequential procedure for estimation of mean of an Inverse Gaussian distribution.Vol.23, PP. 87-89.

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