Cmjv01i02p0145

J. Comp. & Math. Sci. Vol. 1(2), 145-154 (2010).

A Probability model for the risk of vulnerability to HIV/AIDS infection among female migrants HIMANSHU PANDEY and RAJENDRA TIWARI Department of Mathematics, and Statistics, D.D.U. Gorakhpur University, Gorakhpur (India) ABSTRACT The main objective of this paper is to developed an inflated probability model for described and analysis, how the female migrant are more vulnerable to HIV/AIDS. The suitability of the model is tested through observed data.

Key Words: Inflated Probability Model, Displaced Geometric Distribution, Method of Moments, M LE.

INTRODUCTION Women are working in almost all types of jobs, such as technical, professional and non-professional in both private and public sectors. So, the traditional role of women as house wives has gradually changed into working women and housewives (Reddy, 15 ; Anand2). They have also started actively participating in the socio-economic development of the country. They are working in almost all types of jobs either that are in Public or Private Sectors. Today, in an increasingly globalized economy, migration often provides an employment opportunities giving rise to an unprecedented flow of migrants, including increasing numbers of female migrants (Jhingarn; Bhatt; Desai)12. The reason for migration is recognized that women

more within countries in response to the inequitable distribution of resources, services and opportunities.

Migration, especially in the process of regional economic development, urbanization and industrialization is an important cause and the effect of social and economic change. The sociocultural characteristics of the households are more likely to be affected by female and children migration whereas, the economic level is affected by the male migrants. Thus, it is important to investigate the variation in the number of migrants from a household under this consideration.

MODEL: A probability model for the number of closed boy friends to describe the distribution of single unmarried

Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)

146 Himanshu Pandey et al., J.Comp.&Math.Sci. Vol.1(2), 145-154 (2010). female migrants proposed under the following assumption:

involves three parameters , , p to be estimated from the observed distribution of female migrants.

(i) Let  be the proportion of female migrants having at least one close boy friend.

ESTIMATION:

(ii) Out of  proportion of female migrants, let  be the proportion of female migrants having only one closed boy friends.

Let N be a known quantity. If N is taken to be known then the proposed model (4.2.1) involves three parameters , and p only.

(iii)Number of close boy friends attached with female migrants follows a truncated displaced Geometric distribution. (iv)Let p be the probability of close boy friends attached with young unmarried female migrants, they are more vulnerable to HIV/AIDS infection.

Let the random variable x denotes the number of closed boy friends.

From the above assumptions, the probability model is given by

The parameters ,  and p are estimated by equating zeroth and first cell theoretical frequencies to the observed frequencies of the respective cells and theoretical mean equal to observed mean as follows:

1 

 

P[ X  0 ]  1   ;

;

f0 f

f1 f

(3.1)

(3.2)

K=0

P[ X  1]   K=1

(1   )pqK  2 P[ X  K ]  1  qN ;

METHOD OF MOMENT:

K=2, 3,……..N (2.1.)

The above probability model

 1  qN 1 NqN 1    (1   )  N p { 1  q } 1  qN  

1  X 1 qN 

(3.3)

Where f 0 =Number of Observed zeroth

Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)

Himanshu Pandey et al., J.Comp.&Math.Sci. Vol.1(2), 145-154 (2010). 147 cell, f

1

=Number of Observed first

log L  f 0 log1  f 1 log  f 2 log

cell, f = Total number of observations.. and X =Observed of the distrilog L  fmean 0 log 1  f 1 log bution.

The expected frequencies of the corresponding cells are obtained after getting the estimated values of the parameters by using the above expressions (3.1), (3.2) and (3.3).

1   p   f 2 log  N 1  q  

  f  f 0  f 1  f 2  log    

 p1           METHOD OF MAXIMUM LIKELIHOOD: 1 q N   







Let x be a random variable from a sample of f observation with the probability function (2.1) where f 0 denote the number of observation in zenoth cell, f

1

denote the number of

observation in first cell and f denote the total number of observations. Then the likelihood function for the given sample can be expressed as:

p(1 )  L  (1) f0 () f1  N   1 q 

 p(1 )       1 qN  

Partially differentiating (3.5) with respect to ,  and p respectively and equating to zero. We get the following equations.

 f f f LogL  0  1  2   1    ( f  f0  f1  f2 )  



f2

f0 f  f0  1 

 0

f  f 0  f1  f 2

(3.4)

Expression for logarithm of likelihood  function is.

(3.5)

f f  log L  1  2    1 

(3.6)

1 

f 1 f 2  f  f 0  f 1  f 2    1  1 

Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)

  

148

Himanshu Pandey et al., J.Comp.&Math.Sci. Vol.1(2), 145-154



f 1  

f

 f

0

 f

1 

 0

(3.7)

f  log L  2  p p 

p



1 

2

 0

(3.8)

After solving the equations (3.6), (3.7) and (3.8), we get the following estimating equations.

 

f  f f

f 1   f  f And

The second order derivations of log L follows from equations (3.6), (3.7) and (3.8) respectively.

 f 2 0 log L     2  2  1   

     f2 

f  f0 f1 f2  p  1  q N   p 

2



f

f



0

2

 f 2 log L     2  

And

2

  

1 2

(3.12)



f

f0f

1   

2

1

  

 f2 2 LogL   2 p 2  p

( f  f 0  f1  f 2 )    (3.14) 2 N  (1  q )  p

(3.9)



(3.10) Now





lo g L  2 lo g L       2

f2 p (3.11)  N 1 q f  f 0  f 12 l o g L  2 lo g L   0    

The asymptotic variance of (, , p) is obtained by investing the information matrix whose elements are negatives of second order of the likelihood function.



  

(3.13)

0

0

(2010).



(3.15)

lo g L  2 lo g L   0  p p 2

Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)

(3.16)

Himanshu Pandey et al., J.Comp.&Math.Sci. Vol.1(2), 145-154 (2010).



lo g L  2 lo g L   0 p  p 2

nd

(3.17) Since

E f

0

 12   21

  f 1   

149

 f 1   f  1 E  f 2   f  p 1   1  q N  E

  2 LogL  E       f

  2 LogL  E       0 f

And

(3.20)

E f  f 0  f 1  f 2   f 1 1Therefore p 1q by inverting the information matrix, the expression for the asymptotic

1

 f  f  f  1 1 p1q N    

variances of the ˆ and ˆ can be obtained as:

Then the expected value of the second partial derivatives of log L can be obtained by using the three different cases as:

V ˆ  

Case1: When P is taking known from the method of moment then.

V

 11

  2 LogL  E    2   1 1      f 1     (3.18)

 22

  2 LogL  E    2   1 1      f 1    

1   f 

 22 11  2 2  

2 12

  

(3.21)

 ˆ  

1   f 

 11 11 22  

2 21

  

(3.22) Case 2: When  is taking known from the method of moment then?



11

  2 LogL E   2   f

  

1   1      1  

(3.19) Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)

(3.23)

150

Himanshu Pandey et al., J.Comp.&Math.Sci. Vol.1(2), 145-154 (2010).

 22 

  2 LogL  E   p 2    f

 1     1   1  q N   p

1

1  q

N

 1 1     1    

    p  

(3.28)

  2 LogL  E   p 2    f

 22

(3.24) And

 1     2 LogL  E 1 q N    p  12  21  0 1 f  (3.25)  1 

q

Therefore by inverting the information matrix, the expression for the asymptotic variances of as:

V  ˆ



ˆ

1   f 

 22 11 22  

2 12

  

1   f 

 11



22

11

 

2 21

 12   21



22

  2 LogL  E   2      f



    p   (3.29)

 2 E  log    p   f

 L 

 2  E  LogL    p     0 (3.30) f

  

(3.27) Case 3: When  is taking known from the method of moment then?

N

p

And

(3.26)

 pˆ  

1  q

and pˆ can be obtained

And

V

  p



 

Therefore, by inverting the information matrix, the expression for the asymptotic variances of ˆ and pˆ can be obtained as:

  22 1 V ˆ   2  (3.31) f   11 22   12 

 

Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)

Himanshu Pandey et al., J.Comp.&Math.Sci. Vol.1(2), 145-154 (2010).

1 And V  p ˆ  f

  11    11  22  

 2  21 

(3.32) APPLICATION: The suitability of the proposed probability model (2.1) is tested through the survey of 362 unmarried working women, randomly selected from 12 working women's hostels in Delhi. The list of the hostels was obtained from Social Welfare Department, YWCA and warden's of the hostels.

Details about the data are given in Jain, et.al.10.

Tables 1 show the distribution of the observed and expected frequencies for unmarried single female migrants according to their close boy friends.

Table 2 show that the estimated values of the parameter and variances for observed and expected number of unmarried female migrants having close boy friends.

The estimated value of the risk of parameters ,  and p for proposed model (2.1) are 0.8039, 0.4364 and 0.4757 respectively by the method of moment and the estimated value of the parameters ,  and p for proposed model (2.1) are 0.8039, 0.4364 and 0.4668 respectively by the method of

Table 1. Observed and Expected numbers of unmarried single female migrants according to their closed boy friends EXPECTED Number of Method of Method of closed boy Observed Moments Maximum friends Likelihood 0 71 71.00 71.00 1 127 127.00 127.00 2 80 81.27 80.01 3 55 42.59 42.66 4 19 22.32 22.75 5 10 17.82 18.58 Total

ˆ ˆ pˆ ˆ

d. f

2

362

151

362 0.8039

362 0.8039

0.4364

0.4364

0.4757

0.4668

7.5614

8.1497

2

2

Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)

152

Himanshu Pandey et al., J.Comp.&Math.Sci. Vol.1(2), 145-154 (2010).

Table 2. Asymptotic Variances of the parameters (, , p ) in three different cases Case 1: When p is taking known from the method of moment then the asymptotic variance of  and  will be:



11

=6.3434



22

=3.2685

V ˆ  =0.0004355

 11

22

=20.7334

V ˆ =0.0008452



2 12

=

Case 2:

2 21

 

=0

When  is taking known from the method of moment then the asymptotic variance of  and p will be:



11

=6.3434



22

=1.9804

V ˆ  =0.0004355

 11

22

=12.5625

V  pˆ  =0.0013949



2 12

=

Case 3:

2 21

=0

When  is taking known from the method of moment then the asymptotic variance of  and p will be:



11

=3.2685



22

=1.9804

V ˆ =0.0008452

 11

22

=6.4729

V  pˆ  =0.0013949



2 12

=

2 21

 

=0

maximum likelihood. The higher value of  indicates that the risk of HIV/ AIDS among female migrant having at least one close boy friend is greater.

From the table 1, it is found that the observed values of 2 are insignificant

at 2 per cent level of significance and hence indicating the suitability of the model.

The proposed distribution describes satisfactorily that the unmarried

Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)

Himanshu Pandey et al., J.Comp.&Math.Sci. Vol.1(2), 145-154 (2010). 153 female migrants having at least one boy friend are more vulnerable to HIV/ AIDS. By increasing the life style of living and working conditions and by providing the adequate facilities to unmarried female migrant the vulnerability to HIV/ AIDS infections in them can be reduced.

7.

8.

9.

ACKNOWLEDGEMENTS First author is thankful to U.G.C., New Delhi for providing a financial support by MRP-37-546/09 (SR).

10.

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