A Probability model for the risk of vulnerability to HIV/AIDS infection among female migrants

8/10/2019 A Probability model for the risk of vulnerability to HIV/AIDS infection among female migrants

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A Probability model for the risk of vulnerabilityto HIV/AIDS infection among female migrants

HIMANSHU PANDEY and RAJ ENDRA TIWARI

Department of Mathematics, and Statistics, D.D.U.

Gorakhpur University, Gorakhpur (India)

ABSTRACT

The main objective of this paper is to developed an inflatedprobability model for described and analysis, how the femalemigrant are more vulnerable to HIV/AIDS. The suitability of themodel is tested through observed data.

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

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

INTRODUCTION

Women are working in almostall types of jobs, such as technical,professional and non-professional inboth private and public sectors. So, thetraditional role of women as housewives has gradually changed into working

women and housewives (Reddy,15;

Anand2). They have also started actively

participating in the socio-economicdevelopment of the country. They areworking in almost all types of jobs eitherthat are in Public or Private Sectors. Today,in an increasingly globalized economy,migration often provides an employmentopportunities giving rise to an unpre-cedented flow of migrants, includingincreasing numbers of female migrants

(J hingarn; Bhatt; Desai)12. The reason

for migration is recognized that women

more within countries in response tothe inequitable distribution ofresources, services and opportunities.Migration, especially in the process ofregional economic development,urbanization and industrialization is animportant cause and the effect ofsocial and economic change. The socio-cultural characteristics of the householdsare more likely to be affected by female

and children migration whereas, theeconomic level is affected by the malemigrants. Thus, it is important toinvestigate the variation in the numberof migrants from a household underthis consideration.

MODEL:

A probability model for thenumber of closed boy friends to describethe distribution of single unmarried

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


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

cell, 1f = Number of Observed first

cell,f = Total number of observations..

and X =Observed mean of the distri-bution.

The expected frequencies of the

corresponding cells are obtained after

getting the estimated values of the

parameters by using the above expre-

ssions (3.1), (3.2) and (3.3).

METHOD OF MAXIMUM LIKELIHOOD:

Let x be a random variable from

a sample of f observation with the

probability function (2.1) where 0f

denote the number of observation in

zenoth cell, 1f denote the number of

observation in first cell and f denote

the total number of observations. Thenthe likelihood function for the givensample can be expressed as:

2

10

1)1()()1(

f

Nff

qpL

210

1

)1(ffff

Nq

p

(3.4)

Expression for logarithm of likelihoodfunction is.

0 1log log 1 logL f f

2

1log

1 N

pf

q

0 1 2 logf f f f

1

1 N

p

q

(3.5)Partially differentiating (3.5) with respectto , and p respectively and equatingto zero. We get the following equations.

1

210 fffLogL

)( 210 ffff

00

1

fff

0 (3.6)

1 2log1

f fL

0 1 21

f f f f



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0 11

1

f f ff

0 (3.7)

2log

fL

p p

0 1 2

1 N

f f f f

q p

0 (3.8)

After solving the equations (3.6), (3.7)and (3.8), we get the following estima-ting equations.

0f f

f

(3.9)

1

0

f

f f

(3.10)

And2

0 11 N

fp

q f f f

(3.11)

The asymptotic variance of (,, p) is obtained by investing the infor-mation matrix whose elements arenegatives of second order of thelikelihood function.

The second order derivations oflog Lfollows from equations (3.6), (3.7)and (3.8) respectively.

20

22log

1

fL

02

f f

(3.12)

20 11

22 2log

1

f f ffL

(3.13)

And

2

2

2

2

p

fLogL

p

2

210

)1(

)(

pq

ffff

N(3.14)

Now

2 lo g L

2 lo g0

L

(3.15)

2 2lo g lo g0

L L

p p

(3.16)

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



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2 2lo g lo g0

L L

p p

(3.17)Since

0 1E f f

1E f f

1

2 1 1 NE f f p q

And

0 1 2 1E f f f f f

1

1 1 Np q

Then the expected value of thesecond partial derivatives of log L canbe obtained by using the threedifferent cases as:

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

1

1

12

2

11f

LogL

E

(3.18)

1

1

12

2

22f

LogLE

(3.19)

nd

2

2112

f

LogLE

0

2

f

LogLE

(3.20)

Therefore by inverting the informationmatrix, the expression for the asymptotic

variances of the and can be

obtained as:

222

11 2 2 12

1V

f

(3.21)

112

1 1 2 2 2 1

1Vf

(3.22)

Case 2:When is taking knownfrom the method of moment then?

f

LogLE

2

2

11

1

1

1

(3.23)




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f

p

LogLE

2

2

22

1 1 1

1 1N Nq p q p

(3.24)And

0

2

2112

f

p

LogLE

(3.25)

Therefore by inverting the informationmatrix, the expression for the asymptotic

variances of and p can be obtainedas:

2 22

1 1 2 2 1 2

1V

f

And (3.26)

1 1

21 1 2 2 2 1

1V p

f

(3.27)

Case 3:When is taking knownfrom the method of moment then?

f

LogLE

2

2

22

1 1

1

(3.28)

f

p

LogLE

2

2

22

1

1 Nq

1 1

1 Np q p

(3.29)And

log2

2112

f

Lp

E

0

2

f

LogL

p

E

(3.30)

Therefore, by inverting the informationmatrix, the expression for the asymptotic

variances of and p can be obtained

as:

222

11 22 12

1Vf

(3.31)




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And

112

11 22 21

1V p

f

(3.32)

APPLICATION:

The suitability of the proposedprobability model (2.1) is testedthrough the survey of 362 unmarriedworking women, randomly selectedfrom 12 working women's hostels inDelhi. The list of the hostels was obtainedfrom Social Welfare Department,

YWCA and warden's of the hostels.Details about the data are given in

J ain, et.al.10.

Tables 1 show the distributionof the observed and expected frequen-cies for unmarried single female migrantsaccording to their close boy friends.

Table 2 show that the estimated valuesof the parameter and variances for

observed and expected number ofunmarried female migrants havingclose boy friends.

The estimated value of the riskof parameters , and p for proposedmodel (2.1) are 0.8039, 0.4364 and0.4757 respectively by the method ofmoment and the estimated value ofthe parameters , and p for proposedmodel (2.1) are 0.8039, 0.4364 and0.4668 respectively by the method of

Table 1. Observed and Expected numbers of unmarried singlefemale migrants according to their closed boy friends

EXPECTEDNumber of Method of Method of closed boy Observed Moments Maximum

friends Likelihood0 71 71.00 71.001 127 127.00 127.002 80 81.27 80.013 55 42.59 42.66

4 19 22.32 22.755 10 17.82 18.58

Total 362 362 362

0.8039 0.8039

0.4364 0.4364

p 0.4757 0.4668

2 7.5614 8.1497

.d f 2 2




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Table 2. Asymptotic Variances of the parameters ( , , p) inthree 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 =0.0004355

11 22 = 20.7334 V =0.0008452

212 = 221 = 0

Case 2: When is taking known from the method of moment then the asymptotic variance of and pwill be:

11 =6.3434

22 =1.9804 V =0.0004355

11 22 = 12.5625 V p =0.0013949

212 =

221 = 0

Case 3: When is taking known from the method of moment then the asymptotic variance of and pwill be:

11 =3.2685

22 =1.9804

V =0.0008452

11 22 =6.4729 V p =0.0013949

212 = 221 = 0

maximum likelihood. The higher valueof indicates that the risk of HIV/AIDS among female migrant having atleast one close boy friend is greater.From the table 1, it is found that the

observed values of 2are insignificant

at 2 per cent level of significance andhence indicating the suitability of themodel.

The proposed distribution des-cribes satisfactorily that the unmarried




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female migrants having at least oneboy friend are more vulnerable to HIV/AIDS. By increasing the life style of livingand working conditions and by providingthe adequate facilities to unmarriedfemale migrant the vulnerability to HIV/

AIDS infections in them can bereduced.

ACKNOWLEDGEMENTS

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

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A Probability model for the risk of vulnerability to HIV/AIDS infection among female migrants