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.
J. Comp. & Math. Sci. Vol. 1(2), 145-154 (2010).
I NTR O DUCTI O N
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-econom ic 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 unpre- cedented 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 dis tribut ion of resources, services and opportunities.
Migration, especially in the process of re gional e conomic deve lopm ent , urbanization and industrialization is an important cause and the effect of social and economic change. The socio- cultural 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:
(i) Let be the proportion of female migrants having at least one close boy friend.
(ii) Out of proportion of female migrants, let be the proportion of female migrants having only one closed boy friends.
(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 infec- tion.
Let the random variable x denotes the number of closed boy friends.
From the above assumptions, the probability model is given by
0 ] 1 [ X
P
; K=0
] 1 [ X
P
; K=1
N K
q K pq
X
P
1 ) 1 ] ( [
2
; K=2, 3,……..N (2 .1.) The above probability model
involves three parameters , , p to be estimated from the observed distribution of female migrants.
ESTIMATION:
Let N be a known quantity. If N is taken to be known then the proposed model (4.2.1) involves three para- meters , and p only.
METHOD OF MOMENT:
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:
f f
01
(3.1)f f
1
(3.2)q Nq q
p q
N N N
N
1 } 1 { ) 1 1 (
1 1
q
N X
1
1
(3.3)
Where f 0=Number of Observed zeroth
Himanshu Pandey et al., J.Comp.&Math.Sci. Vol.1(2), 145-154 (2010). 147
cell, f 1=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 f 0 denote the number of observation in zenoth cell, f 1denote 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:
2 1
0
1 ) 1 ) (
( ) 1 (
f N f
f
q L p
2 1 0
1 ) 1
(
f f f fq
Np
(3.4)Expression for logarithm of likelihood function is.
0 1 2
log L f log 1 f log f log
0 1 2
log log 1 log log 1
1
NL f f f p
q
f f
0f
1f
2 log
1
1
Np
q
(3.5 )Partially differentiating (3.5) with respect to , and p respectively and equating to zero. We get the following equations.
1
2 1
0
f f
LogL f
) ( f f
0 f
1 f
2
0 0
1
f f
f
0
(3.6)1 2
log 1 1
f f
L
0 1 2
1 2
1 1
f f f f
f f
Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)
0 1
1
1
f f f
f
0
(3.7)log f
2p L p
0 1 2
2
1
Nf f f f
f
p p q p
0
(3.8)After solving the equations (3.6), (3.7) and (3.8), we get the following estima- ting equations.
0
f f
f
(3.9)
1
0
f
f f
(3.10)And 2
0 1
1
Nf p
q f f f
(3.11)The asymptotic variance of (,
, p) is obtained by investing the infor- mation matrix whose elements are ne gatives of sec ond order of t he likelihood function.
The second order derivations of log L follows from equations (3.6), (3.7) and (3.8) respectively.
2 0
2
log
2 21 L f
0
2 2 2
f f
(3 .1 2 )
2 1 0 1
2 log 2 2
1
f f f
L f
(3.13)
And
2 2 2
2
p LogL f
p
2 2 1 0
) 1
(
) (
p q
f f f f
N (3.14)
Now
2
l o g L
2l o g L
2
l o g
2l o g
L L 0
(3.15)2
lo g
2l o g
L L 0
p p
(3 .1 6 ) 148 Himanshu Pandey et al., J.Comp.&Math.Sci. Vol.1(2), 145-154 (2010).
2
lo g
2lo g
L L 0
p p
(3.17) Since
0 1
E f f
E f
1 f
2 1 1
N
1E f f p q
And
0 1 2 1 1 1
E f f f f f p q
11 1 1
NE f f f f f p q
Then the expected value of the second partial derivatives of log L can be obt aine d by us ing t he t hre e different cases as:
Case1: When P is taking known from the method of moment then.
1
1
2
1
2
11
f
E LogL
(3.18)
11
2 1
2
22 f
E LogL
(3.19) nd
2
21 12
f
E LogL
0
2
f
E LogL
(3.20)Therefore by inverting the information matrix, the expression for the asymptotic variances of the ˆ and ˆ can be obtained as:
2 2 21 1 2 2 1 2
ˆ 1
V f
(3.21)
11 21 1 2 2 2 1
ˆ 1
V f
(3.22) Case 2: When is taking known from the method of moment then?
f
E LogL
2 2
11
1 1
1
(3.23) Himanshu Pandey et al., J.Comp.&Math.Sci. Vol.1(2), 145-154 (2010). 149
Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)
f p E LogL
2 2
22
1 1 1
1 q N p 1 q N p
(3.24) And
0
2
21
12
f
p E LogL
(3.25) Therefore by inverting the information matrix, the expression for the asymptotic variances of
ˆ
and ˆp can be obtained as:
2 2 21 1 2 2 1 2
ˆ 1
V
f
And (3.26)
1 1 21 1 2 2 2 1
ˆ 1
V p
f
(3.27) Case 3: When is taking known from the method of moment then?
f
E LogL
2 2
22
1 1
1
(3.28)f p E LogL
2 2
22
1
1 q
Np
1 1
1
Nq p q p
(3.29)And
log
2
21 12
f
p L
E
0
2
f
p LogL
E
(3.30) Therefore, by inverting the information matrix, the expression for the asymptotic variances of ˆ and ˆp can be obtained as:
22 211 22 12
ˆ 1
V f
(3.31)150 Himanshu Pandey et al., J.Comp.&Math.Sci. Vol.1(2), 145-154 (2010).
And
11 211 22 21
ˆ 1 V p f
(3 .32 ) APPLICATION:
The suitability of the proposed proba bilit y model (2.1 ) is t e s te d 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 f rom S oc ia l Welfare Depart me nt , 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 frequen- cies 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 362 362 362
ˆ 0.8039 0.8039
ˆ 0.4364 0.4364
ˆp 0.4757 0.4668
ˆ 2
7.5614 8.1497
.d f 2 2
Himanshu Pandey et al., J.Comp.&Math.Sci. Vol.1(2), 145-154 (2010). 151
Journal of Computer and Mathematical Sciences Vol. 1 Issue 2, 31 March, 2010 Pages (103-273)
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 =0.000435511 22=20.7334 V
ˆ =0.00084522
12= 221 =0
Case 2: 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 =0.000435511 22=12.5625 V p
ˆ =0.00139492
12= 221 =0
Case 3: 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 =0.000845211 22=6.4729 V p
ˆ =0.00139492
12= 221 =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 des- cribes satisfactorily that the unmarried 152 Himanshu Pandey et al., J.Comp.&Math.Sci. Vol.1(2), 145-154 (2010).
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/
A I DS inf e c tions in t he m c a n be reduced.
ACKNOWLEDGEMENTS
First author is thankful to U.G.C., New Delhi for providing a financial support by MRP-37-546/09 (SR).
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