Childhood mortality is one of the important indicators of a nation’s general medical and public health conditions, and consequently, the country’s level of socioeconomic development. Its decline is therefore not only desirable but also indicative of an improvement in general living standards. The infant mortality rate, probability of dying before one year of age expressed per 1000 livebirths and underfive mortality rate, probability of dying between birth and age 5 expressed per 1000 livebirths have been considered as measures of children’s wellbeing for many years. In India, 2.1 million children die before their fifth birthday (child death). Half of these children die even before they are 28 days old, accounting for onefourth of global infant deaths. Among the 9.7 million child deaths worldwide annually, onethird occur in India. While around 4 million children die within the first 28 days of life across the world every year,^{1, 2}, and reduction in child death or enhancement in the probability of survival of children is one of the known reasons for reduction infertility. That implies a reduction in infant and child morality which is likely the most important aim of the millennium development goals, as children are the most important assets of a nation. Studies on earlyage mortality in the last five decades are mostly confined to infant death, but, it has been realized that child death also needs to be examined in addition to infant mortality.
A number of parametric models for the study of the age patterns of mortality have been developed over the year. In order to get this pattern through modeling, the first parametric model for the duration of mortality is developed by ^{3}. The firstever attempt to study infant mortality through parametric modeling is proposed by ^{4} who used a hyperbolic function for the study of infant mortality. Another attempt to represent mortality across the entire age range was the eight parameter nonlinear model in, “Age pattern of mortality” ^{5}. Later, ^{6} proposed a logarithmic approximation and the Weibull function is recommended by ^{7}. After that several attempts have been made to study age at infant mortality through mathematical models. In these circumstances, a number of attempts have been made to study the age pattern of the mortality mathematical model ^{8, 9, 10, 11}. Modified PolyaAeppli and PoissonGamma distribution for child death is discussed by ^{12}. A modified truncated Geometric distribution was developed for the number of child death ^{13}. BetaBinomial model and inflated Binomial distribution are proposed for the number of child death for fixed parity i.e. fixed number of children ever born ^{14, 15}. Further ^{16} developed a model by mixing Poisson distribution with the BetaBinomial distribution to study the variation of child death in society. PoissonLindley distribution is proposed for the study of a number of child deaths and compare with PoissonExponential and PoissonGamma distribution ^{17}. Recently a mixture probability distribution is developed for the study of the pattern of child death ^{18}. In the present study, we have proposed a probability model for child death which gives a clear picture of its pattern. The beauty of this model is that we can estimate the risk of child death as the above mentioned previous models but this model also provides the estimate of average children ever born to the females, which is the added benefit of the proposed model. This study proposes a mathematical model to explain the pattern of child death for all females in the most populous state of India i.e. Uttar Pradesh along with other major states of the different parts of India.
Since probability models provide concise and clear representations of extensive data sets in a better way in recent years increased attention has been paid to the proposition and derivation of probability models for the distribution of infant mortality. Let x denote the number of child death to a female and the distribution of a number of child death to females of given parity k and risk of child death p follows a binomial distribution.
As we discussed in the literature, that the number of parity (number of children ever born to a female) k itself a random variable and following a displaced Poisson distribution with probability mass function as;
Therefore, the joint density of k and x for the given p is given by,
Hence, the marginal distribution of x;
On simplifying this marginal distribution we get;
Further, in this model, we assume that the probability of death of a child varies from female to female but remains constant at each birth for a given female. We assume that the probability of death of child ‘p’ follows a beta distribution with parameter a and b which is given by;
The joint density of x is given as;
After simplifying this reduces to;
Hence, the marginal distribution of x is given by;
It can be verified easily that
In this study method of the moment has been used as an estimation procedure to estimate the unknown parameters λ, a, and b of the model considered for a number of child deaths to females of all parity. The method of moments generally provides an estimate which is consistent but not as efficient as the method of maximum likelihood. But the method of moments is often used because it leads to very simple computations than the maximum likelihood method. Also, the complexities in mathematical derivation involved in the maximum likelihood method. The first three moments of the probability model considered here are as follows
Let
From equation (12) we have,
This implies that
Using equation (12) and (13) we have;
putting the values of b from (16);
solving the above equation, we get the value of ‘a’,
Now from equation (14) & (15), we have;
Using equation (15);
Simplifying equation (21), we get;
Or,
Substituting the value from a equation (20) in the above equation and after simplifying the above equation, we have;
The value of δ and η can be calculated from the data directly by using the equation (19) and (21) respectively. After obtaining and substituting these values in equation (23) we get a sevendegree polynomial equation in λ, which is an unknown and analytical solution of λ can be obtained. Once we get the value of λ the value of other parameters a and b can be easily obtained by using equations (20) and (16) respectively. After obtaining the estimates of the parameters the expected frequencies can be obtained by using the proposed probability model.
To illustrate the application of the model discussed above the data collected in NFHSII (199899) ^{19} and NFHSIII (200506) ^{20} for the state Uttar Pradesh has been considered. For other major states data from NFHSIII (200506) has been used only. In this data information on all live births to a woman and their survival status at the time of survey has been collected. In case of death of child age at death for each child has been recorded separately. Women, who have no births in the preceding five years from the reference date of the survey, have been considered with the assumption that these females have already completed their family size. By this, we can avoid censored observations from the data. Childless females have been considered keeping the point in mind that all females should be taken for the experience of any child death.
The data set of the state Uttar Pradesh contains information on females. For NFHSIII data the whole state is divided into three categories namely Total, Urban and Rural according to their place of residence. The total number of women considered was 4673 and among them, the distribution of females according to their place of residence is 2106 (urban) & 2567 (rural). For NFHSII data the state of Uttar Pradesh contains information of 4385 females with 1003 urban and 3382 rural females. Also to check the suitability of the proposed model according to age groups the data of Uttar Pradesh is divided according to the present age group 3039 and 4049 years respectively and also according to a place of residence i.e. rural and urban.
The observed and expected number of frequencies of females according to the number of child death for the different domains of Uttar Pradesh i.e. Uttar Pradesh urban, rural, and as a whole are shown by
No. of Child Deaths 
NFHSIII 
NFHSII 

Observed Frequency 
Expected Frequency 
Observed Frequency 
Expected Frequency 

0 
1620 
1620.89 
750 
751.04 
1 
325 
325.82 
157 
154.57 
2 
107 
104.25 
60 
57.33 
3 
40 
36.03 
14 
23.51 
4 
6 
12.57 
15 
9.83 
5+ 
8 
6.44 
7 
6.72 
Total 
2106 
2106 
1003 
1003 
Parameters 

Parameters 


ChiSq. (df) 
4.324 (2) 
ChiSq. (df) 
6.742 (2) 

pvalue 
0.115 
pvalue 
0.034 

Risk of child death 
0.0494 
Risk of child death 
0.0551 
No. of Child Deaths 
NFHSIII 
NFHSII 

Observed Frequency 
Expected Frequency 
Observed Frequency 
Expected Frequency 

0 
1663 
1649.70 
2288 
2248.99 
1 
516 
541.99 
528 
609.69 
2 
227 
220.34 
301 
269.25 
3 
98 
92.10 
143 
130.28 
4 
43 
38.07 
71 
64.32 
5 
11 
15.33 
25 
31.55 
6+ 
9 
9.47 
14 
15.18 
7+ 


12 
12.74 
Total 
2567 
2567 
3382 
3382 
Parameters 

Parameters 


ChiSq. (df) 
3.818 (3) 
ChiSq. (df) 
18.796 (4) 

pvalue 
0.282 
pvalue 
0.001 

Risk of child death 
0.0799 
Risk of child death 
0.0718 
No. of Child Deaths 
NFHSIII 
NFHSII 

Observed Frequency 
Expected Frequency 
Observed Frequency 
Expected Frequency 

0 
3283 
3273.44 
3038 
3000.24 
1 
841 
864.47 
685 
762.90 
2 
334 
323.49 
361 
327.23 
3 
138 
128.42 
157 
154.47 
4 
49 
51.16 
86 
74.45 
5 
15 
20.04 
30 
35.63 
6 
8 
7.64 
15 
16.71 
7+ 
5 
4.34 
13 
13.37 
Total 
4673 
4673 
4385 
4385 
Parameters 

Parameters 


ChiSq. (df) 
3.167 (3) 
ChiSq. (df) 
14.883 (4) 

pvalue 
0.367 
pvalue 
0.005 

Risk of child death 
0.0632 
Risk of child death 
0.0686 
No. of Child Deaths 
NFHSIII 
NFHSII 

Observed Frequency 
Expected Frequency 
Observed Frequency 
Expected Frequency 

0 
1167 
1159.31 
969 
962.46 
1 
325 
340.74 
287 
305.61 
2 
120 
113.95 
130 
113.94 
3 
39 
37.75 
41 
42.85 
4+ 
18 
17.25 
14 
15.74 



8 
8.40 
Total 
1669 
1669 
1449 
1449 
Parameters 

Parameters 


ChiSq. (df) 
1.173 (1) 
ChiSq. (df) 
3.733 (2) 

pvalue 
0.279 
pvalue 
0.155 

Risk of child death 
0.0836 
Risk of child death 
0.0814 
No. of Child Deaths 
NFHSIII 
NFHSII 

Observed Frequency 
Expected Frequency 
Observed Frequency 
Expected Frequency 

0 
918 
925.52 
801 
784.89 
1 
414 
403.97 
353 
382.80 
2 
189 
187.58 
217 
209.50 
3 
86 
84.95 
115 
114.23 
4 
33 
36.76 
72 
60.28 
5 
12 
15.09 
26 
30.41 
6+ 
11 
9.13 
12 
14.58 
7+ 


12 
11.31 
Total 
1663 
1663 
1608 
1608 
Parameters 

Parameters 


ChiSq. (df) 
1.734 (3) 
ChiSq. (df) 
6.341 (4) 

p value 
0.629 
pvalue 
0.175 

Risk of child death 
0.1314 
Risk of child death 
0.1565 
States 
Parameters 
chisquare 
Degree of freedom 
Risk of child death 





Northern States 

Uttar Pradesh 
6.7262 
0.5112 
7.5836 
3.2568 
4 
0.0632 

Bihar 
6.3683 
0.4298 
7.3630 
3.3813 
4 
0.0552 

Madhya Pradesh 
4.5315 
0.3213 
3.1414 
0.1659 
2 
0.0928 

Rajasthan 
5.6221 
0.3317 
5.2801 
5.0062 
4 
0.0591 

Eastern States 

Orissa 
4.0157 
0.4699 
7.7448 
1.1698 
1 
0.0572 

West Bengal 
4.5174 
0.2537 
6.7115 
1.3115 
1 
0.0364 

Western States 

Gujarat 
4.0025 
0.3172 
6.9107 
0.3739 
1 
0.0439 

Maharashtra 
4.7816 
0.2372 
8.5353 
7.2130 
1 
0.0270 

Southern States 

Andhra Pradesh 
4.0157 
0.3080 
8.9412 
2.0712 
4 
0.0333 

Karnataka 
4.2088 
0.3253 
9.2339 
2.6655 
1 
0.0340 

Kerala 
1.9095 
0.1497 
8.9482 
0.4122 
 
0.0165 

Tamil Nadu 
2.9724 
0.3282 
7.8670 
3.2133 
 
0.0400 
From the results discussed above, we may conclude that the model proposed here may be considered to be suitable to describe the distribution of a number of child death to females. The remarkable utility of this model is that it provides a clear visualization about the unobservable risk of child deaths and also provides the average children ever born to female through the distribution of females according to the number of child deaths. Thus, this model provides a new dimension for comparing the risk of child death to females in different regions of Uttar Pradesh and other states.
In this study, we suggest a ‘model for child death’ which we believe will describe the pattern of mortality adequately for a wide variety of experiences. Although the model may not always give a fit close enough for certain purposes it reproduces the risk of child death and the mean number of children ever born. The model is appropriate and applicable to other similar data. The mathematical model gives an adequate representation of child deaths among all females of Uttar Pradesh and preliminary studies give different experiences indicates its wide applicability. The model can be used to calculate the average number of children ever born through the distribution of a number of child death. One may modify the proposed model by taking some truncated probability distribution for the number of children ever born to the females, which would be done separately in the future.