A Displaced Probability Model for the Child Mortality in a Family

Transcription

1 International Journal o Statistics and Applications 4, 4(): DOI:.59/j.statistics.44. A Displaced Probability Model or the Child Mortality in a Family Himanshu Pandey, Ashutosh Pandey,*, Vivek Kumar Shukla Department o Mathematics and Statisistics, DDU Gorakhpur University, Gorakhpur, U.P., India Department o Management, School o Management Sciences, Lucknow U.P., India Abstract In Present paper, a displaced probability model has been proposed to study the distribution o amilies according to the number o child death within the irst ive years o lie. The model involved several parameters, which is estimated with the help o method o maximum likelihood and method o moments and itted to observed survey data to draw important conclusions. Keywords Probability model, Child-Mortality, MLE. Introduction Child mortality, also known as under-5 mortality, reers to the death o inants and children under the age o ive. In,.9 million children under ive died, down rom 7. million in, 8. million in 9, and.4 million in 99. About hal o child deaths occur in Sub-Saharan Arica. Reduction o child mortality is the ourth o the United Nations' Millennium Development Goals. Child Mortality Rate is the highest in low-income countries, such as most countries in Sub-Saharan Arica. A child's death is emotionally and physically damaging or the mourning parents. Many deaths in the third world go unnoticed since many poor amilies cannot aord to register their babies in the government registry. In demography child mortality are useul as a sensitive index o a nation s health conditions and as guided or the structuring o public health programmes. Child Mortality is interrelated to social, cultural, economic, physiological and other actor. The high rate o inant and child mortality shows a low level development o the health programme and also or the nation s. Inant and Child mortality has been o interest o researchers and demographers because o its apparent relationship with the acceptance o modern contraceptive means (Kabir et al, 99). Some eects have been made to estimate the current levels o child mortality by using data available rom the dierent survey and other speciic sources. Hill and Devid (989) have suggested an approach or estimating child mortality rom all births which have taken place in last ive * Corresponding author: ashutosh.srmcem@gmail.com (Ashutosh Pandey) Published online at Copyright 4 Scientiic & Academic Publishing. All Rights Reserved years beore the survey. However, the estimates obtained though this method also suer rom the problem o under reporting Pathak, Pandey, and Mishra, (99). In this Circumstances, some o the earlier studies about child mortality by using model (Chauhan, 997; Gavrilova, 99; Goldblatt, 989; Heligman and Pollard, 98; Krishna, 99; Ronald and Carter, 99; Thiele, 97). Initially, Keyitz (977) used a hyperbolic unction to study the inant and child mortality. Later Arnold (98) used pareto distribution and Krishnan and Yin (99) applied inite range model or the same. The direct measures o mortality being not reliable, the problem may be overcome by the recently developed model building approaches which make it possible to obtain estimates rom inormation other than vital statistics. In present paper a probability model has been proposed to study the distribution o amilies according to the number o child deaths within the irst ive years o lie.. The Model Let x denote the number o child deaths in a amily at the survey point. Then the distribution o x is derived under the ollowing assumption.. Only those amilies are considered in which at least one birth prior to the survey has occurred.. At the survey point, a amily either has experienced a child loss or not. Let α and ( α) be the respective proportions.. Out o α proportion o amilies, Let β be the proportion o amilies in which only one child death has occurred. 4. Remaining ( β)α proportion o amilies, experiencing multiple child deaths, ollows a displaced log distribution with parameter λ according to the

2 9 Himanshu Pandey et al.: A Displaced Probability Model or the Child Mortality in a Family number o child deaths. Under these assumptions the probability distribution o X is given by.. Estimation P X = = ( α), k= P X = = αβ, k= P X = k = α β ( )λk k ln ( λ), k=,,4 (.) Method o moment: We shall discuss the method o moments to estimate the three parameters α, β, λ and o the probability unction (.). Let (x, ) be a requency distribution whose parameters to be estimated rom the observed distributions o amilies according to the number o child deaths. The ollowing estimation technique is employed or estimation or rest o the parameter. Equating proportions o zeroth cell, irst cell requency and simple mean to their corresponding observed values L [ Taking Log both sides we get log L log( ) ( )( ) ln( ) respectively, which convert into the ollowing equations. = α (.) = αβ (.) X (.) ln( ) Where = Observed value o zeroth cell requency = Observed value o irst cell requency = Total number o observations = i i X = Sample mean o the observed values. Method o Maximum Likelihood: The proposed model involves our parameters α, β, λ to be estimated rom the observed distribution o amilies according to the number o child deaths but it cannot be possible to estimate all these simultaneously by this method, so the value o N has been taken as method o moments. From the population (.), the likelihood unction L is given by, ln( ) ] [ ] ( ) (.4) log log log log( ) log log( ) log{ln( )} Now, partially dierentiating w. r. t. α, β and λ respectively and equating to zero. log L log L log ( )ln( ) log( ) log log{ ln( } log{ln( ) ( )ln( ){ ln( )} L (.8) Equating to zero equation (.), (.7) and (.8) and solving the estimate o α, β and λ can easily be obtained as: ( ) ˆ ( )ln( ) ˆ ( ) ( )ln( ){ ln( )} The second partial derivatives o Log L can be obtained as: (.5) (.) (.7) ( ) log L (.9) ( )

3 International Journal o Statistics and Applications 4, 4(): ( ) log L (.) ( ) ln( ) ln( ) ( ) ln( ) log L ln( ) ( ) ln( ) ln( Now partial derivative o Here, log L log L, α β and log L λ log L α β w.r.to β, λ and α respectively we get as: = log L β λ E = ( α) E = αβ ( )λ E = α( β) ln ( λ) ( ) E. ( ) ln( ) (.) = log L λ α = (.) Using the above acts, the expected value o the second partial derivatives can be obtained as: = E log L α = α + α = E log L β = α β + β (.) (.4) = E log L λ (.5) ( ). ( ) ln( ) ( ). ln( ) The covariance between the estimators becomes zero since E log L α β ln( ) ln( ) ln( ) ( )ln( ) ( )ln( ) ln( = E log L β λ Thus the asymptotic variances o the estimator can be obtained as: 4. Application V α =, V β = = E log L α λ and V λ = = (.) (.7) The suitability o the proposed model is examined to the sample data collected under a survey entitled Eect o breasteeding on ertility in North Rural India in 995 and another is entitled A Demographic survey on ertility and mortality in rural Nepal: A Study o Palpa and Rupandehi Districts in. On the other hand, one set o data has been taken rom a household sample survey in Brazil in 987 (Sastry,997). The parameters o the proposed model have been estimated by the method o moment and method o maximum likelihood. The estimated values o dierent parameters are given in tables to or the child deaths. The estimated value o α.75 and.75 are or India, Nepal, North East Brazil by Method o Moments and MLE respectively. Which shows the proportion o amilies experienced a child loss, was ound slightly higher or India than North East Brazil which is.8 and.8 by Method o Moments and MLE respectively, and in Nepal the value o α is.9 and.8 by Method o Moments and MLE respectively. The estimate o β are (.5857,.758,.5) and (.5855,.757,.5) by Method o Moments and MLE respectively, respectively or India, Nepal, North East Brazil. Which shows the proportion o amilies having only one child death was ound greater or Nepal (.757) as compared to other countries and the estimated values or λ are.79,.5 and.59 by the method o moment and.9,.579 and

4 9 Himanshu Pandey et al.: A Displaced Probability Model or the Child Mortality in a Family.548 by the maximum likelihood respectively or the above mentioned countries. The graph to shows the deviation o expected values rom the observed values in the three types o data in two dierent methods o estimation techniques. From tables to it is ound that the value o χ are insigniicant at 5% level o signiicance or all set o data. The proposed model suitably described the pattern o child mortality or dierent data in Indian subcontinents. Table. Distribution o observed and expected number o amilies according to the number o child deaths in Eastern Uttar Pradesh (India) Number o child dead Observed number o amilies Child dead in Eastern Uttar Pradesh Expected number o amilies Method o Moment Method o Maximum Likelihood Total α β λ V(α) V β V(λ) χ d X F L M

5 International Journal o Statistics and Applications 4, 4(): Table. Distribution o observed and expected number o amilies according to the number o child deaths in Nepal Number o child dead Observed number o amilies Child dead in Nepal Expected number o amilies Method o Moment Method o Maximum Likelihood Total α β λ V(α) V β V(λ) χ d X F L M

6 94 Himanshu Pandey et al.: A Displaced Probability Model or the Child Mortality in a Family Table. Distribution o observed and expected number o amilies according to the number o child deaths in North East Brazil Number o child dead Observed number o amilies Child dead in North East Brazil Expected number o amilies Method o Moment Method o Maximum Likelihood Total α β λ V(α) V β V(λ) χ d X F L M Demography India Vol., No., pp.-74. REFERENCES [] Arnold, B.C.(99), Pareto Distributions, Vol. 5 in Statistical Distributions, Fairland (M.D): International Co-operative Publishing house. [] Chauhan, R.K. (997), Graduation o Inant Deaths by Age. [] Goldblatt, P.O. (989), Mortality by Social Class, Population Trends, No. 5, Summer. [4] Heligman L. and Pollard, J.H.(98), Age pattern o mortality. Journal o Institute o Actuaries 7: [5] Hill, A.G. and Devid, H.P. (989). Measuring Child Mortality in the Third World: Neglected Sources and Novel Approaches, IUSSP Proceeding o International Conerence,

7 International Journal o Statistics and Applications 4, 4(): New Delhi, India. [] Keyit, N.(977), Applied Mathematical Demography. John Wiley, New York. [7] Krishnan, P. (99), Mortality Modeling with order Statistics, Edmonton: Population Research Laboratory, Department o Sociology, University o Alberta, Research Discussion paper No. 95. (ISSN 7-47). [8] Krishnan, P. and Jin. Y. (99): A Statistical Model o Inant Mortality Janasamkhya, Vol., No., pp-7-7. [9] Pathak, K.B.; Pandey, A.and Mishra, U.S.(99). On Estimating current Levels o Fertility and Child Mortality rom the Data on Open Birth Interval and Survival Status o the last Child. [] Ronald, D.Lee and Lawrence, R. Carter, (99): Modelling and Forcasting U.S. mortality. Journal o American Statistical Associations, 87: [] Srivastava,S. (): Some Probability Models or Number o Child Deaths in a Families, Unpublished Ph.D. Thesis in Statistics, Banaras Hindu University, Varanasi India. [] Thiele, P.N. (97), On mathematical Formula to express the rate o Mortality throughout the whole o lie. Journal o Institute o Actuarie.