Statistical Estimation under Length Biased Distributions

In simple random sampling, there is an equal chance of selection for each unit in the population. However, there are some practical situations where this might not be ideal. We consider the case of length-biased sampling, whereby the chance of selection of a unit is proportional to the length of the unit, by which the length-biased density is defined. Specifically we consider the case of a continuous random variable Y with pdf f(YiO), where, f(YiO) is of an exponential family of distributions and 0 is a k-parameter vector to be estimated. We show the bias of the MLE of 8 to be of order nand use the jackknife estimation technique to eliminate the leading term in the bias of the MLE of 8. It is shown that the Jackknife estimator has the same normal distribution as the MLE in large samples. The estimation study is extended to the regression problem where the mean of the sufficient statistics of the exponential family of distributions depends on a set of covariates. Both the bias and asymptotic distribution of the MLE and the Jackknife are provided. The Fisher information from the length-biased distribution and the original distribution are compared through the A-optimalily and D-optimality which are functions of the eigenvalues of the Fisher information matrices. Data from the Demographic and Health Surveys is analyzed by maximum likelihood methods and Jackknife methods. The response variables is the first order birth interval and the potential covariates are the age at marriage and duration of marriage for the mothers. Simulation studies from the lognormal and gamma distributions indicate a reduction in the bias of the MLE's for the scale parameter of the lognormal distribution. Both the MLE and the Jackknife estimates are approximately normal. The RMSE for the MLE and the Jackknife estimates are given.