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Comparison of the New Estimators: The Semi-Parametric Likelihood Estimator, SPW, and the Conditional Weighted Pseudo Likelihood Estimator, WPCE

The analysis of sample-based studies involving sampling designs for small sample size, is challenging because the sample selection probabilities (as well as the sample weights) is dependent on the response variable and covariates. The study has focused on using systems of weighted regression estimating equations, using different modified weights, to estimate the coefficients of Weighted Likelihood Estimators. Usually, the design-consistent (Weighted) estimators are obtained by solving (sample) weighted estimating equations. They are then used to construct estimates which have better relative efficiencies and smaller finite small sample bias than the estimates from the Horvitz-Thompson Weighted Estimator with unmodified weight, option A. The purpose of our study is to compare derived Estimators of the weighted regression estimating equations for estimating the coefficients of Weighted Likelihood Estimators, the Semi-Parametric Weighted Likelihood Estimator, SPW and the Weighted Conditional Pseudo Likelihood Estimator, WCPE with the conventional Horvitz-Thompson Weighted Likelihood Estimator, using relative efficiency, sample bias and Standard Error for small sample size. The constructed estimates from the system of weighted regression estimating equations, using different modified weights, are actually the Weighted Likelihood Estimators. The study compared the two new estimators, the Semi-parametric weighted estimator, SPW and the Weighted Conditional Pseudo Likelihood estimator, WCPE, for both the unmodified and modified Weights, which were found to have better relative efficiency and smaller finite small sample bias than the estimates from conventional Horvitz-Thompson Weighted Estimator, for both generated and for real data. The outcome of the tests show strong similarity in performance to those obtained using the simulated data. Estimates were constructed which have better relative efficiencies and smaller finite small sample bias than the estimates from the Horvitz-Thompson Weighted Estimator with unmodified weight, option A.