A study of a modification of moment estimation of a two parameter gamma distribution has been carried out. The scale parameter has been estimated using the Modified Moment Estimates (MMEs) and their performance is compared to the Maximum Likelihood Estimates (MLEs). Modified Moment Estimates are easy to compute relative to Maximum Likelihood Estimates. To adjust their bias, the Jackknife technique has been used. The efficiency of these estimators is compared by calculating their respective variances. The R-has been used to carry out the Monte Carlo simulation of data. The results obtained show that the Modified Moment Estimates perform as well as the Maximum Likelihood Estimates when bias as well as variation of data is taken into account.

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Abstract:Mixture experiments are special type of response surface designs where the factors under study are proportions of the ingredients of a mixture. In response surface designs the main interest of the experimenter may not always be in the response at individual locations, but the differences between the responses at various locations is of great interest. Most of the studies on estimation of slope (rate of change) have concentrated in Central Composite Designs (CCD) yet mixture experiments are intended to show the response for all possible formulations of the mixture and to identify optimal proportions for each of the ingredients at different locations. Slope optimal mixture designs for third degree Kronecker model were studied in order to obtained optimal formulations for all possible ingredients in simplex centroid. Weighted Simplex Centroid Designs (WSCD) and Uniformly Weighted Simplex Centroid Designs (UWSCD) mixture experiments were obtained in order to identify optimal proportions for each of the ingredients formulation. Derivatives of the Kronecker model mixture experiment were used to obtain Slope Information Matrices (SIM) for four ingredients. Maximal parameters of interest for third degree Kronecker model were considered. D-, E-, A-, and T-optimal criteria and their efficiencies for both WSCD and UWSCD third degree Kronecker model were obtained. UWSCD was found to be more efficient than WSCD for almost all the points in the simplex designs, therefore recommended for more optimal results in mixture experiments.