Estimation of a Conditional Mean in a Linear Regression Model

Consider the two linear regression models of Y_ij on X_ij, namely Y_ij = β_io + β_il X_ij + ε_ij,j = 1,2,…,n_i, i = 1,2, where ε_ij are assumed to be normally distributed with zero mean and common unknown variance σ². The estimated value of a mean of Y₁ for a given value of X₁ is made to depend on a preliminary test of significance of the hypothesis β₁₁ = β₂₁. The bias and the mean square error of the estimator for the conditional mean of Y₁ are given. The relative efficiency of the estimator to the usual estimator is computed and is used to determine a proper choice of the significance level of the preliminary test.     

Shrinkage Pre-Test Estimator of the Intercept Parameter for a Regression Model with Multivariate Student-t Errors

In the presence of an uncertain prior information about the value of the slope parameter, the estimation of the intercept parameter of a simple regression model with a multivariate Student-t error distribution is investigated. The unrestricted, restricted and shrinkage preliminary test maximum likelihood estimators are defined. The expressions for the bias and the mean square error of the three estimators are provided and the relative efficiences are analyzed. A maximin criterion is established, and graphs are constructed for an arbitrary number of degrees of freedom (D.F.) as well as sample sizes. A criterion to select optimal significance level is also discussed.     

On a Martingale Approach for the Estimation of the Parameters of a Bivariate Carrier Borne Epidemic Model in Presence of a Linear Relationship Between Carriers and Suspectibles

The present investigation is to waive the restriction of PICARD (1980) viz. x₀ + y₀ = n (the size of the population) for the obvious reason that the size of the susceptible population exposed to risk will depend on the growth rate of carriers and to evolve a simple methodology for the estimation of parameters of Downton's bivariate model. A lienar relationship of X_t on Y_t enabled to predict the size of the susceptible population exposed to the risk by a given number of carriers at any time. The same linear equation, when incorporated in Downton's bivariate model has generated the marginal process of Y_t. Using several Martingales constructed on the basis of the above marginal process, the minimum number of carriers that can bring out an epidemic and various parameters of Downton's model have been estimated.