The generalized preliminary test estimator when different sets of stochastic restrictions are available

Abstract

Arumairajan and Wijekoon (Commun Stat—Theor Methods, in press, 2014) proposed a generalized preliminary test stochastic restricted estimator (GPTSRE) to represent the preliminary test estimators when stochastic restrictions are available in addition to sample model. The aim of this paper is to define the GPTSRE when two different sets of competing stochastic restrictions are available. Moreover the conditions for superiority of GPTSRE based on one stochastic restriction over the other are derived with respect to mean square error (MSE) matrix criterion. Furthermore the estimator GPTSRE is theoretically compared with almost unbiased ridge estimator and almost unbiased Liu estimator in the MSE matrix sense. Finally a Monte Carlo simulation study and numerical example are done to illustrate the theoretical findings.

Appendix

Lemma 1

Wang et al. (2006) Let $n\times n$ matrices $M>0$, $N>0$ (or $N\ge 0)$, then $M>N$ if and only if $\lambda _1 ( {NM^{-1}})<1$.where $\lambda _1 ( {NM^{-1}})$ is the largest eigenvalue of the matrix $NM^{-1}$.

Lemma 2

Trenkler and Toutenburg (1990) Let $\hat{\beta }_1$ and $\hat{\beta }_2$ be two linear estimator of $\beta$. Suppose that $D=D( {\hat{\beta }_1 })-D( {\hat{\beta }_2 })$ is positive definite then $\Delta =MSE( {\hat{\beta }_1 })-MSE( {\hat{\beta }_2 })$ is nonnegative definite if and only if ${b}'_2 ( {D+b_1 {b}'_1 })^{-1}b_2 \le 1$, where $b_j$ denotes the bias vector of $\hat{\beta }_j,j=1,2.$

Lemma 3

Baksalary and Trenkler (1991) Let C be a nonnegative definite matrix and $c_1$, $c_2$ be linearly independent vectors. Furthermore for some generalized inverse $C^-$ of C, let $f_{ij} ={c}'_i C^-c_j$; $i=1,2,\,j=1,2$ and let

$$\begin{aligned} s=\frac{{c}'_2 (I-CC^-{)}'(I-CC^-)c_2 }{{c}'_1 (I-CC^-)(I-CC^-)c_1 } \end{aligned}$$

where $c_1{\,\in \,}\mathfrak {R}(C)$ and $\mathfrak {R}(.)$ denote the column space of the corresponding matrix. Then we have $C+c_1 {c}'_1 -c_2 {c}'_2 \ge 0$ if and only if

• (a) $c_1 \in \mathfrak {R}(C),\,c_2 \in \mathfrak {R}(C)$ and $(f_{11} +1)(f_{22} -1)\le f_{12}^2$ or

• (b) $c_1 \notin \mathfrak {R}(C),\,c_2 \in \mathfrak {R}(C,c_1 )$ and $(c_2 -sc_1 {)}'C^-(c_2 -sc_1 )\le 1-s^2$

and all expressions in (a) and (b) are independent of the choice of $C^-$.