Sequential point estimation of regression parameters in a linear model
The linear model \(y_ i=x_ i'\beta +\epsilon_ i\), \(i=1,2,...\), where \(x_ i\) and \(\beta\) are \(p\times 1\) vectors, \(\epsilon_ i's\) are disturbance terms, independent and normally distributed with mean 0 and variance \(\sigma^ 2\), is considered. A sequential procedure is proposed for the point estimation of \(\beta\) under the family of loss functions \[ L^*_ n(\alpha,t)=A[({\tilde \beta}_ n-\beta)'(X_ n'X_ n)({\tilde \beta}_ n-\beta)/n]^{\alpha /2}+Cn^ t, \] where A,\(\alpha\),C,t are known constants, \(X_ n=(x_ 1,x_ 2,...,x_ n)'\). The procedure consists of the estimator \({\tilde \beta}_ n=(X_ n'X_ n)^{-1}X_ n'Y_ n\), where \(Y_ n=(y_ 1,y_ 2,...,y_ n)'\), and the stopping time \(N\equiv N(\sigma)\), the smallest positive integer \(n\geq m\) \((\geq p+2)\) for which \(n\geq [\alpha K(p,\alpha){\hat \sigma}_ n^{\alpha}/2Ct]^{2/(2t+\alpha)}\), where \(K(p,\alpha)=A2^{\alpha /2}\Gamma ((p+2)/2)\Gamma (p/2).\) In the paper the asymptotic behavior of the risk-efficiency and the regret for this sequential procedure have been established. The results concerning the asymptotic behaviour of the regret generalize those derived by \textit{N. Mukhopadhyay} [Sequential estimation of regression parameter in Gauss-Markov set-up. J. Indian Stat. Assoc. 12, 39-43 (1974)]. The results concerning the asymptotic behaviour of the risk-efficiency have been obtained also by \textit{Y. H. Wang} [J. Am. Stat. Assoc. 75, 977- 983 (1980; Zbl 0455.62064)].
scientific article; zbMATH DE number 4001252
scientific article