Edgeworth expansion and bootstrap approximation for M-estimators of linear regression parameters with increasing dimensions

In this study, we consider two different inference problems in linear regression models. The first problem deals with the model y[subscript] j = x[subscript]spj'[beta] + [epsilon][subscript] j; j = 1,2, ...,n, where y[subscript]1,y[subscript]2, ...,y[subscript] n are observations; [epsilon][subscript]1,[epsilon][subscript]2, ...,[epsilon][subscript] n are independent and identically distributed random variables with a common distribution function; x[subscript]1,x[subscript]2, ...,x[subscript] n are known, nonrandom p-vectors; and [beta] is px1 vector of parameters. Edgeworth expansions for the standardized as well as studentized linear combinations of least squares estimators are obtained without assuming normal errors. The number of parameters p is allowed to increase with n and essentially under the condition p = O(n[superscript](1[over]2-[delta])) as n→[infinity] for any [delta] > 0. These results extend the results of Qumsiyeh (1986). It is also shown that the bootstrap method is second order correct for the studentized statistics improving the results of Bickel and Freedman (1983);The second model is a p-population model given by Y[subscript]i = X[subscript] iB[subscript] i + [epsilon][subscript]i; i = 1,2, ...,p, where Y[subscript] i is the n[subscript] ix1 vector of observations from the i[superscript]th population; [epsilon][subscript] i is a n[subscript] ix1 random vector; [beta][subscript] i is the kx1 vector of parameters; and X[subscript] i is n[subscript] i xk known nonrandom matrix. Here k is a fixed positive integer and n[subscript] i≥ 1 denotes the i[superscript]th sample size for i = 1,2, ...,p. This is an extension of Ringland (1980) to a general regression model. Edgeworth expansions and bootstrap approximations of the M-estimator corresponding to some score function [psi] of the linear regression parameters are obtained under some regularity conditions on [psi] and on the error distribution function. This extends the results of Lahiri (1990) from fixed to increasing dimensionality. Results of this part remain valid under the condition p[superscript]3[over] N→ 0 as N→[infinity], where N = [sigma][subscript]spi=1p n[subscript] i is the number of observations.