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Strong consistency of least squares estimate in multiple regression when the error variance is infinite. (English) Zbl 0913.62024

Summary: Let \(Y_i=x_i'\beta+e_i\), \(1\leq i\leq n\), \(S_n=\sum^n_{i=1}x_ix_i'\). Suppose that the random errors \(e_1,e_2,\dots\) are i.i.d., with a common distribution \(F\) belonging to the class \[ {\mathcal F}_r=\left\{F:\int^\infty_{-\infty}xdF=0,\;0<\int^\infty_{-\infty}| x|^rdF<\infty\right\} \] for some \(r\in[1,2)\). We obtain a sufficient condition for the strong consistency of the least squares estimate (LSE) \(\widehat\beta_n\) of \(\beta\). The condition is necessary in the following sense: If the condition is not satisfied, then for some \(F\in{\mathcal F}_r\), \(\widehat\beta_n\) fails to converge a.s. to \(\beta\).

MSC:

62F12 Asymptotic properties of parametric estimators
62J05 Linear regression; mixed models
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