On a global deviation measure for an estimate of the regression line

V. D. Konakov

Teoriâ veroâtnostej i ee primeneniâ, Tome 22 (1977) no. 4, pp. 879-888

Résumé

Let $X_1,X_2,\dots$ be a sequence of independent identically distributed random vectors with values in the Euclidean plane. We prove that the limiting distribution for a properly normalized quadratic functional $$ \int(r(x)-\hat r_n(x))^2\hat h_n^2(x)p(x)\,dx $$ is normal $(0,\sigma^2)$, where $r_n(x)$ is an estimate of the regression line $r(x)$ of the form (1). We obtain also the limiting distribution in case of a sequence of «local» alternatives of the form (7). Finally, for the rate of convergence of moments, we have $$ |\nu_{n,2k}-\nu_{2k}|\le c_1(k,\sigma)n^{-\frac{1}{2}+\delta},\qquad |\nu_{n,2k+1}|\le c_2(k,\sigma)n^{-\frac{1}{4}+\delta}, $$ where $c_1(k,\sigma)$ and $c_2(k,\sigma)$ are some constants which depend on the order $k$ of the moment and variance $\sigma^2$.