Abstract:
Thr $(X, Y),\left(X_i, Y_i\right) i=1,2, \cdots$ be i.i.d. $R^p \times R^q$-valued random vectors wilh common joint distribution $G(x, y)$ and joint density $g(x, y)$. Let $h(x)$ be the marginal density of $X$ and let $f(y \mid x)=g(x, y) / h(x)$ be the conditional density of $Y$ on $X$. In this pilys we propose the following type of conditional density estimators (calle I double -irnel estimators) $$ f_n(y \mid x)=\sum_{i=1}^n K_1\left(\frac{x-X_i}{a_n}\right) K_2\left(\frac{y-Y_i}{b_n}\right) /\left\{b_n^q \sum_{j=1}^n K_1\left(\frac{x-X_j}{a_n}\right)\right\} $$ whiro $E_1$ and $K_3$ are probability density functions on $R^p$ and $R^q$ respectively, and both $a_n$ and $l$, are sequences of small positive numbers. Denote $$ \begin{gathered} g_n(x, y)=\left(n a_n^p b_n^q\right)^{-1} \sum_{i=1}^n K_1\left(\frac{x-X_i}{a_n}\right) K_2\left(\frac{y-Y_i}{b_n}\right), \\ h_n(x)=\left(n a_n^p\right)^{-1} \sum_{i=1}^n K_1\left(\frac{x-X_i}{a_n}\right) \end{gathered} $$ the $\left.11 f_{n^{\prime}} \mid x\right)=g_n(x, 3) / h_n(x)$. If $a_n=b_n, g_n(x, y)$ is Rosenblatt estimator of joint densily $g(a,!)$. In the paper we obtain both pointwise and uniform strong consistency of $g_n(x$, $y)$ and $\because(y \mid x)$.
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