AISM 52, 630-645

## Laws of iterated logarithm and related asymptotics for estimators of conditional density and mode

### K.L. Mehra1, Y.S. Ramakrishnaiah2 and P. Sashikala3

1Department of Mathematical Sciences, University of Alberta, Edmonton, Canada T6G 2G1
2Department of Statistics, Osmania University, Hyderabad-500007, India
3Department of Statistics, B.B.C.I.T. Kachiguda, Hyderabad-500027, India

(Received September 29, 1997; revised February 4, 1999)

Abstract.    Let $(X_i,Y_i)$ be a sequence of i.i.d. random vectors in $R^{(2)}$ with an absolutely continuous distribution function $H$ and let $g_x(y)$, $y\in\R^{(1)}$ denote the conditional density of $Y$ given $X=x\in\Lambda(F)$, the support of $F$, assuming that it exists. Also let $M(x)$ be the (unique) conditional mode of $Y$ given $X=x$ defined by $M(x)= \arg\max_y(g_x(y))$. In this paper new classes of smoothed rank nearest neighbor (RNN) estimators of $g_x(y)$, its derivatives and $M(x)$ are proposed and the laws of iterated logarithm (pointwise), uniform a.s. convergence over $-\infty<y<\infty$ and $x\in$ a compact $C\subseteq\Lambda(F)$ and the asymptotic normality for the proposed estimators are established. Our results and proofs also cover the Nadayara-Watson (NW) case. It is shown using the concept of the relative efficiency that the proposed RNN estimator is superior (asymtpotically) to the corresponding NW type estimator of $M(x)$, considered earlier in literature.

Key words and phrases:    Conditional density, conditional mode, smooth rank nearest neighbor estimators, law of iterated logarithm, uniform strong convergence.

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