LAWS OF ITERATED LOGARITHM AND RELATED ASYMPTOTICS FOR ESTIMATORS OF CONDITIONAL DENSITY AND MODE

K. L. MEHRA^{1}, Y. S. RAMAKRISHNAIAH^{2} AND P. SASHIKALA^{3}

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 ∈ R^{(1)} denote the conditional density of Y given X = x ∈ Λ(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 −∞ < y < ∞ and x ∈ a compact C ⊆ Λ(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.