Abstract
Let (X i , Y i ) be a sequence of i.i.d. random vectors in R with an absolutely continuous distribution function H and let g x (y), y∈ R 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 (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.
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Mehra, K.L., Ramakrishnaiah, Y.S. & Sashikala, P. Laws of Iterated Logarithm and Related Asymptotics for Estimators of Conditional Density and Mode. Annals of the Institute of Statistical Mathematics 52, 630–645 (2000). https://doi.org/10.1023/A:1017517124707
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DOI: https://doi.org/10.1023/A:1017517124707