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

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 (F), the support of F, assuming that it exists. Also let M(x) be the (unique) conditional mode of Y given X = x defined...

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Bibliographic Details
Published in:Annals of the Institute of Statistical Mathematics 2000-12, Vol.52 (4), p.630-645
Main Authors: MEHRA, K. L, RAMAKRISHNAIAH, Y. S, SASHIKALA, P
Format: Article
Language:English
Subjects:
Algorithms
Estimating techniques
Exact sciences and technology
Mathematics
Nonparametric inference
Parametric inference
Probability and statistics
Sciences and techniques of general use
Statistical methods
Statistics
Studies
Theory
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Summary: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 (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 (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. [PUBLICATION ABSTRACT]
ISSN:0020-3157
1572-9052
DOI:10.1023/A:1017517124707