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Limiting law results for a class of conditional mode estimates for functional stationary ergodic data

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Abstract

The main purpose of the present work is to establish the functional asymptotic normality of a class of kernel conditional mode estimates when functional stationary ergodic data are considered. More precisely, consider a random variable (X,Z) taking values in some semi-metric abstract space E × F. For a real function φ defined on F and for each x ∈ E, we consider the conditional mode, say ⊝φ(x), of the real random variable φ(Z) given the event “X = x”. While estimating the conditional mode function by Θ̂φ,n(x), using the kernel-type estimator, we establish the limiting law of the family of processes {Θ̂φ(x) - Θφ(x)} (suitably normalized) over Vapnik–Chervonenkis class C of functions φ. Beyond ergodicity, no other assumption is imposed on the data. This paper extends the scope of some previous results established under mixing condition for a fixed function φ. From this result, the asymptotic normality of a class of predictors is derived and confidence bands are constructed. Finally, a general notion of bootstrapped conditional mode constructed by exchangeably weighting samples is presented. The usefulness of this result will be illustrated in the construction of confidence bands.

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Author information

Authors and Affiliations

  1. L.M.A.C., Sorbonne Univ., Univ. de Technológie de Compiègne, Compiègne, France

    S. Bouzebda

  2. Dept. Statist., United Arab Emirates Univ., Al Ain, UAE

    M. Chaouch

  3. L.S.T.A., Univ. Paris VI, Paris; EISTI, Cergy-Pontoise, France

    N. Laïb

Authors
  1. S. Bouzebda
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  2. M. Chaouch
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  3. N. Laïb
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Correspondence to S. Bouzebda.

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Bouzebda, S., Chaouch, M. & Laïb, N. Limiting law results for a class of conditional mode estimates for functional stationary ergodic data. Math. Meth. Stat. 25, 168–195 (2016). https://doi.org/10.3103/S1066530716030029

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  • DOI: https://doi.org/10.3103/S1066530716030029

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