Adaptive estimation of the transition density of a Markov chain

Lacour, Claire

doi:10.1016/j.anihpb.2006.09.003

Cited by 20 publications

(40 citation statements)

References 22 publications

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“…The control of the bias term follows from Lemma 5.4 recalled in Brunel et al (2007) and following from results stated in Hochmuth (2002), Lacour (2007) andNilkol'skii (1975).…”

Section: E Brunel F Comte and C Lacourmentioning

confidence: 94%

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Minimax estimation of the conditional cumulative distribution function

2010

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Consider an i.i.d. sample (Xi, Yi), i = 1, . . . , n of observations and denote by F (y|x) the conditional cumulative distribution function of Yi given Xi = x. We provide a data driven nonparametric strategy to estimate F . We prove that, in term of the integrated mean square risk on a compact set, our estimator performs a squared-bias variance compromise. We deduce from this an upper bound for the rate of convergence of the estimator, in a context of anisotropic function classes. A lower bound for this rate is also proved, which implies the optimality of our estimator. Then our procedure can be adapted to positive censored random variables Yi's, i.e. when only Zi = inf(Yi, Ci) and δi = ½ {Y i ≤C i } are observed, for an i.i.d. censoring sequence (Ci) 1≤i≤n independent of (Xi, Yi) 1≤i≤n . Simulation experiments and a real data example illustrate the method.

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“…The control of the bias term follows from Lemma 5.4 recalled in Brunel et al (2007) and following from results stated in Hochmuth (2002), Lacour (2007) andNilkol'skii (1975).…”

Section: E Brunel F Comte and C Lacourmentioning

confidence: 94%

“…It is proved in Section 7.5 of Lacour (2007) that, if 2 j 1 +j 2 = o(n), P πε (Λ n (ε * K , ε) > e −λ ) ≥ p 0 (the context is slightly different but it is sufficient to replace X i+1 by Y i and to bound f by f ∞ ).…”

Section: E Brunel F Comte and C Lacourmentioning

confidence: 98%

Minimax estimation of the conditional cumulative distribution function

2010

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“…A definition of the -mixing coefficients (in general and in the case of a Markov chain) can be found in Doukhan [17]. A lot of Markov chains satisfy Assumptions A4, see examples in Section 2.2 in Lacour [23].…”

Section: Notations and Assumptionsmentioning

confidence: 99%

Adaptive estimation of the transition density of a particular hidden Markov chain

Lacour

2008

Journal of Multivariate Analysis

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We study the following model of hidden Markov chain: $Y_i=X_i+\varepsilon_i$, $ i=1,\dots,n+1$ with $(X_i)$ a real-valued positive recurrent and stationary Markov chain and $(\varepsilon_i)_{1\leq i\leq n+1}$ a noise independent of the sequence $(X_i)$ having a known distribution. We present an adaptive estimator of the transition density based on the quotient of a deconvolution estimator of the density of $X_i$ and an estimator of the density of $(X_i,X_{i+1})$. These estimators are obtained by contrast minimization and model selection. We evaluate the $L2$ risk and its rate of convergence for ordinary smooth and supersmooth noise with regard to ordinary smooth and supersmooth chains. Some examples are also detailed

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“…To that end, we prove new deviation inequalities for bifurcating Markov chains that we develop independently in a more general setting, when S is not necessarily restricted to R. Note also that when P 0 = P 1 , we have Q = P 0 = P 1 as well and we retrieve the usual framework of nonparametric estimation of Markov chains when the observation is based on (Y i ) 1≤i≤n solely. We are therefore in the line of combining and generalising the study of Clémençon [15] and Lacour [33,34] that both consider adaptive estimation for Markov chains when S ⊆ R.…”

Section: 2mentioning

confidence: 99%

“…As for the case d = 2 and d = 3, the structure of BMC comes into play and we need to prove a specific optimality result, stated in Theorems 9 and 10. We rely on classical lower bound techniques for density estimation and Markov chains (Hoffmann [31], Clémençon [15], Lacour [33,34]). …”

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confidence: 99%

Adaptive estimation for bifurcating Markov chains

Penda¹,

Hoffmann²,

Olivier³

2017

Bernoulli

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Abstract. In a first part, we prove Bernstein-type deviation inequalities for bifurcating Markov chains (BMC) under a geometric ergodicity assumption, completing former results of Guyon and Bitseki Penda, Djellout and Guillin. These preliminary results are the key ingredient to implement nonparametric wavelet thresholding estimation procedures: in a second part, we construct nonparametric estimators of the transition density of a BMC, of its mean transition density and of the corresponding invariant density, and show smoothness adaptation over various multivariate Besov classes under L p -loss error, for 1 ≤ p < ∞. We prove that our estimators are (nearly) optimal in a minimax sense. As an application, we obtain new results for the estimation of the splitting size-dependent rate of growth-fragmentation models and we extend the statistical study of bifurcating autoregressive processes.Mathematics Subject Classification (2010): 62G05, 62M05, 60J80, 60J20, 92D25,

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Adaptive estimation of the transition density of a Markov chain

Cited by 20 publications

References 22 publications

Minimax estimation of the conditional cumulative distribution function

Minimax estimation of the conditional cumulative distribution function

Adaptive estimation of the transition density of a particular hidden Markov chain

Adaptive estimation for bifurcating Markov chains

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