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Kernel estimation for stationary density of Markov chains with general state space
Abstract
Let {X
n
}
n
≥0 be a Markov chain with stationary distributionf(x)ν(dx), ν being a σ-finite measure onE⊂R
d
. Under strict stationarity and mixing conditions we obtain the consistency and asymptotic normality for a general class of
kernel estimates off(). When the assumption of stationarity is dropped these results are extended to geometrically ergodic chains.
... Finally, we derive the strong consistency of the drift estimator. The proof is based following some ideas in Campos and Dorea (2005). In order to do that let us introduce some notation. ...
... By Proposition 4.1 in Campos and Dorea (2005), the sequence {X i } is ϕ mixing with ϕ(n) = 2γρ n (being γ as in (6)). Let x ∈ S and x i be such that x − x i < h d+2 . ...
We consider the estimation of the drift and the level sets of the stationary distri- bution of a Brownian motion with drift, reflected in the boundary of a compact set $S\subsetR^d$ , departing from the observation of a trajectory of this process. We obtain the uniform consistency and rates of convergence for the proposed kernel based estimators. This problem has relevant applications in ecology, in estimating the home-range and the core-area of an animal based on tracking data. Recently, the problem of estimat- ing the domain of a reflected Brownian motion was considered in Cholaquidis, et al. (2016), when the stationary distribution is uniform and the estimation of the core-area, defined as a level set of the stationary distribution, is meaningless. As a by-product of our results, we obtain an estimation of the drift function. In order to prove our re- sults, some new theoretical properties of the reflected Brownian motion with drift are obtained, under fairly general assumptions. These properties allow us to perform the estimation for flexible regions close to reality. The theoretical findings are illustrated on simulated and real data examples.
... Over the years, kernel type estimators for Markov processes have received significant attention. See, for example, Athreya and Atuncar (1998), Hili (2001), Campos and Dorea (2005) and Lacour (2008). ...
La serie de Documentos de Investigación del Banco de México divulga resultados preliminares de trabajos de investigación económica realizados en el Banco de México con la finalidad de propiciar el intercambio y debate de ideas. El contenido de los Documentos de Investigación, así como las conclusiones que de ellos se derivan, son responsabilidad exclusiva de los autores y no reflejan necesariamente las del Banco de México. The Working Papers series of Banco de México disseminates preliminary results of economic research conducted at Banco de México in order to promote the exchange and debate of ideas. The views and conclusions presented in the Working Papers are exclusively of the authors and do not necessarily reflect those of Banco de México.
Bayesian analysis often requires the researcher to employ Markov Chain Monte Carlo (MCMC) techniques to draw samples from a posterior distribution which in turn is used to make inferences. Currently, several approaches to determine convergence of the chain as well as sensitivities of the resulting inferences have been developed. This work develops a Hellinger distance approach to MCMC diagnostics. An approximation to the Hellinger distance between two distributions f and g based on sampling is introduced. This approximation is studied via simulation to determine the accuracy. A criterion for using this Hellinger distance for determining chain convergence is proposed as well as a criterion for sensitivity studies. These criteria are illustrated using a dataset concerning the Anguilla australis, an eel native to New Zealand.
For a homogeneous and uniformly ergodic Markov chain, with transition kernel
$$P(x, A) = \int_{A} f(y|x)\hbox{d}y, x \in E \subset R^{d}$$, we analyse some reliability measures and failure rates associated with the transition probabilities. Sufficient conditions
for strong consistency are obtained for estimates based on kernel density estimators.
In this work, sequences of random variables are considered satisfying certain mixing conditions. After the relevant definitions are presented, some alternative characterizations are discussed. Also, illustrative examples are given for each case considered. Finally , various moment in equalities are extensively discussed in a systematic manner. These equalities are interesting on their own right and also useful in statistical applications. Certain such applications will be presented in a separate report to avoid overloading the present one
Let X1,…,Xn+1 be the first n+1 random variables from a strictly stationary Markov process which satisfies certain additional regularity conditions. On the basis of these random variables, a nonparametric estimate of the one-step transition distribution function is shown to be uniformly (in the main argument) strongly consistent and asymptotically normal. Furthermore, for any p∈(0, 1), a natural nonparametric estimate of the p-th quantile of the distribution function just mentioned is proven to be strongly consistent. The class of Markov processes studied includes the Markov processes usually considered in the literature; namely, processes which either satisfy Doeblin’s hypothesis, or, more generally, are geometrically ergodic.
Let p(·) be a density with respect to a σ-finite measure ν on , where . In this note, we propose a general class of kernel estimates for p(·). It is shown that our results on strong consistency and asymptotic normality include the classical results for continuous densities on and extend some results of kernel estimators for discrete distributions.
The purpose of this paper is to study the problem of estimation of the stationary density and the transition density of a real-valued recurrent Markov chain. By using techniques of regenerative processes we are able to significantly reduce the strong hypotheses on the Markov chain such as Doeblin recurrence, stationarity, and mixing that were imposed in all the earlier works. We assume that the Markov chain satisfies a much weaker condition known as Harris recurrence. Our results hold for any initial distribution and we assume no mixing.
We derive exponential inequalities for the oscillation of functions of random variables about their mean. This is illustrated on the Kolmogorov-Smirnov statistic, the total variation distance for empirical measures, the Vapnik-Chervonenkis distance, and various performance criteria in nonparametric density estimation. We also derive bounds for the variances of these quantities. In this paper, we collect various extensions of Hoeding's inequality and highlight their applica- tions in the nonparametric estimation of densities and distribution functions. For completeness, the proofs of the inequalities are sketched as well. In the last section, we present new bounds for the variance of functions of independent random variables. The inequalities are illustrated on examples in nonparametric estimation, and are shown to be sharper than those obtained from the Efron-Stein inequality.
Let { X n} be a Markov chain that is either f -mixing or satisfies the Poisson equation.In this note we obtain the convergence rate under L 1 -criterion for bounded functions of the X k 's. And in the hidden Markov model setup { (X n ,Y n ) }we study the kernel estimate of the density of the observed variables { Y n }when a 'stable' status is reached.
[eng] Transportation costs and monopoly location in presence of regional disparities. . This article aims at analysing the impact of the level of transportation costs on the location choice of a monopolist. We consider two asymmetric regions. The heterogeneity of space lies in both regional incomes and population sizes: the first region is endowed with wide income spreads allocated among few consumers whereas the second one is highly populated however not as wealthy. Among the results, we show that a low transportation costs induces the firm to exploit size effects through locating in the most populated region. Moreover, a small transport cost decrease may induce a net welfare loss, thus allowing for regional development policies which do not rely on inter-regional transportation infrastructures. cost decrease may induce a net welfare loss, thus allowing for regional development policies which do not rely on inter-regional transportation infrastructures. [fre] Cet article d�veloppe une statique comparative de l'impact de diff�rents sc�narios d'investissement (projet d'infrastructure conduisant � une baisse mod�r�e ou � une forte baisse du co�t de transport inter-r�gional) sur le choix de localisation d'une entreprise en situation de monopole, au sein d'un espace int�gr� compos� de deux r�gions aux populations et revenus h�t�rog�nes. La premi�re r�gion, faiblement peupl�e, pr�sente de fortes disparit�s de revenus, tandis que la seconde, plus homog�ne en termes de revenu, repr�sente un march� potentiel plus �tendu. On montre que l'h�t�rog�n�it� des revenus constitue la force dominante du mod�le lorsque le sc�nario d'investissement privil�gi� par les politiques publiques conduit � des gains substantiels du point de vue du co�t de transport entre les deux r�gions. L'effet de richesse, lorsqu'il est associ� � une forte disparit� des revenus, n'incite pas l'entreprise � exploiter son pouvoir de march� au d�triment de la r�gion l
Estimation of transition distribution function and its quantiles in Markov processes: Strong consistency and asymptotic normality, Nomparametrie Functional Estimation and Related Topics Moment inequalities for mixing sequences of random variables
- G C Roussas
Roussas, G. C. (1991). Estimation of transition distribution function and its quantiles in Markov processes: Strong consistency and asymptotic normality, Nomparametrie Functional Estimation and Related Topics (ed. G. G. Roussas), 443-462, Kluwer Academic Publishers, Dordrecht. Roussas, G. G. and Ioannides, D. (1987). Moment inequalities for mixing sequences of random variables, Stochastic Analysis and Applications, 5(1), 61-120.
Asymptotic analysis of kernel type estimators for densities associated with general Markov chains
- V S M Campos
- V. S. M. Campos
Density estimates and Markov sequences
- M Rosenblatt
- M. Rosenblatt