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Estimation of the hazard rate function with a reduction of bias and variance at the boundary

Bożena Janiszewska, Roman Różański (2005)

Discussiones Mathematicae Probability and Statistics

In the article, we propose a new estimator of the hazard rate function in the framework of the multiplicative point process intensity model. The technique combines the reflection method and the method of transformation. The new method eliminates the boundary effect for suitably selected transformations reducing the bias at the boundary and keeping the asymptotics of the variance. The transformation depends on a pre-estimate of the logarithmic derivative of the hazard function at the boundary.

Estimation variances for parameterized marked Poisson processes and for parameterized Poisson segment processes

Tomáš Mrkvička (2004)

Commentationes Mathematicae Universitatis Carolinae

A complete and sufficient statistic is found for stationary marked Poisson processes with a parametric distribution of marks. Then this statistic is used to derive the uniformly best unbiased estimator for the length density of a Poisson or Cox segment process with a parametric primary grain distribution. It is the number of segments with reference point within the sampling window divided by the window volume and multiplied by the uniformly best unbiased estimator of the mean segment length.

Estimators of the asymptotic variance of stationary point processes - a comparison

Michaela Prokešová (2011)

Kybernetika

We investigate estimators of the asymptotic variance σ 2 of a d –dimensional stationary point process Ψ which can be observed in convex and compact sampling window W n = n W . Asymptotic variance of Ψ is defined by the asymptotic relation V a r ( Ψ ( W n ) ) ∼ σ 2 | W n | (as n → ∞ ) and its existence is guaranteed whenever the corresponding reduced covariance measure γ red ( 2 ) ( · ) has finite total variation. The three estimators discussed in the paper are the kernel estimator, the estimator based on the second order intesity of the point process and the...

Extremal behaviour of stationary processes: the calibration technique in the extremal index estimation

D. Prata Gomes, Maria Manuela Neves (2010)

Discussiones Mathematicae Probability and Statistics

Classical extreme value methods were derived when the underlying process is assumed to be a sequence of independent random variables. However when observations are taken along the time and/or the space the independence is an unrealistic assumption. A parameter that arises in this situation, characterizing the degree of local dependence in the extremes of a stationary series, is the extremal index, θ. In several areas such as hydrology, telecommunications, finance and environment, for example, the...

Hazard rate model and statistical analysis of a compound point process

Petr Volf (2005)

Kybernetika

A stochastic process cumulating random increments at random moments is studied. We model it as a two-dimensional random point process and study advantages of such an approach. First, a rather general model allowing for the dependence of both components mutually as well as on covariates is formulated, then the case where the increments depend on time is analyzed with the aid of the multiplicative hazard regression model. Special attention is devoted to the problem of prediction of process behaviour....

Irregular sampling and central limit theorems for power variations : the continuous case

Takaki Hayashi, Jean Jacod, Nakahiro Yoshida (2011)

Annales de l'I.H.P. Probabilités et statistiques

In the context of high frequency data, one often has to deal with observations occurring at irregularly spaced times, at transaction times for example in finance. Here we examine how the estimation of the squared or other powers of the volatility is affected by irregularly spaced data. The emphasis is on the kind of assumptions on the sampling scheme which allow to provide consistent estimators, together with an associated central limit theorem, and especially when the sampling scheme depends on...

LAMN property for hidden processes : the case of integrated diffusions

Arnaud Gloter, Emmanuel Gobet (2008)

Annales de l'I.H.P. Probabilités et statistiques

In this paper we prove the Local Asymptotic Mixed Normality (LAMN) property for the statistical model given by the observation of local means of a diffusion process X. Our data are given by ∫01X(s+i)/n dμ(s) for i=0, …, n−1 and the unknown parameter appears in the diffusion coefficient of the process X only. Although the data are neither markovian nor gaussian we can write down, with help of Malliavin calculus, an explicit expression for the log-likelihood of the model, and then study the asymptotic...

Local Asymptotic Normality Property for Lacunar Wavelet Series multifractal model

Jean-Michel Loubes, Davy Paindaveine (2011)

ESAIM: Probability and Statistics

We consider a lacunar wavelet series function observed with an additive Brownian motion. Such functions are statistically characterized by two parameters. The first parameter governs the lacunarity of the wavelet coefficients while the second one governs its intensity. In this paper, we establish the local and asymptotic normality (LAN) of the model, with respect to this couple of parameters. This enables to prove the optimality of an estimator for the lacunarity parameter, and to build optimal...

Local Asymptotic Normality Property for Lacunar Wavelet Series multifractal model*

Jean-Michel Loubes, Davy Paindaveine (2012)

ESAIM: Probability and Statistics

We consider a lacunar wavelet series function observed with an additive Brownian motion. Such functions are statistically characterized by two parameters. The first parameter governs the lacunarity of the wavelet coefficients while the second one governs its intensity. In this paper, we establish the local and asymptotic normality (LAN) of the model, with respect to this couple of parameters. This enables to prove the optimality of an estimator for the lacunarity parameter, and to build optimal...

Local estimation of the Hurst index of multifractional brownian motion by increment ratio statistic method

Pierre Raphaël Bertrand, Mehdi Fhima, Arnaud Guillin (2013)

ESAIM: Probability and Statistics

We investigate here the central limit theorem of the increment ratio statistic of a multifractional Brownian motion, leading to a CLT for the time varying Hurst index. The proofs are quite simple relying on Breuer–Major theorems and an original freezing of time strategy. A simulation study shows the goodness of fit of this estimator.

Local superefficiency of data-driven projection density estimators in continuous time.

Denis Bosq, Delphine Blanke (2004)

SORT

We construct a data-driven projection density estimator for continuous time processes. This estimator reaches superoptimal rates over a class F0 of densities that is dense in the family of all possible densities, and a «reasonable» rate elsewhere. The class F0 may be chosen previously by the analyst. Results apply to Rd-valued processes and to N-valued processes. In the particular case where square-integrable local time does exist, it is shown that our estimator is strictly better than the local...

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