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Expansions for the distribution of M-estimates with applications to the Multi-Tone problem

Christopher S. Withers, Saralees Nadarajah (2011)

ESAIM: Probability and Statistics

We give a stochastic expansion for estimates θ ^ that minimise the arithmetic mean of (typically independent) random functions of a known parameterθ. Examples include least squares estimates, maximum likelihood estimates and more generally M-estimates. This is used to obtain leading cumulant coefficients of θ ^ needed for the Edgeworth expansions for the distribution and densityn1/2θ0) to magnitude n−3/2 (or to n−2 for the symmetric case), where θ0 is the true parameter value and n is typically the...

Expansions for the distribution of M-estimates with applications to the Multi-Tone problem

Christopher S. Withers, Saralees Nadarajah (2012)

ESAIM: Probability and Statistics

We give a stochastic expansion for estimates that minimise the arithmetic mean of (typically independent) random functions of a known parameter θ. Examples include least squares estimates, maximum likelihood estimates and more generally M-estimates. This is used to obtain leading cumulant coefficients of needed for the Edgeworth expansions for the distribution and density n1/2 ( of − θ0) to magnitude n−3/2 (or to n−2 for the symmetric case), where θ0 is the true parameter value and n is typically...

Exponential deficiency of convolutions of densities

Iosif Pinelis (2012)

ESAIM: Probability and Statistics

If a probability density p(x) (x ∈ ℝk) is bounded and R(t) := ∫e〈x, tu〉p(x)dx < ∞ for some linear functional u and all t ∈ (0,1), then, for each t ∈ (0,1) and all large enough n, the n-fold convolution of the t-tilted density p ˜ t ˜pt := e〈x, tu〉p(x)/R(t) is bounded. This is a corollary of a general, “non-i.i.d.” result, which is also shown to enjoy a certain optimality property. Such results and their corollaries stated in terms of the absolute integrability of the corresponding characteristic...

Exponential deficiency of convolutions of densities∗

Iosif Pinelis (2012)

ESAIM: Probability and Statistics

If a probability density p(x) (x ∈ ℝk) is bounded and R(t) := ∫e〈x, tu〉p(x)dx < ∞ for some linear functional u and all t ∈ (0,1), then, for each t ∈ (0,1) and all large enough n, the n-fold convolution of the t-tilted density p ˜ t := e〈x, tu〉p(x)/R(t) is bounded. This is a corollary of a general, “non-i.i.d.” result, which is also shown to enjoy a certain optimality property. Such results and their corollaries stated in terms of the absolute integrability of the corresponding characteristic...

Extremal and additive processes generated by Pareto distributed random vectors

Kosto V. Mitov, Saralees Nadarajah (2014)

ESAIM: Probability and Statistics

Pareto distributions are most popular for modeling heavy tailed data. Here, we obtain weak limits of a sequence of extremal and a sequence of additive processes constructed by a series of Bernoulli point processes with bivariate Pareto space components. For the limiting processes we derive the one dimensional distributions in explicit forms. Some of the main properties of these distributions are also proved.

Extreme order statistics in an equally correlated Gaussian array

Mateusz Wiśniewski (1994)

Applicationes Mathematicae

This paper contains the results concerning the weak convergence of d-dimensional extreme order statistics in a Gaussian, equally correlated array. Three types of limit distributions are found and sufficient conditions for the existence of these distributions are given.

Generalized regression estimation for continuous time processes with values in functional spaces

Bertrand Maillot, Christophe Chesneau (2021)

Commentationes Mathematicae Universitatis Carolinae

We consider two continuous time processes; the first one is valued in a semi-metric space, while the second one is real-valued. In some sense, we extend the results of F. Ferraty and P. Vieu in ``Nonparametric models for functional data, with application in regression, time-series prediction and curve discrimination'' (2004), by establishing the convergence, with rates, of the generalized regression function when a real-valued continuous time response is considered. As corollaries, we deduce the...

Goodness of fit tests with weights in the classes based on ( h , φ ) -divergences

Elena Landaburu, Leandro Pardo (2000)

Kybernetika

The aim of the paper is to present a test of goodness of fit with weigths in the classes based on weighted h , φ -divergences. This family of divergences generalizes in some sense the previous weighted divergences studied by Frank et al [frank] and Kapur [kapur]. The weighted h , φ -divergence between an empirical distribution and a fixed distribution is here investigated for large simple random samples, and the asymptotic distributions are shown to be either normal or equal to the distribution of a linear...

Goodness-of-fit tests based on K φ -divergence

Teresa Pérez, Julio A. Pardo (2003)

Kybernetika

In this paper a new family of statistics based on K φ -divergence for testing goodness-of-fit under composite null hypotheses are considered. The asymptotic distribution of this test is obtained when the unspecified parameters are estimated by maximum likelihood as well as minimum K φ -divergence.

Hurwicz's estimator of the autoregressive model with non-normal innovations

Youcef Berkoun, Hocine Fellag (2011)

Applicationes Mathematicae

Using the Bahadur representation of a sample quantile for m-dependent and strong mixing random variables, we establish the asymptotic distribution of the Hurwicz estimator for the coefficient of autoregression in a linear process with innovations belonging to the domain of attraction of an α-stable law (1 < α < 2). The present paper extends Hurwicz's result to the autoregressive model.

Inference about stationary distributions of Markov chains based on divergences with observed frequencies

María Luisa Menéndez, Domingo Morales, Leandro Pardo, Igor Vajda (1999)

Kybernetika

For data generated by stationary Markov chains there are considered estimates of chain parameters minimizing φ –divergences between theoretical and empirical distributions of states. Consistency and asymptotic normality are established and the asymptotic covariance matrices are evaluated. Testing of hypotheses about the stationary distributions based on φ –divergences between the estimated and empirical distributions is considered as well. Asymptotic distributions of φ –divergence test statistics are...

Kolmogorov-Smirnov two-sample test based on regression rank scores

Martin Schindler (2008)

Applications of Mathematics

We derive the two-sample Kolmogorov-Smirnov type test when a nuisance linear regression is present. The test is based on regression rank scores and provides a natural extension of the classical Kolmogorov-Smirnov test. Its asymptotic distributions under the hypothesis and the local alternatives coincide with those of the classical test.

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