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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...

Mixture decompositions of exponential families using a decomposition of their sample spaces

Guido F. Montúfar (2013)

Kybernetika

We study the problem of finding the smallest m such that every element of an exponential family can be written as a mixture of m elements of another exponential family. We propose an approach based on coverings and packings of the face lattice of the corresponding convex support polytopes and results from coding theory. We show that m = q N - 1 is the smallest number for which any distribution of N q ...

Nuevos modelos de distribuciones de extremos basados en aproximaciones en las ramas.

Enrique Castillo, Eladio Moreno, Jaime Puig-Pey (1983)

Trabajos de Estadística e Investigación Operativa

En este trabajo se presenta una metodología que permite clasificar funciones de distribución absolutamente continuas unidimensionales atendiendo a sus ramas. La idea básica es que, en las ramas la función de distribución difiere en un infinitésimo del valor uno o cero dependiendo de la rama de interés. La principal ventaja de esta clasificación es su aplicación a la teoría de distribuciones de extremos. En esta línea se obtienen nuevas familias de distribuciones de extremos. Entre ellas, las clásicas...

On the compound Poisson-gamma distribution

Christopher Withers, Saralees Nadarajah (2011)

Kybernetika

The compound Poisson-gamma variable is the sum of a random sample from a gamma distribution with sample size an independent Poisson random variable. It has received wide ranging applications. In this note, we give an account of its mathematical properties including estimation procedures by the methods of moments and maximum likelihood. Most of the properties given are hitherto unknown.

On two fragmentation schemes with algebraic splitting probability

M. Ghorbel, T. Huillet (2006)

Applicationes Mathematicae

Consider the following inhomogeneous fragmentation model: suppose an initial particle with mass x₀ ∈ (0,1) undergoes splitting into b > 1 fragments of random sizes with some size-dependent probability p(x₀). With probability 1-p(x₀), this particle is left unchanged forever. Iterate the splitting procedure on each sub-fragment if any, independently. Two cases are considered: the stable and unstable case with p ( x ₀ ) = x ₀ a and p ( x ₀ ) = 1 - x ₀ a respectively, for some a > 0. In the first (resp. second) case, since smaller...

Optimization of the size of nonsensitiveness regions

Eva Lešanská (2002)

Applications of Mathematics

The problem is to determine the optimum size of nonsensitiveness regions for the level of statistical tests. This is closely connected with the problem of the distribution of quadratic forms.

Parameter estimation of S-distributions with alternating regression.

I-Chun Chou, Harald Martens, Eberhard O. Voit (2007)

SORT

We propose a novel 3-way alternating regression (3-AR) method as an effective strategy for the estimation of parameter values in S-distributions from frequency data. The 3-AR algorithm is very fast and performs well for error-free distributions and artificial noisy data obtained as random samples generated from S-distributions, as well as for traditional statistical distributions and for actual observation data. In rare cases where the algorithm does not immediately converge, its enormous speed...

Poisson perturbations

Andrew D. Barbour, Aihua Xia (2010)

ESAIM: Probability and Statistics

Stein's method is used to prove approximations in total variation to the distributions of integer valued random variables by (possibly signed) compound Poisson measures. For sums of independent random variables, the results obtained are very explicit, and improve upon earlier work of Kruopis (1983) and Čekanavičius (1997); coupling methods are used to derive concrete expressions for the error bounds. An example is given to illustrate the potential for application to sums of dependent random variables. ...

Polynomial expansions of density of power mixtures

Denys Pommeret (2007)

ESAIM: Probability and Statistics

For any given random variable Y with infinitely divisible distribution in a quadratic natural exponential family we obtain a polynomial expansion of the power mixture density of Y. We approach the problem generally, and then consider certain distributions in greater detail. Various applications are indicated and the results are also applied to obtain approximations and their error bounds. Estimation of density and goodness-of-fit test are derived.

Randomized goodness of fit tests

Friedrich Liese, Bing Liu (2011)

Kybernetika

Classical goodness of fit tests are no longer asymptotically distributional free if parameters are estimated. For a parametric model and the maximum likelihood estimator the empirical processes with estimated parameters is asymptotically transformed into a time transformed Brownian bridge by adding an independent Gaussian process that is suitably constructed. This randomization makes the classical tests distributional free. The power under local alternatives is investigated. Computer simulations...

Second order asymptotic distribution of the R φ -divergence goodness-of-fit statistics

María Del Carmen Pardo (2000)

Kybernetika

The distribution of each member of the family of statistics based on the R φ -divergence for testing goodness-of-fit is a chi-squared to o ( 1 ) (Pardo [pard96]). In this paper a closer approximation to the exact distribution is obtained by extracting the φ -dependent second order component from the o ( 1 ) term.

Some discrete exponential dispersion models: Poisson-Tweedie and Hinde-Demétrio classes.

Célestin C. Kokonendji, Simplice Dossou-Gbété, Clarice G. B. Demétrio (2004)

SORT

In this paper we investigate two classes of exponential dispersion models (EDMs) for overdispersed count data with respect to the Poisson distribution. The first is a class of Poisson mixture with positive Tweedie mixing distributions. As an approximation (in terms of unit variance function) of the first, the second is a new class of EDMs characterized by their unit variance functions of the form μ + μp, where p is a real index related to a precise model. These two classes provide some alternatives...

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