Instructional Uses of Approximate Convolutions and Their Graphs

A technique is presented for approximating convolution densities f * g by forming the convolution of step-function approximations of f and g. This technique, amenable to the electronic calculator or computer, provides suitably accurate graphs for illustrating concepts in probability and statistics. Several examples are included.     

Gaussian deconvolution via differentiation

Suppose that ? is a Gaussian density and that g = f * ?, where * denotes convolution. From observations with density g, one wishes to estimate f. We analyze an estimate which is a linear combination of estimates of derivatives of g and show that this estimate converges in an L₂ norm at a rate which is compatible with the pointwise optimal rate established by Fan (1991).     

Root n consistent and optimal density estimators for moving average processes

Abstract.  The marginal density of a first order moving average process can be written as a convolution of two innovation densities. Saavedra & Cao [Can. J. Statist. (2000), 28, 799] propose to estimate the marginal density by plugging in kernel density estimators for the innovation densities, based on estimated innovations. They obtain that for an appropriate choice of bandwidth the variance of their estimator decreases at the rate 1/ n . Their estimator can be interpreted as a specific U -statistic. We suggest a slightly simplified U -statistic as estimator of the marginal density, prove that it is asymptotically normal at the same rate, and describe the asymptotic variance explicitly. We show that the estimator is asymptotically efficient if no structural assumptions are made on the innovation density. For innovation densities known to have mean zero or to be symmetric, we describe improvements of our estimator which are again asymptotically efficient.     

Asymptotic Expansions for i.i.d. Sums Via Lower-order Convolutions

In this article, we introduce new asymptotic expansions for probability functions of sums of independent and identically distributed random variables. Results are obtained by efficiently employing information provided by lower-order convolutions. In comparison with Edgeworth-type theorems, advantages include improved asymptotic results in the case of symmetric random variables and ease of computation of main error terms and asymptotic crossing points. The first-order estimate can perform quite well against the corresponding renormalized saddlepoint approximation and, pointwise, requires evaluation of only a single convolution integral. While the new expansions are fairly straightforward, the implications are fortuitous and may spur further related work.     

Relationships among moments of order statistics in samples from two related outlier models and some applications

Balakrishnan (1987a) has recently shown that the moments of order statistics in samples drawn from a continuous population with pdf f(x) symmetric about zero comprising a single outlier with pdf g(x) also symmetric about zero can be expressed in terms of the moments of order statistics in samples drawn from the population obtained by folding the pdf f(x) at zero and the moments of order statistics in samples drawn from the population obtained by folding the pdf f(x) at zero comprising a single outlier with pdf obtained by folding g(x) at zero. The cumulative round off error involved in the numerical evaluation of the moments of order statistics from the symmetric outlier model, using a table of the moments of order statistics from the folded population and the moments of order statistics from the folded outlier model, has also been studied by Balakrishnan (1987a) and shown to be not serious. Making use of these results we study here the robustness of some estimators of th location and scale parameters of a double exponential distribution.     

Improved Estimators of the Population Mean for Missing Data

For two dependent random variables X and Y with distributions of convolution equivalence, sufficient conditions are given under which the distribution of the minimum min (X, Y) is still of convolution equivalence. We further extend the result to the multivariate case.     

A note on efficient density estimators of convolutions

It is already known that the convolution of a bounded density with itself can be estimated at the root-n rate using the two asymptotically equivalent kernel estimators: (i) Frees estimator ( Frees, 1994) and (ii) Saavedra and Cao estimator ( Saavedra and Cao, 2000). In this work, we investigate the efficiency of these estimators of the convolution of a bounded density. The efficiency criterion used in this work is that of a least dispersed regular estimator described in Begun et al. (1983). This concept is based on the Hájek–Le Cam convolution theorem for locally asymptotically normal (LAN) families.     

Tests of fit for a lognormal distribution

The problem of goodness of fit of a lognormal distribution is usually reduced to testing goodness of fit of the logarithmic data to a normal distribution. In this paper, new goodness-of-fit tests for a lognormal distribution are proposed. The new procedures make use of a characterization property of the lognormal distribution which states that the Kullback–Leibler measure of divergence between a probability density function (p.d.f) and its r-size weighted p.d.f is symmetric only for the lognormal distribution [Tzavelas G, Economou P. Characterization properties of the log-normal distribution obtained with the help of divergence measures. Stat Probab Lett. 2012;82(10):1837–1840]. A simulation study examines the performance of the new procedures in comparison with existing goodness-of-fit tests for the lognormal distribution. Finally, two well-known data sets are used to illustrate the methods developed.     

On multimodal distributions

As a sequel to khinchine's definition of unimodality a multimodal distribution function is defined. A characterization for such a distribution is given. The convolution of two such distributions is studied.     

GOODNESS OF FIT STATISTICS BASED ON THE SPACINGS OF COMPLETE OR CENSORED SAMPLES

A goodness-of-fit statistic Z is defined in terms of the spacings generated by the order statistics of a complete or a censored sample from a distribution of the type (l/)f((x-μ)/), μ and unknown. The distribution of Z is studied, mostly through Monte Carlo methods. The power properties of Z for testing Exponential, Uniform, Normal, Gamma and Logistic distributions are discussed; Z is shown to be more powerful than the Smith & Bain (1976) correlation statistic, except for testing Uniform, Normal and Logistic (symmetric distributions) against symmetric alternatives. The statistic Z is generalized to test the goodness-of-fit from κ 2 independent complete or censored samples.     

Robust statistics for testing mean vectors of multivariate distributions

We develop a ‘robust’ statistic T² _R, based on Tiku's (1967, 1980) MML (modified maximum likelihood) estimators of location and scale parameters, for testing an assumed meam vector of a symmetric multivariate distribution. We show that T² _R is one the whole considerably more powerful than the prominenet Hotelling T² statistics. We also develop a robust statistic T² _D for testing that two multivariate distributions (skew or symmetric) are identical; T² _D seems to be usually more powerful than nonparametric statistics. The only assumption we make is that the marginal distributions are of the type (1/σ_k)f((x-μ_k)/σ_k) and the means and variances of these marginal distributions exist.     

Efficient estimation in a nonlinear counting-process regression model

Suppose we observe i.i.d. copies of X, C, Y, where X is a counting process, C is a censoring process talcing only values 0 and 1, and Y is a covariate process. Assume that the intensity process of X is of the form C(s)a(s, Y(s)) with a unknown, but that the distribution of X, C, Y is unspecified otherwise. McKeague and Utikal proposed an estimator for the doubly cumulative hazard f f a(s, y) ds dy and determined its asymptotic distribution. We show that the estimator is regular and efficient in the sense of a Hájek-Inagaki convolution theorem for partially specified models.     

Convolution of fuzzy distributions in decision-making

This paper deals with a decision-making problem that has hardly been addressed so far. Frequently, management is challenged to deal with sums of random variables, the distribution of which is not fully known. For example, a high return on an investment project may be “more likely” than a low one in the first period but “less likely” in the second period; however, the investment decision has to be based on the sum of the two retums. The paper contains some essential theorems concerning the convolution of distribution functions for the case of fuzzy random variables. Moreover, an example is given illustrating the incorporation of new Linear Partial Information (LPI) and the transition from an a priori to a posteriori convolution of fuzzy distribution functions.     

The non-null distribution of the beta statistic in the test of the univariate general linear hypothesis,when the error distribution is spherically symmetric

We obtain the Mellin transform of the Beta statistic used in the test of the univariate general linear hypothesis, assuming that the error distribution is spherically symmetric. From this, the non-null distribution of the statistic is obtained. The normal-errors representation of the Beta as a central Beta with random d.f. is shown to hold iff the error distribution is a normal scale mixture. Closed form expressions for the density are given, without employing this assumption.     

A Remedy for Kernel Estimation Under Random Design

Two common kernel-based methods for non-parametric regression estimation suffer from well-known drawbacks when the design is random. The Gasser-Müller estimator is inadmissible due to its high variance while the Nadaraya-Watson estimator has zero asymptotic efficiency because of poor bias behavior. Under asymptotic consideration, the local linear estimator avoids these two drawbacks of kernel estimators and achieves minimax optimality. However, when based on compact support kernels its finite sample behavior is disappointing because sudden kinks may show up in the estimate.

This paper proposes a modification of the kernel estimator, called the binned convolution estimator leading to a fast O(n) method. Provided the design density is continously differentiable and the conditional fourth moments exist the binned convolution estimator has asymptotic properties identical with those of the local linear estimator.     

Bayesian Non-Parametric Estimation of Smooth Hazard Rates for Seismic Hazard Assessment

Abstract.  Hazard rate estimation is an alternative to density estimation for positive variables that is of interest when variables are times to event. In particular, it is here shown that hazard rate estimation is useful for seismic hazard assessment. This paper suggests a simple, but flexible, Bayesian method for non-parametric hazard rate estimation, based on building the prior hazard rate as the convolution mixture of a Gaussian kernel with an exponential jump-size compound Poisson process. Conditions are given for a compound Poisson process prior to be well-defined and to select smooth hazard rates, an elicitation procedure is devised to assign a constant prior expected hazard rate while controlling prior variability, and a Markov chain Monte Carlo approximation of the posterior distribution is obtained. Finally, the suggested method is validated in a simulation study, and some Italian seismic event data are analysed.     

The distribution by age of the frequency of first marriage in a female cohort

The authors present 2 methods for the approximation of a representative schedule recording first marriage frequencies by age. Both treatments are mathematically complex. One method achieves a very close approximation with a simple closed form frequency function, which is the limiting distribution of the convolution of an infinite number of exponentially distributed components. The other method achieves an equal approximation by the convolution of a normal distribution of age of entry into a marriageable state and as few as 3 exponentially distributed delays. This latter convolution provides a feasible model of nuptiality, a model receiving surprising empirical support.     

Some remarks on numerical convolution

McConalogue (1980) describes an extension t o the Clérour-McConalogue cubic splining algorithm which permits the convolution of distributions whose densities exhibit a singularity at the origin. Reasons for generalising the original algorithm are discussed and a brief survey of other methods for numerical convolution is given. An example illustrates an application of the extended algorithm.     

A Note on Robustness of Normal Variance Estimators Under t-Distributions

ABSTRACT

In many real life problems one assumes a normal model because the sample histogram looks unimodal, symmetric, and/or the standard tests like the Shapiro-Wilk test favor such a model. However, in reality, the assumption of normality may be misplaced since the normality tests often fail to detect departure from normality (especially for small sample sizes) when the data actually comes from slightly heavier tail symmetric unimodal distributions. For this reason it is important to see how the existing normal variance estimators perform when the actual distribution is a t-distribution with k degrees of freedom (d.f.) (t _k -distribution). This note deals with the performance of standard normal variance estimators under the t _k -distributions. It is shown that the relative ordering of the estimators is preserved for both the quadratic loss as well as the entropy loss irrespective of the d.f. and the sample size (provided the risks exist).     

Some Utilizations of Lambert W Function in Distribution Theory

We introduce a new class of positive infinitely divisible probability laws calling them 𝔏_γ distributions. Their cumulant-generating functions (cgf) are expressed in terms of the principal branch of the Lambert W function. The probability density functions (pdfs) of 𝔏_γ laws are bounded resembling pdf of a Lévy stable distribution. The exponential dispersion model constructed starting from an 𝔏_γ distribution admits the inverse Gaussian approximation. The natural exponential family constructed starting from an 𝔏_γ distribution constitutes the reciprocal of the natural exponential family generated by a spectrally negative stable law with α = 1. We derive new results on 𝔏_γ laws and the related exponential dispersion models, including their convolution and scaling closure properties. We generate another exponential dispersion model starting from an exponentially compounded 𝔏_γ law. This distribution emerges in the Poisson mixture representation of a generalized Poisson law. We extend the Poisson approximation for the scaled Neyman type A exponential dispersion model. We derive saddlepoint-type approximations for some of these exponential dispersion models. The role of the Lambert W function is emphasized.