Asymptotic normality of estimators for parameters of a multivariate skew-normal distribution

In this paper, asymptotic normality is established for the parameters of the multivariate skew-normal distribution under two parametrizations. Also, an analytic expression and an asymptotic normal law are derived for the skewness vector of the skew-normal distribution. The estimates are derived using the method of moments. Convergence to the asymptotic distributions is examined both computationally and in a simulation experiment.     

On the skew-normal calibration model

In this article, we present the EM-algorithm for performing maximum likelihood estimation of an asymmetric linear calibration model with the assumption of skew-normally distributed error. A simulation study is conducted for evaluating the performance of the calibration estimator with interpolation and extrapolation situations. As one application in a real data set, we fitted the model studied in a dimensional measurement method used for calculating the testicular volume through a caliper and its calibration by using ultrasonography as the standard method. By applying this methodology, we do not need to transform the variables to have symmetrical errors. Another interesting aspect of the approach is that the developed transformation to make the information matrix nonsingular, when the skewness parameter is near zero, leaves the parameter of interest unchanged. Model fitting is implemented and the best choice between the usual calibration model and the model proposed in this article was evaluated by developing the Akaike information criterion, Schwarz’s Bayesian information criterion and Hannan–Quinn criterion.     

Variable selection in joint location and scale models of the skew-normal distribution

A regression model with skew-normal errors provides a useful extension for ordinary normal regression models when the data set under consideration involves asymmetric outcomes. Variable selection is an important issue in all regression analyses, and in this paper, we investigate the simultaneously variable selection in joint location and scale models of the skew-normal distribution. We propose a unified penalized likelihood method which can simultaneously select significant variables in the location and scale models. Furthermore, the proposed variable selection method can simultaneously perform parameter estimation and variable selection in the location and scale models. With appropriate selection of the tuning parameters, we establish the consistency and the oracle property of the regularized estimators. Simulation studies and a real example are used to illustrate the proposed methodologies.     

Large sample confidence intervals for the skewness parameter of the skew-normal distribution based on Fisher's transformation

The skew-normal model is a class of distributions that extends the Gaussian family by including a skewness parameter. This model presents some inferential problems linked to the estimation of the skewness parameter. In particular its maximum likelihood estimator can be infinite especially for moderate sample sizes and is not clear how to calculate confidence intervals for this parameter. In this work, we show how these inferential problems can be solved if we are interested in the distribution of extreme statistics of two random variables with joint normal distribution. Such situations are not uncommon in applications, especially in medical and environmental contexts, where it can be relevant to estimate the distribution of extreme statistics. A theoretical result, found by Loperfido [7 Loperfido, N. 2002. Statistical implications of selectively reported inferential results. Statist. Probab. Lett., 56: 13–22. [Crossref], [Web of Science ®] , [Google Scholar]], proves that such extreme statistics have a skew-normal distribution with skewness parameter that can be expressed as a function of the correlation coefficient between the two initial variables. It is then possible, using some theoretical results involving the correlation coefficient, to find approximate confidence intervals for the parameter of skewness. These theoretical intervals are then compared with parametric bootstrap intervals by means of a simulation study. Two applications are given using real data.     

Shrinkage estimation of location parameters in a multivariate skew-normal distribution

Abstract

This paper studies decision theoretic properties of Stein type shrinkage estimators in simultaneous estimation of location parameters in a multivariate skew-normal distribution with known skewness parameters under a quadratic loss. The benchmark estimator is the best location equivariant estimator which is minimax. A class of shrinkage estimators improving on the best location equivariant estimator is constructed when the dimension of the location parameters is larger than or equal to four. An empirical Bayes estimator is also derived, and motivated from the Bayesian procedure, we suggest a simple skew-adjusted shrinkage estimator and show its dominance property. The performances of these estimators are investigated by simulation.     

On characteristic functions of the wrapped lognormal and the wrapped Weibull distributions

The characteristic function plays a prominent role in determining the pdf of a circular model using the trigonometric moments. The characteristic functions of the wrapped lognormal and the wrapped Weibull distributions cannot be expressed in a closed form. Hence, numerical evaluation of the same is presented along with graphs. Also, certain population characteristics of the wrapped lognormal and the wrapped Weibull distributions are presented.     

Statistical inference for the size-biased Weibull distribution

In this paper, statistical inferences for the size-biased Weibull distribution in two different cases are drawn. In the first case where the size r of the bias is considered known, it is proven that the maximum-likelihood estimators (MLEs) always exist. In the second case where the size r is considered as an unknown parameter, the estimating equations for the MLEs are presented and the Fisher information matrix is found. The estimation with the method of moments can be utilized in the case the MLEs do not exist. The advantage of treating r as an unknown parameter is that it allows us to perform tests concerning the existence of size-bias in the sample. Finally a program in Mathematica is written which provides all the statistical results from the procedures developed in this paper.     

Multivariate Saddlepoint test for the wrapped normal model

In this article we show the effectiveness and the accuracy of the test statistic based on the expnnent of the saddlepoint approximation for the density of M-estimators, proposed by Robinson, Ronchetti and Young (1999), for testing simultaneous hypotheses on the mean and on the variance of a wrapped normal distribution. We base this test statistic on the trigonometric method of moments estimator proposed by Gatto and Jammalamadaka (l999b), which admits the M-estimator representation necessary for this test. This test statistic has an approximate chi-squared distribution, asympiotically up to the second order, and the high accuracy of this approximation is shown by numerical simulations.     

The unit-inverse Gaussian distribution: A new alternative to two-parameter distributions on the unit interval

A new two-parameter distribution over the unit interval, called the Unit-Inverse Gaussian distribution, is introduced and studied in detail. The proposed distribution shares many properties with other known distributions on the unit interval, such as Beta, Johnson S_B, Unit-Gamma, and Kumaraswamy distributions. Estimation of the parameters of the proposed distribution are obtained by transforming the data to the inverse Gaussian distribution. Unlike most distributions on the unit interval, the maximum likelihood or method of moments estimators of the parameters of the proposed distribution are expressed in simple closed forms which do not need iterative methods to compute. Application of the proposed distribution to a real data set shows better fit than many known two-parameter distributions on the unit interval.     

Estimation of the Length Distribution of Marine Populations in the Gaussian-multinomial Setting using the Method of Moments

In this paper, the problem of estimation of the length distribution of marine populations in the Gaussian-multinomial model is considered. For the purpose of the mean and covariance parameter estimation, the method of moments estimators are developed. That is, minimum variance linear unbiased estimator for the mean frequency vector is derived and a consistent estimator for the covariance matrix of the length observations is presented. The usefulness of the proposed estimators is illustrated with an analysis of real cod length measurement data.     

An autoregressive model based on the generalized hyperbolic distribution

We define a nonlinear autoregressive time series model based on the generalized hyperbolic distribution in an attempt to model time series with non-Gaussian features such as skewness and heavy tails. We show that the resulting process has a simple condition for stationarity and it is also ergodic. An empirical example with a forecasting experiment is presented to illustrate the features of the proposed model.     

On the probability distribution of economic growth

Three important and significantly heteroscedastic gross domestic product series are studied. Omnipresent heteroscedasticity is removed and the distributions of the series are then compared to normal, normal mixture and normal–asymmetric Laplace (NAL) distributions. NAL represents a skewed and leptokurtic distribution, which is in line with the Aghion and Howitt [1 Aghion, P. and Howitt, P. 1992. A model of growth through creative destruction. Econometrica, 60: 323–351. [Crossref], [Web of Science ®] [Google Scholar]] model for economic growth, based on Schumpeter's idea of creative destruction. Statistical properties of the NAL distributions are provided and it is shown that NAL fits the data better than the alternatives.     

The performance of the linear and quadratic discriminant functions for three types of non-normal distribution

The purpose of thls paper is to investlgate the performance of the LDF (linear discrlmlnant functlon) and QDF (quadratic dlscrminant functlon) for classlfylng observations from the three types of univariate and multivariate non-normal dlstrlbutlons on the basls of the mlsclasslficatlon rate. The theoretical and the empirical results are described for unlvariate distributions, and the empirical results are presented for multivariate distributions. It 1s also shown that the sign of the skewness of each population and the kurtosis have essential effects on the performance of the two discriminant functions. The variations of the populatlon speclflc mlsclasslflcatlon rates are greatly depend on the sample slze. For the large dlmenslonal populatlon dlstributlons, if the sample sizes are sufflclent, the QDF performs better than the LDF. We show the crlterla of a cholce between the two discriminant functions as an application.     

Tests for generalized exponential laws based on the empirical Mellin transform

Recently, many standard families of distributions have been generalized by exponentiating their cumulative distribution function (CDF). In this paper, test statistics are constructed based on CDF–transformed observations and the corresponding moments of arbitrary positive order. Simulation results for generalized exponential distributions show that the proposed test compares well with standard methods based on the empirical distribution function.     

Small sample properties of parameter estimation in the dirichlet distribution

A numerically feasible algorithm is proposed for maximum likelihood estimation of the parameters of the Dirichlet distribution. The performance of the proposed method is compared with the method of moments using bias ratio and squared errors by Monte Carlo simulation. For these criteria, it is found that even in small samples maximum likelihood estimation has advantages over the method of moments.     

Some computational aspects of the generalized von Mises distribution

This article deals with some important computational aspects of the generalized von Mises distribution in relation with parameter estimation, model selection and simulation. The generalized von Mises distribution provides a flexible model for circular data allowing for symmetry, asymmetry, unimodality and bimodality. For this model, we show the equivalence between the trigonometric method of moments and the maximum likelihood estimators, we give their asymptotic distribution, we provide bias-corrected estimators of the entropy, the Akaike information criterion and the measured entropy for model selection, and we implement the ratio-of-uniforms method of simulation.     

A method for estimating parameters and quantiles of the three-parameter inverse Gaussian distribution based on statistics invariant to unknown location

The inverse Gaussian (IG) distribution, also known as the Wald distribution, is a long-tailed positively skewed distribution and a well-known lifetime distribution. In this paper, we propose an efficient method of estimation for the parameters and quantiles of the three-parameter IG distribution, which is based on statistics invariant to unknown location. Through a Monte Carlo simulation study, we then show that the proposed method performs well compared with other prominent methods in terms of bias and variance. Finally, we present two illustrative examples.     

Further Results on the Limiting Distribution of GMM Sample Moment Conditions

In this article, we examine the limiting behavior of generalized method of moments (GMM) sample moment conditions and point out an important discontinuity that arises in their asymptotic distribution. We show that the part of the scaled sample moment conditions that gives rise to degeneracy in the asymptotic normal distribution is T-consistent and has a nonstandard limiting distribution. We derive the appropriate asymptotic (weighted chi-squared) distribution when this degeneracy occurs and show how to conduct asymptotically valid statistical inference. We also propose a new rank test that provides guidance on which (standard or nonstandard) asymptotic framework should be used for inference. The finite-sample properties of the proposed asymptotic approximation are demonstrated using simulated data from some popular asset pricing models.     

The asymptotic distribution of the likelihood ratio when the model is incorrect

Classical results on the asymptotic distribution of the likelihood ratio statistic rely on the assumption that the model chosen to construct the test statistic be correct. The model is said to be correct if it contains the true distribution of the observations. In this paper the asymptotic distribution of the likelihood ratio statistic is derived without the condition that the model need be correct.     

Estimating the parameters of Poisson-exponential models

This paper proposes two methods of estimation for the parameters in a Poisson-exponential model. The proposed methods combine the method of moments with a regression method based on the empirical moment generating function. One of the methods is an adaptation of the mixed-moments procedure of Koutrouvelis & Canavos (1999). The asymptotic distribution of the estimator obtained with this method is derived. Finite-sample comparisons are made with the maximum likelihood estimator and the method of moments. The paper concludes with an exploratory-type analysis of real data based on the empirical moment generating function.