Inference for the Generalized Normal Laplace Distribution

The generalized normal Laplace distribution has been used in financial modeling because of its skewness and excess kurtosis. To estimate its parameters, we use a method based on minimizing the quadratic distance between the real and imaginary parts of the empirical and theoretical characteristic functions. The quadratic distance estimator (QDE) obtained is shown to be robust, consistent, and with an asymptotically normal distribution. The goodness-of-fit test statistics presented follow an asymptotic chi-square distribution. The performance of the QDE is illustrated by simulation results and an application to financial data.     

Efficient estimators for the good family

We consider the problem of estimating the two parameters of the discrete Good distribution. We first show that the sufficient statistics for the parameters are the arithmetic and the geometric means. The maximum likelihood estimators (MLE's) of the parameters are obtained by solving numerically a system of equations involving the Lerch zeta function and the sufficient statistics. We find an expression for the asymptotic variance-covariance matrix of the MLE's, which can be evaluated numerically. We show that the probability mass function satisfies a simple recurrence equation linear in the two parameters, and propose the quadratic distance estimator (QDE) which can be computed with an ineratively reweighted least-squares algorithm. the QDE is easy to calculate and admits a simple expression for its asymptotic variance-covariance matrix. We compute this matrix for the MLE's and the QDE for various values of the parameters and see that the QDE has very high asymptotic efficiency. Finally, we present a numerical example.     

ESTIMATION FOR THE POSITIVE STABLE LAWS, I

Estimation of scale and index parameters of positive stable laws is considered. Maximum likelihood estimation is known to be efficient, but very difficult to compute, while methods based on the sample characteristic function are computationally easy, but have uncertain efficiency properties.
In this paper an estimation method is presented which is reasonably easy to compute, and which has good efficiency properties, at least when the index α (0, 0.5). The method is based on an expression for the characteristic function of the logarithm of a positive stable random variable, and is derived by relating the stable estimation problem to that of location/scale estimation in extreme-value-distribution families, for which efficient methods are known.
The proposed method has efficiency which →1 as α→,but on the other hand, efficiencies deteriorate after α >0.5, and in fact appear to →0 as α+ 1.     

Some properties of modified integrated squared error estimators for the stable laws

The estimation procedure of Paulson, Holcomb and Leitch (1975) for the parameters of the stable laws is shown to be similar in spirit to the modified X²minimum procedure. This observation suggests that a class of modified integrated squared error procedures may be developed for the stable laws as well as much more generally. For the stable case, some influence curves, asymptotic covariances, and efficiencies are given, and the robustness of maximum likelihood estimators is discussed.     

On von Mises type conditions for p-max stable laws, rates of convergence and generalized log Pareto distributions

In this article we obtain an alternative formulation of the von Mises type conditions for p-max stable laws in terms of generalized log Pareto distributions (glogPds). Relationship between the rate of convergence of extremes and the remainder terms in the von Mises type conditions is investigated. It is shown that the rate of convergence in the von Mises type conditions for p-max stable laws determines the distance of the underlying distribution function from a glogPd.     

Weighted similarity tests for location-scale families of stable distributions

ABSTRACT

The class of stable distributions plays a central role in the study of asymptotic behavior of normalized partial sums, the same role performed by normal distribution among those with finite second moment. In this note, by exploiting the connection between stable laws and regularly varying functions, we present weighted similarity tests for stable location-scale families. The proposed weight functions are based on the 2nd-order Mallows distance between the empirical distribution and the root stable distribution. And the resulting statistics converge weakly to functionals of Brownian bridge.     

Some Simple Method of Estimation for the Parameters of the Discrete Stable Distribution with the Probability Generating Function

In this article, we develop a method to estimate the two parameters of the discrete stable distribution. By minimizing the quadratic distance between transforms of the empirical and theoretical probability generating functions, we obtain estimators simple to calculate, asymptotically unbiased, and normally distributed. We also derive the expression for their variance–covariance matrix. We simulate several samples of discrete stable distributed datasets with different parameters, to analyze the effect of tuncation on the right tail of the distribution.     

Modified weighted squared error estimation procedures with special emphasis on the stable laws

Two families of parameter estimation procedures for the stable laws based on a variant of the characteristic function are provided. The methodology which produces viable computational procedures for the stable laws is generally applicable to other families of distributions across a variety of settings. Both families of procedures may be described as a modified weighted chi-squared minimization procedure, and both explicitly take account of constraints on the parameter space. Influence func-tions for and efficiencies of the estimators are given. If x₁, x₂, …x_n random sample from an unknown distribution F , a method for determining the stable law to which F is attracted is developed. Procedures for regression and autoregres-sion with stable error structure are provided. A number of examples are given.     

Wavelet-based estimation for multivariate stable laws

In this article, an estimation problem for multivariate stable laws using wavelets has been studied. The method of applying wavelets, which has already been done, to estimate parameters in univariate stable laws, has been extended to multivariate stable laws. The proposed estimating method is based on a nonlinear regression model on wavelet coefficients of characteristic functions. In particular, two parametric sub-classes of stable laws are considered: the class of multivariate stable laws with discrete spectral measure, and sub-Gaussian laws. Using a simulation study, the proposed method has been compared with well-known estimation procedures.     

Homogeneity tests among groups for microsatellite data

We propose a homogeneity test among groups on a quadratic distance measure. The underlying mutation process in the microsatellite loci is studied using the stepwise mutation model. Asymptotic normality of the test statistic is proved under very mild regularity conditions. Resampling methods, such as jackknife, are used in the application to build confidence intervals for the difference in allelic variation between and within groups. The method is applied in a real data to test whether there are differences in the distribution of the repeated sequence among groups defined by ethnicity and alcoholism index (ALDX1).     

Statistical methods of risk assessment for continuous variables

Adverse health effects for continuous responses are not as easily defined as adverse health effects for binary responses. Kodell and West (1993) developed methods for defining adverse effects for continuous responses and the associated risk. Procedures were developed for finding point estimates and upper confidence limits for additional risk under the assumption of a normal distribution and quadratic mean response curve with equal variances at each dose level. In this paper, methods are developed for point estimates and upper confidence limits for additional risk at experimental doses when the equal variance assumption is relaxed. An interpolation procedure is discussed for obtaining information at doses other than the experimental doses. A small simulation study is presented to test the performance of the methods discussed.     

Imputation techniques for incomplete data in quadratic discriminant analysis

We have compared the efficacy of five imputation algorithms readily available in SAS for the quadratic discriminant function. Here, we have generated several different parametric-configuration training data with missing data, including monotone missing-at-random observations, and used a Monte Carlo simulation to examine the expected probabilities of misclassification for the two-class quadratic statistical discrimination problem under five different imputation methods. Specifically, we have compared the efficacy of the complete observation-only method and the mean substitution, regression, predictive mean matching, propensity score, and Markov Chain Monte Carlo (MCMC) imputation methods. We found that the MCMC and propensity score multiple imputation approaches are, in general, superior to the other imputation methods for the configurations and training-sample sizes we considered.     

Regression for non-Euclidean data using distance matrices

Regression methods for common data types such as measured, count and categorical variables are well understood but increasingly statisticians need ways to model relationships between variable types such as shapes, curves, trees, correlation matrices and images that do not fit into the standard framework. Data types that lie in metric spaces but not in vector spaces are difficult to use within the usual regression setting, either as the response and/or a predictor. We represent the information in these variables using distance matrices which requires only the specification of a distance function. A low-dimensional representation of such distance matrices can be obtained using methods such as multidimensional scaling. Once these variables have been represented as scores, an internal model linking the predictors and the responses can be developed using standard methods. We call scoring as the transformation from a new observation to a score, whereas backscoring is a method to represent a score as an observation in the data space. Both methods are essential for prediction and explanation. We illustrate the methodology for shape data, unregistered curve data and correlation matrices using motion capture data from an experiment to study the motion of children with cleft lip.     

Analysis of multivariate repeated measures data with a Kronecker product structured covariance matrix

In this article we consider a set of t repeated measurements on p variables (or characteristics) on each of the n individuals. Thus, data on each individual is a p ×t matrix. The n individuals themselves may be divided and randomly assigned to g groups. Analysis of these data using a MANOVA model, assuming that the data on an individual has a covariance matrix which is a Kronecker product of two positive definite matrices, is considered. The well-known Satterthwaite type approximation to the distribution of a quadratic form in normal variables is extended to the distribution of a multivariate quadratic form in multivariate normal variables. The multivariate tests using this approximation are developed for testing the usual hypotheses. Results are illustrated on a data set. A method for analysing unbalanced data is also discussed.     

Mixture designs for five components in orthogonal blocks

The methods developed by John and Draper et al. of partitioning the blends (runs) of four mixture components into two or more orthogonal blocks when fitting quadratic models are extended to mixtures of five components. The characteristics of Latin squares of side five are used to derive rules for reliably and quickly obtaining designs with specific properties. The designs also produce orthogonal blocks when higher order models are fitted.     

Improved estimation of the stable laws

Fitting general stable laws to data by maximum likelihood is important but difficult. This is why much research has considered alternative procedures based on empirical characteristic functions. Two problems then are how many values of the characteristic function to select, and how to position them. We provide recommendations for both of these topics. We propose an arithmetic spacing of transform variables, coupled with a recommendation for the location of the variables. It is shown that arithmetic spacing, which is far simpler to implement, closely approximates optimum spacing. The new methods that result are compared in simulation studies with existing methods, including maximum-likelihood. The main conclusion is that arithmetic spacing of the values of the characteristic function, coupled with appropriately limiting the range for these values, improves the overall performance of the regression-type method of Koutrouvelis, which is the standard procedure for estimating general stable law parameters.     

Inference for a leptokurtic symmetric family of distributions represented by the difference of two gamma variates

We introduce a family of leptokurtic symmetric distributions represented by the difference of two gamma variates. Properties of this family are discussed. The Laplace, sums of Laplace and normal distributions all arise as special cases of this family. We propose a two-step method for fitting data to this family. First, we perform a test of symmetry, and second, we estimate the parameters by minimizing the quadratic distance between the real parts of the empirical and theoretical characteristic functions. The quadratic distance estimator obtained is consistent, robust and asymptotically normally distributed. We develop a statistical test for goodness of fit and introduce a test of normality of the data. A simulation study is provided to illustrate the theory.     

Semi-doubly optimal concentric circles fitting with presence of heteroscedasticity

Previous studies focus on homogeneous and isotropic assumptions about the noisy data. Many methods have been developed recently for fitting concentric circles to data. In this paper, these statistical assumptions have been relaxed. To the best of our knowledge, only one iterative method has been recently developed. Due to its complexity, no such algorithm is available to compute the reliable maximum likelihood estimator (MLE). Accordingly, we have developed four new methods that outperform the existing methods including the orthogonal distance regression (ODR). We also discuss which of these methods is superior according to the four principles: statistical efficiency, accuracy, robustness, and computational efficiency. Numerical experiments on synthetic and real images have been conducted to validate our findings.     

A Comparison of Two Group Classification Approaches to Fat-tailed and Skewed Data

The problem of two-group classification has implications in a number of fields, such as medicine, finance, and economics. This study aims to compare the methods of two-group classification. The minimum sum of deviations and linear programming model, linear discriminant analysis, quadratic discriminant analysis and logistic regression, multivariate analysis of variance (MANOVA) test-based classification and the unpooled T-square test-based classification methods, support vector machines and k-nearest neighbor methods, and combined classification method will be compared for data structures having fat-tail and/or skewness. The comparison has been carried out by using a simulation procedure designed for various stable distribution structures and sample sizes.