Inference Based on Order Statistics from the Generalized Pareto Distribution and Application

ABSTRACT

In this article, we derive exact explicit expressions for the single, double, triple, and quadruple moments of order statistics from the generalized Pareto distribution (GPD). Also, we obtain the best linear unbiased estimates of the location and scale parameters (BLUE's) of the GPD. We then use these results to determine the mean, variance, and coefficients of skewness and kurtosis of certain linear functions of order statistics. These are then utilized to develop approximate confidence intervals for the generalized Pareto parameters using Edgeworth approximation and compare them with those based on Monte Carlo simulations. To show the usefulness of our results, we also present a numerical example. Finally, we give an application to real data.     

Multi-Center Clinical Trials with Random Enrollment: Theoretical Approximations

ABSTRACT

We consider the problem of analyzing multi-center clinical trials when the number of patients at each center and on each treatment arm is random and follows the Poisson distribution. Theoretical approximations are made for the first two moments of the mean square errors (MSE's) for three different estimators of treatment effect difference that are commonly used in multi-center clinical trials. To construct these approximations, approximations are needed for the harmonic mean and negative moments of the Poisson distribution. This is achieved through the use of recurrence relations. The accuracy of the approximations for the moments of the MSE's were then validated through comparing the theoretical values to those obtained from a simulation study under two different enrollment environments.     

Parameter estimation of the generalized Pareto distribution—Part II

This is the second part of a paper which focuses on reviewing methods for estimating the parameters of the generalized Pareto distribution (GPD). The GPD is a very important distribution in the extreme value context. It is commonly used for modeling the observations that exceed very high thresholds. The ultimate success of the GPD in applications evidently depends on the parameter estimation process. Quite a few methods exist in the literature for estimating the GPD parameters. Estimation procedures, such as the maximum likelihood (ML), the method of moments (MOM) and the probability weighted moments (PWM) method were described in Part I of the paper. We shall continue to review methods for estimating the GPD parameters, in particular methods that are robust and procedures that use the Bayesian methodology. As in Part I, we shall focus on those that are relatively simple and straightforward to be applied to real world data.     

Data-driven smooth test for a location-scale family

A combination of a smooth test statistic and (an approximate) Schwarz's selection rule has been proposed by Inglot, T., Kallenberg, W. C. M. and Ledwina, T. ((1997). Data-driven smooth tests for composite hypotheses. Ann. Statist. 25, 1222–1250) as a solution of a standard goodness-of-fit problem when nuisance parameters are present. In the present paper we modify the above solution in the sense that we propose another analogue of Schwarz's rule and rederive properties of it and the resulting test statistic. To avoid technicalities we restrict our attention to location-scale family and method of moments estimators of its parameters. In a parallel paper [Janic-Wróblewska, A. (2004). Data-driven smooth tests for the extreme value distribution. Statistics, in press] we illustrate an application of our solution and advantages of modification when testing of fit to extreme value distribution.     

Finite-n Corrections to Normal Approximation

ABSTRACT

We show that the sampling distribution of a statistic which follows the Central Limit Theorem can be approximated by a Normal distribution, modified according to the statistic's first few moments. One can thus achieve an impressive accuracy even for relatively small samples.     

Marshall–Olkin extended two-parameter bathtub-shaped lifetime distribution

ABSTRACT

The Marshall–Olkin extended two-parameter bathtub distribution is introduced and its structural properties are investigated, including the compounding representation of the distribution, the shapes of the density and the hazard rate function, the moments and quantiles. Estimation of the model parameters by maximum likelihood is discussed. Applications to some real data sets which motivate the usefulness of the model are provided. Comparison between the proposed model and other commonly used distributions is performed using real data sets. A simulation study is presented to investigate the accuracy of the estimates of the model's parameters.     

Extension Of Dale's Moment Conditions With Application To The Wright–fisher Model

Dale's necessary and sufficient conditions for an array to contain the joint moments for some probability distribution on the unit simplex in R² are extended to the unit simplex in R ^d . These conditions are then used in a computational method, based on linear programming, to evaluate the stationary distribution for the diffusion approximation of the Wright–Fisher model in population genetics. The computational method uses a characterization of the diffusion through an adjoint relation between the diffusion operator and its stationary distribution. Application of this adjoint relation to a set of functions in the domain of the generator leads to one set of constraints for the linear program involving the moments of the stationary distribution. The extension of Dale's conditions on the moments add another set of linear conditions and the linear program is solved to obtain bounds on numerical quantities of interest. Numerical illustrations are given to illustrate the accuracy of the method.

     

Relationships Between Central Moments and Cumulants,with Formulae for the Central Moments of Gamma Distributions

Abstract

Simple expressions are presented that relate cumulants to central moments without involving moments about the origin. These expressions are used to obtain recursive formulae for the central moments of the gamma distribution, with exponential and chi-square distributions as special cases.     

Parameter estimation of the generalized Pareto distribution—Part I

The generalized Pareto distribution (GPD) has been widely used in the extreme value framework. The success of the GPD when applied to real data sets depends substantially on the parameter estimation process. Several methods exist in the literature for estimating the GPD parameters. Mostly, the estimation is performed by maximum likelihood (ML). Alternatively, the probability weighted moments (PWM) and the method of moments (MOM) are often used, especially when the sample sizes are small. Although these three approaches are the most common and quite useful in many situations, their extensive use is also due to the lack of knowledge about other estimation methods. Actually, many other methods, besides the ones mentioned above, exist in the extreme value and hydrological literatures and as such are not widely known to practitioners in other areas. This paper is the first one of two papers that aim to fill in this gap. We shall extensively review some of the methods used for estimating the GPD parameters, focusing on those that can be applied in practical situations in a quite simple and straightforward manner.     

Kurtosis of the logistic-exponential survival distribution

ABSTRACT

In this article, the kurtosis of the logistic-exponential distribution is analyzed. All the moments of this survival distribution are finite, but do not possess closed-form expressions. The standardized fourth central moment, known as Pearson’s coefficient of kurtosis and often used to describe the kurtosis of a distribution, can thus also not be expressed in closed form for the logistic-exponential distribution. Alternative kurtosis measures are therefore considered, specifically quantile-based measures and the L-kurtosis ratio. It is shown that these kurtosis measures of the logistic-exponential distribution are invariant to the values of the distribution’s single shape parameter and hence skewness invariant.     

A generalization of the logistic linear'model

Consider the logistic linear model, with some explanatory variables overlooked. Those explanatory variables may be quantitative or qualitative. In either case, the resulting true response variable is not a binomial or a beta-binomial but a sum of binomials. Hence, standard computer packages for logistic regression can be inappropriate even if an overdispersion factor is incorporated. Therefore, a discrete exponential family assumption is considered to broaden the class of sampling models. Likelihood and Bayesian analyses are discussed. Bayesian computation techniques such as Laplacian approximations and Markov chain simulations are used to compute posterior densities and moments. Approximate conditional distributions are derived and are shown to be accurate. The Markov chain simulations are performed effectively to calculate posterior moments by using the approximate conditional distributions. The methodology is applied to Keeler's hardness of winter wheat data for checking binomial assumptions and to Matsumura's Accounting exams data for detailed likelihood and Bayesian analyses.     

THE GENERALIZED SECANT HYPERBOLIC DISTRIBUTION AND ITS PROPERTIES

ABSTRACT

The properties of a family of distributions generalizing the secant hyperbolic are developed. This family consists of symmetric distributions, with kurtosis ranging from 1.8 to infinity, and includes the logistic as a special case, the uniform as a limiting case, and closely approximates the normal and Student's t-distributions with corresponding kurtosis. A significant difference between this family and Student's t is that for any member of the generalized secant hyperbolic family, all moments are finite. Further, technical difficulties associated with evaluating moments of Student's t (especially for fractional degrees of freedom) are not present with this family. The properties of the maximum likelihood and modified maximum likelihood estimates of the location and scale parameters for complete samples are considered. Examples illustrate the methods developed in this work.     

Copula-Based Models for the Power of Independence Tests

ABSTRACT

We consider independence tests and the methods to evaluate their efficiency. First, we observe that many of the most used independence tests are functions of the empirical copula, which is a sufficient statistic. Hence, the power of these tests, such as the tests based on Spearman's ρ, on Kendall's τ, and on Gini's γ, depend solely on the theoretical copula, and not on the marginal distributions. Then, we consider monotone dependence tests and we propose a parametric model to define the power function. Such a model is based on a path of copulas, from the copula of discordance to the copula of concordance, and can be characterized by the copula of the underlying joint distribution. Moreover, we introduce a consistent estimator of the path of copulas. Finally, we provide some examples of applications, and in particular, a bootstrap-plug-in estimator of the power curve, all useful for power comparison.     

Alternative Tests for Parameter Stability

ABSTRACT

We propose new tests for parameter stability based on estimates computed from a sequence of subsamples moving forward and backward across the sample. We obtain a sequence of moving estimates tests and we derive their asymptotic null distribution based on the functional central limit theorem. The critical values are approximated using Durbin's method. Our simulation results show that these tests have comparable size and slightly higher power in detecting structural change than other competing tests.     

The truncated generalized poisson distribution and its estimation

A discrete probability model always gets truncated during the sampling process and the point of truncation depends upon the sample size. Also, the generalized Poisson distribution cannot be used with full justification when the second parameter is negative. To avoid these problems a truncated generalized Poisson distribution is defined and studied. Estimation of its parameters by moments method, maximum likelihood method and a mixed method are considered. Some examples are given to illustrate the effect on the parameters’ estimates when a non-truncated GPD is used instead of a truncated GPD.     

A NEW TEST FOR EQUALITY OF VARIANCES FOR k NORMAL POPULATIONS

ABSTRACT

A simple test based on Gini's mean difference is proposed to test the hypothesis of equality of population variances. Using 2000 replicated samples and empirical distributions, we show that the test compares favourably with Bartlett's and Levene's test for the normal population. Also, it is more powerful than Bartlett's and Levene's tests for some alternative hypotheses for some non-normal distributions and more robust than the other two tests for large sample sizes under some alternative hypotheses. We also give an approximate distribution to the test statistic to enable one to calculate the nominal levels and P-values.     

A new hybrid estimation method for the generalized pareto distribution

ABSTRACT

The generalized Pareto distribution (GPD) is important in the analysis of extreme values, especially in modeling exceedances over thresholds. Most of the existing methods for estimating the scale and shape parameters of the GPD suffer from theoretical and/or computational problems. A new hybrid estimation method is proposed in this article, which minimizes a goodness-of-fit measure and incorporates some useful likelihood information. Compared with the maximum likelihood method and other leading methods, our new hybrid estimation method retains high efficiency, reduces the estimation bias, and is computation friendly.     

Exponentiality test based on Renyi distance between equilibrium distributions

ABSTRACT

On the basis of Csiszar's φ-divergence discrimination information, we propose a measure of discrepancy between equilibriums associated with two distributions. Proving that a distribution can be characterized by associated equilibrium distribution, a Renyi distance of the equilibrium distributions is constructed that made us to propose an EDF-based goodness-of-fit test for exponential distribution. For comparing the performance of the proposed test, some well-known EDF-based tests and some entropy-based tests are considered. Based on the simulation results, the proposed test has better powers than those of competing entropy-based tests for the alternatives with decreasing hazard rate function. The use of the proposed test is evaluated in an illustrative example.     

A Laguerre polynomial approximation for a goodness-of-fit test for exponential distribution based on progressively censored data

This paper proposes an approximation to the distribution of a goodness-of-fit statistic proposed recently by Balakrishnan et al. [Balakrishnan, N., Ng, H.K.T. and Kannan, N., 2002, A test of exponentiality based on spacings for progressively Type-II censored data. In: C. Huber-Carol et al. (Eds.), Goodness-of-Fit Tests and Model Validity (Boston: Birkhäuser), pp. 89–111.] for testing exponentiality based on progressively Type-II right censored data. The moments of this statistic can be easily calculated, but its distribution is not known in an explicit form. We first obtain the exact moments of the statistic using Basu's theorem and then the density approximants based on these exact moments of the statistic, expressed in terms of Laguerre polynomials, are proposed. A comparative study of the proposed approximation to the exact critical values, computed by Balakrishnan and Lin [Balakrishnan, N. and Lin, C.T., 2003, On the distribution of a test for exponentiality based on progressively Type-II right censored spacings. Journal of Statistical Computation and Simulation, 73 (4), 277–283.], is carried out. This reveals that the proposed approximation is very accurate.