A weighted bootstrap approximation for comparing the error distributions in nonparametric regression

Several procedures have been proposed for testing the equality of error distributions in two or more nonparametric regression models. Here we deal with methods based on comparing estimators of the cumulative distribution function (CDF) of the errors in each population to an estimator of the common CDF under the null hypothesis. The null distribution of the associated test statistics has been approximated by means of a smooth bootstrap (SB) estimator. This paper proposes to approximate their null distribution through a weighted bootstrap. It is shown that it produces a consistent estimator. The finite sample performance of this approximation is assessed by means of a simulation study, where it is also compared to the SB. This study reveals that, from a computational point of view, the proposed approximation is more efficient than the one provided by the SB.     

Enhancement for two commonly-used approximations for the inverse cumulative function of the normal distribution

Two commonly used approximations for the inverse distribution function of the normal distribution are Schmeiser's and Shore's. Both approximations are based on a power transformation of either the cumulative density function (CDF) or a simple function of it. In this note we demonstrate, that if these approximations are presented in the form of the classical one-parameter Box-Cox transformation, and the exponent of the transformation is expressed as a simple function of the CDF, then the accuracy of both approximations may be considerably enhanced, without losing much in algebraic simplicity. Since both approximations are special cases of more general four-parameter systems of distributions, the results presented here indicate that the accuracy of the latter, when used to represent non-normal density functions, may also be considerably enhanced.     

Generalized order statistics from arbitrary distributions and the Markov chain property

The joint and marginal distributions of generalized order statistics based on an arbitrary distribution function are established in terms of the lexicographic distribution function. Furthermore, we show that generalized order statistics and the corresponding number of ties form a two-dimensional Markov chain.     

Bayesian Inference in Generalized Error and Generalized Student-t Regression Models

This study takes up inference in linear models with generalized error and generalized t distributions. For the generalized error distribution, two computational algorithms are proposed. The first is based on indirect Bayesian inference using an approximating finite scale mixture of normal distributions. The second is based on Gibbs sampling. The Gibbs sampler involves only drawing random numbers from standard distributions. This is important because previously the impression has been that an exact analysis of the generalized error regression model using Gibbs sampling is not possible. Next, we describe computational Bayesian inference for linear models with generalized t disturbances based on Gibbs sampling, and exploiting the fact that the model is a mixture of generalized error distributions with inverse generalized gamma distributions for the scale parameter. The linear model with this specification has also been thought not to be amenable to exact Bayesian analysis. All computational methods are applied to actual data involving the exchange rates of the British pound, the French franc, and the German mark relative to the U.S. dollar.     

Comparison of several Birnbaum–Saunders distributions

Birnbaum–Saunders (BS) distribution is widely used in reliability applications to model failure times. For several samples from possible different BS distributions, to prevent wrong conclusions in any further analysis, it is of importance to accompany a formal comparison for characteristic quantities of the distributions, including mean, quantile and reliability function difference. To this end, two test statistics, which are respectively based on the exact generalized p-value approach and the Delta method, are proposed and their behaviours are investigated. Simulation studies are carried out to examine the size and power performance of the newly proposed statistics. An interesting phenomenon is that in the finite sample simulations we conduct, the Delta method-based test almost uniformly outperforms the generalized p-value-based test although its sampling null distribution is simulated by Monte Carlo method. This might suggest that the sampling null distribution of the Delta method-based test statistic would have a fast convergence to its limit. The tests are also applied to analyse a real example on the fatigue life of 6061-T6 aluminium coupons for illustration.     

Comparisons of several Pareto distributions based on record values

In order to avoid wrong conclusions in any further analysis, it is of importance to conduct a formal comparison for characteristic quantities of the distributions. These characteristic quantities we are familiar with include mean, quantity and reliability function, and so on. In this paper, we consider two tests aiming at the comparisons for function of parameters in Pareto distribution based on record values. They are generalized p-value-based test and parametric bootstrap-based test, respectively. The resulting procedures are easy to compute and are applicable to small samples. A simulation study is conducted to investigate and compare the performance of the proposed tests. A phenomenon we note is that generalized p-value-based test almost uniformly outperforms the parametric bootstrap-based test.     

A Dimensional CLT for Non-Central Wilks' Lambda in Multivariate Analysis

Abstract.  We consider the non-central distribution of the classical Wilks' lambda statistic for testing the general linear hypothesis in MANOVA. We prove that as the dimension of the observation vector goes to infinity, Wilks' lambda obeys a central limit theorem under simple growth conditions on the non-centrality matrix. In one case we also prove a stronger result: the saddlepoint cumulative distribution function (CDF) approximation for the standardized version of Wilks' lambda converges uniformly on compact sets to the standard normal CDF. These theoretical results go some way towards explaining why saddlepoint approximations to the distribution of Wilks' lambda retain excellent accuracy in high-dimensional cases.     

Inference for Income Distributions Using Grouped Data

We develop a general approach to estimation and inference for income distributions using grouped or aggregate data that are typically available in the form of population shares and class mean incomes, with unknown group bounds. We derive generic moment conditions and an optimal weight matrix that can be used for generalized method-of-moments (GMM) estimation of any parametric income distribution. Our derivation of the weight matrix and its inverse allows us to express the seemingly complex GMM objective function in a relatively simple form that facilitates estimation. We show that our proposed approach, which incorporates information on class means as well as population proportions, is more efficient than maximum likelihood estimation of the multinomial distribution, which uses only population proportions. In contrast to the earlier work of Chotikapanich, Griffiths, and Rao, and Chotikapanich, Griffiths, Rao, and Valencia, which did not specify a formal GMM framework, did not provide methodology for obtaining standard errors, and restricted the analysis to the beta-2 distribution, we provide standard errors for estimated parameters and relevant functions of them, such as inequality and poverty measures, and we provide methodology for all distributions. A test statistic for testing the adequacy of a distribution is proposed. Using eight countries/regions for the year 2005, we show how the methodology can be applied to estimate the parameters of the generalized beta distribution of the second kind (GB2), and its special-case distributions, the beta-2, Singh–Maddala, Dagum, generalized gamma, and lognormal distributions. We test the adequacy of each distribution and compare predicted and actual income shares, where the number of groups used for prediction can differ from the number used in estimation. Estimates and standard errors for inequality and poverty measures are provided. Supplementary materials for this article are available online.     

Some aspects of statistical inference on the lagrange (generalized) poisson distribution

A generalization of the Poisson distribution was defined by Consul and Jain (Ann. Math. Statist., 41, (1970)) and was obtained as a particular family of Lagrange distributions by Consul and Shenton (SIAM. J. Appl. Math., 23, (1972)). The distribution is subsequently named the generalized Poisson distribution (GPD). This GPD reduces to the Poisson distribution for ? = 0. When the data have a one-way layout structure, the asymptotically locally optimal Neyman's C(d) test is constructed and compared with the conditional test on the hypothesis H_o? = 0. Within the framework of the generalized linear models an appropriate link function is given, and the asymptotic distributions of the estimated parameters are derived.     

A modified Kolmogorov-Smirnov test for the inverse gaussian density with unknown parameters

The Kolmogorov-Smirnov (KS) test is an empirical distribution function (EDF) based goodness-of-fit test that requires the underlying hypothesized density to be continuous and completely specified. When the parameters are unknown and must be estimated from the data, standard tables of the KS test statistic are not valid. Approximate upper tail percentage points of the KS statistic for the inverse Gaussian (IG) distribution with unknown parameters are tabled in this paper.

A study of the power of the KS test for the IG distribution indicates that the test is able todiscriminate between the IG distribution and distributions such as the uniform and exponentialdistributions that are very different in shape, but is relatively unable to discriminate between the IG distribution and distributions that are similar in shape such as the lognormal and Weibull distributions. In modeling settings the former distinction is typically more important to make than the latter distinction.     

Asymmetric generalizations of symmetric univariate probability distributions obtained through quantile splicing

Abstract

Balakrishnan et al. proposed a two-piece skew logistic distribution by making use of the cumulative distribution function (CDF) of half distributions as the building block, to give rise to an asymmetric family of two-piece distributions, through the inclusion of a single shape parameter. This paper proposes the construction of asymmetric families of two-piece distributions by making use of quantile functions of symmetric distributions as building blocks. This proposition will enable the derivation of a general formula for the L-moments of two-piece distributions. Examples will be presented, where the logistic, normal, Student’s t(2) and hyperbolic secant distributions are considered.     

Bayesian Estimation for Deformed Modified Power Series Distribution

In this article, posterior distribution, posterior moments, and predictive distribution for the modified power series distributions deformed at any of a support point under linex and generalized entropy loss function are derived. It is assumed that the prior information can be summarized by a uniform, Beta, two-sided power, Gamma, or generalized Pareto distributions. The obtained results are demonstrated on the generalized Poisson and the generalized negative binomial distribution deformed at a given point.     

Some asymptotic results for hypothesis testing in multivariate failure time data

This paper presents a class of generalized Wald, generalized score and generalized likelihood ratio statistics for hypothesis testing and model selection for multivariate failure time data. These statistics are based on a marginal hazard model with a common baseline hazard function. The large sample distributions of these statistics are examined. It is shown that the proposed test statistics follow asymptotically a weighted sum of independent χ₁² distributions.     

Exact distributional computations for Roy’s statistic and the largest eigenvalue of a Wishart distribution

Computational expressions for the exact CDF of Roy’s test statistic in MANOVA and the largest eigenvalue of a Wishart matrix are derived based upon their Pfaffian representations given in Gupta and Richards (SIAM J. Math. Anal. 16:852–858, 1985). These expressions allow computations to proceed until a prespecified degree of accuracy is achieved. For both distributions, convergence acceleration methods are used to compute CDF values which achieve reasonably fast run times for dimensions up to 50 and error degrees of freedom as large as 100. Software that implements these computations is described and has been made available on the Web.     

A new generalized two-sided class of distributions with an emphasis on two-sided generalized normal distribution

The ordinary-G class of distributions is defined to have the cumulative distribution function (cdf) as the value of the cdf of the ordinary distribution F whose range is the unit interval at G, that is, F(G), and it generalizes the ordinary distribution. In this work, we consider the standard two-sided power distribution to define other classes like the beta-G and the Kumaraswamy-G classes. We extend the idea of two-sidedness to other ordinary distributions like normal. After studying the basic properties of the new class in general setting, we consider the two-sided generalized normal distribution with maximum likelihood estimation procedure.     

Precedence-type test based on Kaplan–Meier estimator of cumulative distribution function

In this paper, we introduce a precedence-type test based on Kaplan–Meier estimator of cumulative distribution function (CDF) for testing the hypothesis that two distribution functions are equal against a stochastically ordered hypothesis. This test is an alternative to the precedence life-test proposed first by Nelson (1963). After deriving the null distribution of the test statistic, we present its exact power function under the Lehmann alternative, and compare the exact power as well as simulated power (under location-shift) of the proposed test with other precedence-type tests. Next, we extend this test to the case of progressively Type-II censored data. Critical values for some combination of sample sizes and progressive censoring schemes are presented. We then examine the power properties of this test procedure and compare them to those of the weighted precedence and weighted maximal precedence tests under a location-shift alternative by means of Monte Carlo simulations. Finally, we present two examples to illustrate all the test procedures discussed here, and then make some concluding remarks.     

Generalized spectral tests for serial dependence

Two tests for serial dependence are proposed using a generalized spectral theory in combination with the empirical distribution function. The tests are generalizations of the Cramér-von Mises and Kolmogorov-Smirnov tests based on the standardized spectral distribution function. They do not involve the choice of a lag order, and they are consistent against all types of pairwise serial dependence, including those with zero autocorrelation. They also require no moment condition and are distribution free under serial independence. A simulation study compares the finite sample performances of the new tests and some closely related tests. The asymptotic distribution theory works well in finite samples. The generalized Cramér-von Mises test has good power against a variety of dependent alternatives and dominates the generalized Kolmogorov-Smirnov test. A local power analysis explains some important stylized facts on the power of the tests based on the empirical distribution function.     

A NOTE ON CHARACTERIZATIONS OF DISTRIBUTIONS BY REGRESSIONS OF NON-ADJACENT GENERALIZED ORDER STATISTICS

We investigate the problem of characterizations of distributions by regressions of generalized order statistics (GOSs) based on a continuous distribution function F . We show that F is uniquely determined if the regressions of a GOS given two other consecutive GOSs are known.     

On the Gini Mean Difference Test for Circular Data

We introduce a new test of isotropy or uniformity on the circle, based on the Gini mean difference of the sample arc-lengths and obtain both the exact and asymptotic distributions under the null hypothesis of circular uniformity. We also provide a table of upper percentile values of the exact distribution for small to moderate sample sizes. Illustrative examples in circular data analysis are also given. It is shown that a “generalized” Gini mean difference test has better asymptotic efficiency than a corresponding “generalized” Rao's test in the sense of Pitman asymptotic relative efficiency.     

A generalization of discrete Weibull distribution

In this paper, we introduce a new family of discrete distributions and study its properties. It is shown that the new family is a generalization of discrete Marshall-Olkin family of distributions. In particular, we study generalized discrete Weibull distribution in detail. Discrete Marshall-Olkin Weibull distribution, exponentiated discrete Weibull distribution, discrete Weibull distribution, discrete Marshall-Olkin generalized exponential distribution, exponentiated geometric distribution, generalized discrete exponential distribution, discrete Marshall-Olkin Rayleigh distribution and exponentiated discrete Rayleigh distribution are sub-models of generalized discrete Weibull distribution. We derive some basic distributional properties such as probability generating function, moments, hazard rate and quantiles of the generalized discrete Weibull distribution. We can see that the hazard rate function can be decreasing, increasing, bathtub and upside-down bathtub shape. Estimation of the parameters are done using maximum likelihood method. A real data set is analyzed to illustrate the suitability of the proposed model.