Another extended Burr III model: some properties and applications

We introduce an extended Burr III distribution as an important model for problems in survival analysis and reliability. The new distribution can be expressed as a linear combination of Burr III distributions and then it has tractable properties for the ordinary and incomplete moments, generating and quantile functions, mean deviations and reliability. The density of its order statistics can be given in terms of an infinite linear combination of Burr III densities. The estimation of the model parameters is approached by maximum likelihood and the observed information matrix is derived. The proposed model is applied to a real data set to illustrate its potentiality.     

Tests for noncorrelation of two multivariate ARMA time series

In many situations, we want to verify the existence of a relationship between multivariate time series. In this paper, we generalize the procedure developed by Haugh (1976) for univariate time series in order to test the hypothesis of noncorrelation between two multivariate stationary ARMA series. The test statistics are based on residual cross-correlation matrices. Under the null hypothesis of noncorrelation, we show that an arbitrary vector of residual cross-correlations asymptotically follows the same distribution as the corresponding vector of cross-correlations between the two innovation series. From this result, it follows that the test statistics considered are asymptotically distributed as chi-square random variables. Two test procedures are described. The first one is based on the residual cross-correlation matrix at a particular lag, whilst the second one is based on a portmanteau type statistic that generalizes Haugh's statistic. We also discuss how the procedures for testing noncorrelation can be adapted to determine the directions of causality in the sense of Granger (1969) between the two series. An advantage of the proposed procedures is that their application does not require the estimation of a global model for the two series. The finite-sample properties of the statistics introduced were studied by simulation under the null hypothesis. It led to modified statistics whose upper quantiles are much better approximated by those of the corresponding chi-square distribution. Finally, the procedures developed are applied to two different sets of economic data.     

A new lifetime model: the Kumaraswamy generalized Rayleigh distribution

The generalized Rayleigh (GR) distribution [V.G. Vodǎ, Inferential procedures on a generalized Rayleigh variate, I, Appl. Math. 21 (1976), pp. 395–412; V.G. Vodǎ, Inferential procedures on a generalized Rayleigh variate, II, Appl. Math. 21 (1976), pp. 413–419] has been applied in several areas such as health, agriculture, biology and other sciences. For the first time, we propose the Kumaraswamy GR (KwGR) distribution for analysing lifetime data. The new density function can be expressed as a mixture of GR density functions. Explicit formulae are derived for some of its statistical quantities. The density function of the order statistics can be expressed as a mixture of GR density functions. We also propose a linear log-KwGR regression model for analysing data with real support to extend some known regression models. The estimation of parameters is approached by maximum likelihood. The importance of the new models is illustrated in two real data sets.     

<Emphasis Type="Italic">Z</Emphasis>-process method for change point problems with applications to discretely observed diffusion processes

The aim of this paper is to develop a general, unified approach, based on some partial estimation functions which we call “Z-process”, to some change point problems in mathematical statistics. The method proposed can be applied not only to ergodic models but also to some models where the Fisher information matrix is random. Applications to some concrete models, including a parametric model for volatilities of diffusion processes are presented. Simulations for randomly time-transformed Brownian bridge process appearing as the limit of the proposed test statistics are performed with computer intensive use.     

关于斜Laplace分布与Levy偏稳定分布的性质

目前有关重尾或偏态数据的统计分析和理论模型相对较少,基于传统的Laplace分布,提出一种处理偏态和重尾数据的新模型——斜Laplace分布,以研究其参数估计方法。利用数理统计知识推导出该分布与一些常见分布(如正态分布、指数分布)间的统计关系,并给出一种可通过设置不同参数值得到不同分布的Levy偏稳定分布及其稳定性。     

A linear model-based test for the heterogeneity of conditional correlations

Current methods of testing the equality of conditional correlations of bivariate data on a third variable of interest (covariate) are limited due to discretizing of the covariate when it is continuous. In this study, we propose a linear model approach for estimation and hypothesis testing of the Pearson correlation coefficient, where the correlation itself can be modeled as a function of continuous covariates. The restricted maximum likelihood method is applied for parameter estimation, and the corrected likelihood ratio test is performed for hypothesis testing. This approach allows for flexible and robust inference and prediction of the conditional correlations based on the linear model. Simulation studies show that the proposed method is statistically more powerful and more flexible in accommodating complex covariate patterns than the existing methods. In addition, we illustrate the approach by analyzing the correlation between the physical component summary and the mental component summary of the MOS SF-36 form across a fair number of covariates in the national survey data.     

Local model uncertainty and incomplete-data bias (with discussion)

Summary.  Problems of the analysis of data with incomplete observations are all too familiar in statistics. They are doubly difficult if we are also uncertain about the choice of model. We propose a general formulation for the discussion of such problems and develop approximations to the resulting bias of maximum likelihood estimates on the assumption that model departures are small. Loss of efficiency in parameter estimation due to incompleteness in the data has a dual interpretation: the increase in variance when an assumed model is correct; the bias in estimation when the model is incorrect. Examples include non-ignorable missing data, hidden confounders in observational studies and publication bias in meta-analysis. Doubling variances before calculating confidence intervals or test statistics is suggested as a crude way of addressing the possibility of undetectably small departures from the model. The problem of assessing the risk of lung cancer from passive smoking is used as a motivating example.     

General results for the beta Weibull distribution

In this paper, we study some mathematical properties of the beta Weibull (BW) distribution, which is a quite flexible model in analysing positive data. It contains the Weibull, exponentiated exponential, exponentiated Weibull and beta exponential distributions as special sub-models. We demonstrate that the BW density can be expressed as a mixture of Weibull densities. We provide their moments and two closed-form expressions for their moment-generating function. We examine the asymptotic distributions of the extreme values. Explicit expressions are derived for the mean deviations, Bonferroni and Lorenz curves, reliability and two entropies. The density of the BW-order statistics is a mixture of Weibull densities and two closed-form expressions are derived for their moments. The estimation of the parameters is approached by two methods: moments and maximum likelihood. We compare the performances of the estimates obtained from both the methods by simulation. The expected information matrix is derived. For the first time, we introduce a log-BW regression model to analyse censored data. The usefulness of the BW distribution is illustrated in the analysis of three real data sets.     

Minimum Scoring Rule Inference

Proper scoring rules are devices for encouraging honest assessment of probability distributions. Just like log‐likelihood, which is a special case, a proper scoring rule can be applied to supply an unbiased estimating equation for any statistical model, and the theory of such equations can be applied to understand the properties of the associated estimator. In this paper, we discuss some novel applications of scoring rules to parametric inference. In particular, we focus on scoring rule test statistics, and we propose suitable adjustments to allow reference to the usual asymptotic chi‐squared distribution. We further explore robustness and interval estimation properties, by both theory and simulations.     

IV估计框架下模型设定检验问题的讨论

 IV估计框架下各种统计量的良好性质依赖于相应的模型设定,如果这些模型设定未能得到数据的支持,其统计推断结论将是不可靠的。如判定计量经济模型是否存在内生性的Hausman检验,实证研究中同一问题的检验结果可能大相径庭。如何通过合理的模型设定检验程序来获得模型参数科学、可靠的估计结果和检验结论呢？本文讨论了工具变量估计框架下的各种模型设定检验问题,明确了各个检验统计量的适用条件及其逻辑联系,给出了工具变量估计框架下模型设定检验的一般步骤。     

Recent and classical goodness-of-fit tests for the Poisson distribution

We give a critical synopsis of classical and recent tests for Poissonity, our emphasis being on procedures which are consistent against general alternatives. Two classes of weighted Cramér–von Mises type test statistics, based on the empirical probability generating function process, are studied in more detail. Both of them generalize already known test statistics by introducing a weighting parameter, thus providing more flexibility with regard to power against specific alternatives. In both cases, we prove convergence in distribution of the statistics under the null hypothesis in the setting of a triangular array of rowwise independent and identically distributed random variables as well as consistency of the corresponding test against general alternatives. Therefore, a sound theoretical basis is provided for the parametric bootstrap procedure, which is applied to obtain critical values in a large-scale simulation study. Each of the tests considered in this study, when implemented via the parametric bootstrap method, maintains a nominal level of significance very closely, even for small sample sizes. The procedures are applied to four well-known data sets.     

Testing symmetry under a skew Laplace model

We develop tests of hypothesis about symmetry based on samples from possibly asymmetric Laplace distributions and present exact and limiting distribution of the test statistics. We postulate that the test statistic derived under the Laplace model is a rational choice as a measure of skewness and can be used in testing symmetry for other, quite general classes of skew distributions. Our results are applied to foreign exchange rates for 15 currencies.     

Goodness-of-Fit for Longitudinal Count Data with Overdispersion

In this article, we develop a method for checking the estimation equations, which is for joint estimation of the regression parameters and the overdispersion parameters, based on one dimension projected covariate. This method is different from the general testing methods in that our proposed method can be applied to high-dimensional response while the classical testing methods can not be extended to high dimension problem simply to construct a powerful test. Furthermore, the properties of the test statistics are investigated and Nonparametric Monte Carlo Test (NMCT) is suggested to determine the critical values of the test statistics under null hypothesis.     

Mantel test for spatial functional data

Statistics for spatial functional data is an emerging field in statistics which combines methods of spatial statistics and functional data analysis to model spatially correlated functional data. Checking for spatial autocorrelation is an important step in the statistical analysis of spatial data. Several statistics to achieve this goal have been proposed. The test based on the Mantel statistic is widely known and used in this context. This paper proposes an application of this test to the case of spatial functional data. Although we focus particularly on geostatistical functional data, that is functional data observed in a region with spatial continuity, the test proposed can also be applied with functional data which can be measured on a discrete set of areas of a region (areal functional data) by defining properly the distance between the areas. Based on two simulation studies, we show that the proposed test has a good performance. We illustrate the methodology by applying it to an agronomic data set.     

Modified likelihood ratio tests for unit gamma regressions

Regression analyses are commonly performed with doubly limited continuous dependent variables; for instance, when modeling the behavior of rates, proportions and income concentration indices. Several models are available in the literature for use with such variables, one of them being the unit gamma regression model. In all such models, parameter estimation is typically performed using the maximum likelihood method and testing inferences on the model''s parameters are usually based on the likelihood ratio test. Such a test can, however, deliver quite imprecise inferences when the sample size is small. In this paper, we propose two modified likelihood ratio test statistics for use with the unit gamma regressions that deliver much more accurate inferences when the number of data points in small. Numerical (i.e. simulation) evidence is presented for both fixed dispersion and varying dispersion models, and also for tests that involve nonnested models. We also present and discuss two empirical applications.     

A new three-parameter extension of the log-logistic distribution with applications to survival data

In this article, a new three-parameter extension of the two-parameter log-logistic distribution is introduced. Several distributional properties such as moment-generating function, quantile function, mean residual lifetime, the Renyi and Shanon entropies, and order statistics are considered. The estimation of the model parameters for complete and right-censored cases is investigated competently by maximum likelihood estimation (MLE). A simulation study is conducted to show that these MLEs are consistent in moderate samples. Two real datasets are considered; one is a right-censored data to show that the proposed model has a superior performance over several existing popular models.     

Inference with a contrast-based posterior distribution and application in spatial statistics

The likelihood function is often used for parameter estimation. Its use, however, may cause difficulties in specific situations. In order to circumvent these difficulties, we propose a parameter estimation method based on the replacement of the likelihood in the formula of the Bayesian posterior distribution by a function which depends on a contrast measuring the discrepancy between observed data and a parametric model. The properties of the contrast-based (CB) posterior distribution are studied to understand what the consequences of incorporating a contrast in the Bayes formula are. We show that the CB-posterior distribution can be used to make frequentist inference and to assess the asymptotic variance matrix of the estimator with limited analytical calculations compared to the classical contrast approach. Even if the primary focus of this paper is on frequentist estimation, it is shown that for specific contrasts the CB-posterior distribution can be used to make inference in the Bayesian way.The method was used to estimate the parameters of a variogram (simulated data), a Markovian model (simulated data) and a cylinder-based autosimilar model describing soil roughness (real data). Even if the method is presented in the spatial statistics perspective, it can be applied to non-spatial data.     

A new lifetime model with decreasing failure rate

In this paper, we introduce a new lifetime distribution by compounding exponential and Poisson–Lindley distributions, named the exponential Poisson–Lindley (EPL) distribution. A practical situation where the EPL distribution is most appropriate for modelling lifetime data than exponential–geometric, exponential–Poisson and exponential–logarithmic distributions is presented. We obtain the density and failure rate of the EPL distribution and properties such as mean lifetime, moments, order statistics and Rényi entropy. Furthermore, estimation by maximum likelihood and inference for large samples are discussed. The paper is motivated by two applications to real data sets and we hope that this model will be able to attract wider applicability in survival and reliability.     

Change-in-mean problem for long memory time series models with applications

Test and estimation procedures for detecting a change in the mean are proposed in infinite moving average long memory time series models. The asymptotic properties of the test statistics and the change-point estimators are investigated. The method is illustrated through the analysis of real data sets from econometrics and climatology.     

Goodness of Fit Test and Latent Distribution Estimation in the Mixed Rasch Model

The Cramér-von Mises test methodology is applied to build a goodness-of fit test for the mixed Rasch model. The Mixed Rasch Model is a probability model of a multivariate discrete random variable driven by an unknown latent continuous variable. The problem of estimation of the unknown fixed difficulty parameters and the latent density function is also considered. The theoretical results are illustrated through simulations and an application to real Quality of Life data.