一类零膨胀广义线性模型极大似然估计的相合性与渐近正态性

讨论了响应变量为单参数指数族且在零点处膨胀的广义线性模型的大样本性质,对其参数进行了极大似然估计,给出了一些正则条件.基于所提出的正则条件,证明了模型参数极大似然估计的相合性与渐近正态性.     

Large sample properties of the mle and mcle for the natural parameter of a truncated exponential family

Summary Consider a truncated exponential family of absolutely continuous distributions with natural parameter θ and truncation parameter γ. Strong consistency and asymptotic normality are shown to hold for the maximum likelihood and maximum conditional likelihood estimates of θ with γ unknown. Moreover, these two estimates are also shown to have the same limiting distribution, coinciding with that of the maximum likelihood estimate for θ when γ is assumed to be known.     

Empirical Likelihood Semiparametric Regression Analysis under Random Censorship

This paper considers large sample inference for the regression parameter in a partly linear model for right censored data. We introduce an estimated empirical likelihood for the regression parameter and show that its limiting distribution is a mixture of central chi-squared distributions. A Monte Carlo method is proposed to approximate the limiting distribution. This enables one to make empirical likelihood-based inference for the regression parameter. We also develop an adjusted empirical likelihood method which only appeals to standard chi-square tables. Finite sample performance of the proposed methods is illustrated in a simulation study.     

Further asymptotic formulas for the non-null distributions of three statistics for multivariate linear hypothesis

Summary New asymptotic expansions of the non-null distributions of the likelihood ratio, Hotelling's and Pillai's statistics for multivariate linear hypothesis are given in terms of normal distribution function and its derivatives, assuming the matrix of noncentrality parameters is of the same order as the sample size.     

A lifetime model with increasing failure rate

This paper deals with a new two-parameter lifetime distribution with increasing failure rate. This distribution is constructed as a distribution of a random sum of independent exponential random variables when the sample size has a zero truncated binomial distribution. Various statistical properties of the distribution are derived. We estimate the parameters by maximum likelihood and obtain the Fisher information matrix. Simulation studies show the performance of the estimators. Also, estimation of the parameters is considered in the presence of censoring. A real data set is analyzed for illustrative purposes and it is noted that the distribution is a good competitor to the gamma, Weibull, exponentiated exponential, weighted exponential and Poisson-exponential distributions for this data set.     

The asymptotic of the necessary sample size in testing the hypotheses on the shape parameter of a distribution close to the normal one

We study the problem of testing the hypothesis on the “approximate normality” formulated in terms of large values of the shape parameter of an asymptotically normal underlying distribution. Considering the examples of gamma-and generalized Birnbaum—Saunders distributions, we propose one way to obtain the asymptotic of the necessary sample size for testing the mentioned hypothesis. Our approach differs from those based on contiguous alternatives or on the use of the large deviations theory for distributions of sums of independent random variables. Our method yields remarkably precise approximate formulas, what is illustrated by numerical data.     

On approximation of the level probabilities for testing ordered parallel regression lines

Approximations for the level probabilities in testing order-restricted hypotheses are examined in this paper for generalized linear models with common slope. It is showed, in particular, that the asymptotic null distribution of the likelihood ratio statistic is a mixture of chi-squared distributions for the cases of simple order and simple tree order. Under a balanced structure, the asymptotic null distribution reduces to the well-known chi-bar-squared distribution. The use of the equal-weights level probabilities is also investigated. This approximation seems to be satisfactory when the sample sizes and the levels of the covariate are not too different among the strata.     

二元极值分布混合模型的矩估计

极值理论在各个领域得到了越来越多的关注和应用, 尤其是多元极值分布. 而矩估计是一种经典的参数估计方法, 计算简单且具有某些优良性, 本文给出边缘为标准指数分布的二元极值混合模型相关参数的矩估计及其渐近方差. 并将其与极大似然估计的渐近方差比较, 结果表明矩估计是一个较好的估计.     

Convergence rates of MLE in a partly linear model

This paper considers the estimation for a partly linear model with case 1 interval censored data. We assume that the error distribution belongs to a known family of scale distributions with an unknown scale parameter. The sieve maximum likelihood estimator (MLE) for the model’s parameter is shown to be strongly consistent, and the convergence rate of the estimator is obtained and discussed.     

A characterization of the one parameter exponential family of distributions by monotonicity of likelihood ratios

Summary Any one parameter exponential family of distributions has monotone likelihood ratios. As the product probabilities of n identical distributions of an exponential family form again an exponential family, it has monotone likelihood ratios for arbitrary n. Furthermore, the members of an exponential family are mutually absolutely continuous. In Part 1, we show that these properties uniquely characterize the exponential family. The application of this result to the theory of testing hypotheses (Part 2) shows that if a family of mutually absolutely continuous distributions has uniformly most powerful tests for arbitrary levels of significance, and arbitrary sample sizes, then it is necessarily an exponential family.The research was done while this author was a Visiting Professor in the Department of Statistics at the University of Chicago. It was supported by Research Grants Nos. NSF-G10368 and NSF-G21058 from the Division of Mathematical, Physical and Engineering Sciences of the National Science Foundation.     

Partial penalized empirical likelihood ratio test under sparse case

A consistent test via the partial penalized empirical likelihood approach for the parametric hypothesis testing under the sparse case, called the partial penalized empirical likelihood ratio (PPELR) test, is proposed in this paper. Our results are demonstrated for the mean vector in multivariate analysis and regression coefficients in linear models, respectively. And we establish its asymptotic distributions under the null hypothesis and the local alternatives of order n^?1/2 under regularity conditions. Meanwhile, the oracle property of the partial penalized empirical likelihood estimator also holds. The proposed PPELR test statistic performs as well as the ordinary empirical likelihood ratio test statistic and outperforms the full penalized empirical likelihood ratio test statistic in term of size and power when the null parameter is zero. Moreover, the proposed method obtains the variable selection as well as the p-values of testing. Numerical simulations and an analysis of Prostate Cancer data confirm our theoretical findings and demonstrate the promising performance of the proposed method in hypothesis testing and variable selection.     

Optimum step-stress partially accelerated life tests for the truncated logistic distribution with censoring

This paper deals with the optimal designing of step-stress partially accelerated life tests (PALTs) in which items are run at both accelerated and use conditions under censoring. It is assumed that the lifetime of the items follow truncated logistic distribution truncated at point zero. Truncated distributions arise when sample selection is not possible in some sub-region of the sample space. The logistic distribution is considered inappropriate for modeling lifetime data because left hand side of its distribution extends to negative infinity, and this could conceivably result in modeling negative times-to-failure. This has necessitated the use of truncated logistic distribution truncated at point zero for modeling lifetime data. Unlike the widely studied exponential, Weibull and lognormal life distributions, the failure rate of truncated logistic distribution is increasing and more realistically bounded below and above by non-zero finite quantity. The optimal change-time for the step PALT is determined by minimizing either the generalized asymptotic variance of maximum likelihood estimates (MLEs) of the acceleration factor and the hazard rate at use condition or the asymptotic variance of MLE of the acceleration factor. Inferential procedure involving model parameters and acceleration factor are studied. Sensitivity analysis is also performed.     

Estimation under failure-censored step-stress life test for the generalized exponential distribution parameters

Accelerated life testing of materials is used to collect failure data quickly when the lifetime of a specimen under use condition is too long. This article considers estimates of the generalized exponential distribution parameters under step-stress partially accelerated life testing with Type-II censoring. The maximum likelihood approach is applied to derive point and asymptotic confidence interval estimations of the model parameters. The performance of the estimators is evaluated numerically for different parameter values and different sample sizes via their mean square error. Also, the average confidence intervals lengths and the associated coverage probabilities are obtained. A simulation study is conducted for illustration.     

On limit distributions emerging in the generalized Birnbaum-Saunders model

We establish that the Birnbaum-Saunders distribution is the equilibrium mixture of the inverse Gaussian distribution and the convolution of this distribution with the chi-square distribution with a single degree of freedom. We give a physical interpretation of this phenomenon in terms of probabilistic models of fatigue life and introduce a general family of so-called crack distributions, which contains, in particular, the normal distribution, the inverse Gaussian distribution, and the Birnbaum-Saunders distributions, as well as others used in applications of the theory of reliability. We pose the problem of isolating these particular distributions lying on the boundary of the parametric space of a crack distribution; to solve this problem, we analyze the asymptotic behavior of the likelihood function of the maximal invariant for a random sample of a crack distribution as the sample size n grows and for large values of the form parameter λ, when the crack distribution is approximated by the normal distribution. In either case, the likelihood function asymptotically depends on the sample data only through the U-statistics Σ _i=1 ⁿ X_i Σ _i=1 ⁿ X _i ⁻¹ . This result allows us to construct asymptotically uniformly most powerful invariant tests for n→∞ or λ→∞; the latter “parametric” scheme of asymptotic analysis of the likelihood is similar to LeCam’s theory of statistical experiments, where large values of n are formally replaced by large values of λ, whereas the sample size n remains fixed. Proceedings of the Seminar on Stability Problems for Stochastic Models, Vologda, Russia, 1998, Part I.     

The generalized linear exponential distribution

This paper deals with a new generalization of the linear exponential distribution. This distribution is called the generalized linear exponential distribution (GLED). Some statistical properties such as moments, modes and quantiles are derived. The failure rate function and the mean residual lifetime are also discussed. The maximum likelihood estimators of the parameters are obtained using a simulation study. Real data are used to determine whether the GLED is better than other well-known distributions in modeling lifetime data or not.     

具有缺失数据的两个伽马分布总体参数的估计与假设检验

主要在数据缺失的情况下研究了伽马分布的参数估计与假设检验,位置参数已知的条件下,给出形状参数的极大似然估计,并证明了形状参数估计的强相合性与渐进正态性,并对两总体参数之差的置信区间和假设检验做出分析,最后做随机模拟验证了其合理性.     

The generalized Gompertz distribution

This paper deals with a new generalization of the exponential, Gompertz, and generalized exponential distributions. This distribution is called the generalized Gompertz distribution (GGD). The main advantage of this new distribution is that it has increasing or constant or decreasing or bathtub curve failure rate depending upon the shape parameter. This property makes GGD is very useful in survival analysis. Some statistical properties such as moments, mode, and quantiles are derived. The failure rate function is also derived. The maximum likelihood estimators of the parameters are derived using a simulations study. Real data set is used to determine whether the GGD is better than other well-known distributions in modeling lifetime data or not.     

The local power of the gradient test

The asymptotic expansion of the distribution of the gradient test statistic is derived for a composite hypothesis under a sequence of Pitman alternative hypotheses converging to the null hypothesis at rate n ^−1/2, n being the sample size. Comparisons of the local powers of the gradient, likelihood ratio, Wald and score tests reveal no uniform superiority property. The power performance of all four criteria in one-parameter exponential family is examined.     

Exponentiated generalized linear exponential distribution

A new generalization of the linear exponential distribution is recently proposed by Mahmoud and Alam [1], called as the generalized linear exponential distribution. Another generalization of the linear exponential was introduced by Sarhan and Kundu  and , named as the generalized linear failure rate distribution. This paper proposes a more generalization of the linear exponential distribution which generalizes the two. We refer to this new generalization as the exponentiated generalized linear exponential distribution. The new distribution is important since it contains as special sub-models some widely well known distributions in addition to the above two models, such as the exponentiated Weibull distribution among many others. It also provides more flexibility to analyze complex real data sets. We study some statistical properties for the new distribution. We discuss maximum likelihood estimation of the distribution parameters. Three real data sets are analyzed using the new distribution, which show that the exponentiated generalized linear exponential distribution can be used quite effectively in analyzing real lifetime data.     

Semi-empiricial Likelihood Confidence Intervals for the Differences of Two Populations Based on Fractional Imputation

Suppose that there are two populations x and y with missing data on both of them, where x has a distribution function F（·） which is unknown and y has a distribution function Gθ（·） with a probability density function gθ（·） with known form depending on some unknown parameter θ. Fractional imputation is used to fill in missing data. The asymptotic distributions of the semi-empirical likelihood ration statistic are obtained under some mild conditions. Then, empirical likelihood confidence intervals on the differences of x and y are constructed.