Problems with Maximum Likelihood Estimation and the 3 Parameter Gamma Distribution

The three parameters involved are scale a , shape 𝜌 , and location s . Maximum likelihood estimators are (\hata, \hat\rho, \hats) . Using recent work on the second order variances, skewness, and kurtosis we establish the facts, that if the location parameter s is to be estimated, then the asymptotic variances only exist if 𝜌 >2, asymptotic skewness only exists if 𝜌 >3, and 2nd order variances and third order fourth central moments only exist if 𝜌 >4. The result of these limitations is that in general very large sample sizes may be needed to avoid inference problems. We also include new continued fractions for the asymptotic covariances of the maximum likelihood estimators considered.     

Parameters estimation for weibull distributed lifetimes under progressive censoring with random removeals

This paper considers the estimation problem when lifetimes are Weibull distributed and are collected under a Type-II progressive censoring with random removals, where the number of units removed at each failure time follows a uniform discrete distribution. The expected time of this censoring plan is discussed and compared numerically to that under a Type II censoring without removal. Maximum likelihood estimator of the parameters and their asymptotic variances are derived.     

Order statistics from the linear-exponential distribution,part I: increasing hazard rate case

For the linear-exponential distribution with increasing hazard rate, exact and explicit expressions for means, product moments and percentage points of order statistics are obtained. Some recurrence relations for both single and product moments of order statistics are also derived. These recurrence relations would enable one to obtain all the higher order moments of order statistics for all sample sizes from those of the lower order     

产品危害事件对品牌资产变动的影响

本文采用达沃和皮洛特的分析框架和两项实验,探索了产品危害事件中公司反应和消费者先前预期是如何影响品牌资产变动的。结果发现,消费者对公司反应的解释取决于其先前预期:出现产品危害事件时,不管公司采用何种反应措施,强预期条件下较弱预期条件下造成的品牌资产损失小。     

On the Variances and Convariances of the Duration State Sizes of Semi-Markov Systems

In the present article we study the asymptotic behaviour of the sequence of vectors with components expectations, variances, and covariances of the state sizes of a semi-Markov system. In this respect, we transform the semi-Markov system into a Markov system with a different though equivalent state space and relate the sequence of the transition probabilities of the respective Markov system as functions of the parameters of the semi-Markov system. Also, we study the asymptotic behavior of the sequence of vectors with components variances and covariances of the duration state sizes, of the related Markov system, under perturbation of the transition probability matrices. We use the results in the study of asymptotic behavior under perturbation of the sequence of vectors with components variances and covariances of the semi-Markov system.     

Approximations of Variances and Covariances for Order Statistics from the Standard Extreme Value Distribution

We consider simple approximations of variances and covariances for order statistics from the standard extreme value distribution. Exact values and simulation results of the variances and covariances for certain sample sizes are used to determine the validity of the suggested approximations.     

Testing for mean and variance differences with samples from distributions that may be non-normal with unequal variances

The problem of comparing several samples to decide whether the means and/or variances are significantly different is considered. It is shown that with very non-normal distributions even a very robust test to compare the means has poor properties when the distributions have different variances, and therefore a new testing scheme is proposed. This starts by using an exact randomization test for any significant difference (in means or variances) between the samples. If a non-significant result is obtained then testing stops. Otherwise, an approximate randomization test for mean differences (but allowing for variance differences) is carried out, together with a bootstrap procedure to assess whether this test is reliable. A randomization version of Levene's test is also carried out for differences in variation between samples. The five possible conclusions are then that (i) there is no evidence of any differences, (ii) evidence for mean differences only, (iii) evidence for variance differences only, (iv) evidence for mean and variance differences, or (v) evidence for some indeterminate differences. A simulation experiment to assess the properties of the proposed scheme is described. From this it is concluded that the scheme is useful as a robust, conservative method for comparing samples in cases where they may be from very non-normal distributions.     

Covariance Analysis for Split-Plot and Split-Block Designs

Commentaries are informative essays dealing with viewpoints of statistical practice, statistical education, and other topics considered to be of general interest to the broad readership of The American Statistician. Commentaries are similar in spirit to Letters to the Editor, but they involve longer discussions of background, issues, and perspectives. All commentaries will be refereed for their merit and compatibility with these criteria.

Proper methodology for the analysis of covariance for experiments designed in a split-plot or split-block design is not found in the statistical literature. Analyses for these designs are often performed incompletely or even incorrectly. This is especially true when popular statistical computer software packages are used for the analysis of these designs. This article provides several appropriate models, ANOVA tables, and standard errors for comparisons from experiments arranged in a standard split-plot, split–split-plot, or split-block design where a covariate has been measured on the smallest size experimental unit.     

New polya and inverse polya distributions of order k

New Polya and inverse Polya distributions of order k are derived by means of generalized urn models and by compounding the binomial and negative binomial distributions of order k of Philippou (1986, 1983) with the beta distribution. It i s noted that the present Polpa distribution of order k includes as special cases a new hypergeometric distribution of order k, a negative one,an inverse one, and a discrete uniform of the same order. The probability generating functions, means and variances of the new distributions are obtained, and five asymptotic results are established relating them to the abovedmentioned binomial and negative binomial distributions of order k, and to the Poisson distribution of the same order of Philippou (1983).Moment estimates are also given and applications are indicated.     

A note on scheffé's method of deriving confidence sets for directions and ratios of normals

Scheffé (1970) introduced a method for deriving confidence sets for directions and ratios of normals. The procedure requires use of an approximation and Scheffé provided evidence that the method performs well for cases in which the variances of the random deviates are known. This paper extends Scheffé's numerical integrations to the case of unknown variances. Our results indicate that Scheffé's method works well when variances are unknown