Modern likelihood inference for the maximum/minimum of a bivariate normal vector

ABSTRACT

We consider the use of modern likelihood asymptotics in the construction of confidence intervals for the parameter which determines the skewness of the distribution of the maximum/minimum of an exchangeable bivariate normal random vector. Simulation studies were conducted to investigate the accuracy of the proposed methods and to compare them to available alternatives. Accuracy is evaluated in terms of both coverage probability and expected length of the interval. We furthermore illustrate the suitability of our proposals by means of two data sets, consisting of, respectively, measurements taken on the brains of 10 mono-zygotic twins and measurements of mineral content of bones in the dominant and non-dominant arms for 25 elderly women.     

Saddlepoint density and distribution functions for the ratio of two linear functions and the product of generalized gamma variates

The generalized gamma distribution is a flexible and attractive distribution because it incorporates several well-known distributions, i.e., gamma, Weibull, Rayleigh, and Maxwell. This article derives saddlepoint density and distribution functions for the ratio of two linear functions of generalized gamma variables and the product of n independent generalized gamma variables. Simulation studies are used to evaluate the accuracy of the saddlepoint approximations. The saddlepoint approximations are fast, easy, and very accurate.     

Bounds on the mean of the maximum of a number of normal random variables

A simple method producing lower and upper bounds on E max(X₁,...,X_n) is presented under assumption that the X_i's are independent normal random variables. Furthermore the upper bounds are determined when the X_i's are normal and positively correlated     

The density of the skew normal sample mean and its applications

This paper focuses on the distribution of the skew normal sample mean. For a random sample drawn from a skew normal population, we derive the density function and the moment generating function of the sample mean. The density function derived can be used for statistical inference on the disease occurrence time of twins in epidemiology, in which the skew normal model plays a key role.     

Computational testing algorithmic procedure of assessment for lifetime performance index of products with two-parameter exponential distribution based on the multiply type II censored sample

Process capability indices have been widely used to evaluate the process performance to the continuous improvement of quality and productivity. Since the lifetime of products may possess a two-parameter exponential distribution, 14 estimators are used to estimate the lifetime performance index based on the multiply type II censored sample. These estimators are utilized to develop the new hypothesis testing algorithmic procedure in the condition of known L. Finally, two practical examples are illustrated to employ the testing algorithmic procedure to determine whether the process is capable.     

Testing high-dimensional normality based on classical skewness and Kurtosis with a possible small sample size

Abstract

By using the idea of principal component analysis, we propose an approach to applying the classical skewness and kurtosis statistics for detecting univariate normality to testing high-dimensional normality. High-dimensional sample data are projected to the principal component directions on which the classical skewness and kurtosis statistics can be constructed. The theory of spherical distributions is employed to derive the null distributions of the combined statistics constructed from the principal component directions. A Monte Carlo study is carried out to demonstrate the performance of the statistics on controlling type I error rates and a simple power comparison with some existing statistics. The effectiveness of the proposed statistics is illustrated by two real-data examples.     

Simulation on probability points for testing of lognormal or weibull distribution with a small sample

Two extensive computer simulated tables of percentage points of the asymptotic test statistics for testing lognormal or Weibull population proposed by Pereira (1978) are discussed. Special attention is given to small sample cases. Some of the most commonly used 16 symmetrical probability points are reported. These points are 0.001, 0.005, 0.01. 002. 0.025. 0.05. 0.10.0.15, 0.85, 0.90, 0.95, 0.975, 0.98, 0.99, 0.995 and 0.999. These empirical Sumulated results can be used to test hypotheses for these two particular populations and are adequate when using a normal approximation.     

The stationary bootstrap for the joint distribution of sum and maximum of stationary sequences

In this work, the stationary bootstrap procedure is used to estimate the joint distribution of sum and maximum of strictly stationary strong mixing sequences. Asymptotic validity is established for stationary bootstrapping of the joint distribution of sum and maximum.     

On minimum distance estimation of location based on the kolmogorov statistic

The asymptotic distribution is derived for the minimum distance estimator of a location parameter based on the Kolmogorov goodness of fit statistic. The distribution is expressed in terms of the distribution of a functional of a Brownian bridge. An upper bound is obtained for the length of the confidence interval based on the Kolmogorov statistic. A simulation study with sample sizes 10 and 20 compares the length of the interval based on the Kolmogorov statistic to the length of the interval based on the maximum likelihood estimator. Another simulation shows the effect of model misspecification on the coverage probabilities of the interval based on the Kolmogorov statistic.     

Exact confidence region for the parameters of the gamma distribution based on two independent samples

The gamma distribution has been discussed by many authors. This article proposes an exact confidence region for the parameters of a two-parameter gamma distribution. The result is based on the fact that the percentiles of the F-distribution, with equal degrees of freedom k, are monotonic in k.     

A note on the mean of the order statistics based on independent exponential random variables

Upper and lower bounds are obtained on the mean of the r-th smallest order statistics based on n independent exponential random variables under a certain condition on r.     

A gamma goodness-of-fit test based on characteristic independence of the mean and coefficient of variation

Characterization theorems in probability and statistics are widely appreciated for their role in clarifying the structure of the families of probability distributions. Less well known is the role characterization theorems have as a natural, logical and effective starting point for constructing goodness-of-fit tests. The characteristic independence of the mean and variance and of the mean and the third central moment of a normal sample were used, respectively, by Lin and Mudholkar [1980. A simple test for normality against asymmetric alternatives. Biometrika 67, 455–461] and by Mudholkar et al. [2002a. Independence characterizations and testing normality against skewness-kurtosis alternatives. J. Statist. Plann. Inference 104, 485–501] for developing tests of normality. The characteristic independence of the maximum likelihood estimates of the population parameters was similarly used by Mudholkar et al. [2002b. Independence characterization and inverse Gaussian goodness-of-fit. Sankhya A 63, 362–374] to develop a test of the composite inverse Gaussian hypothesis. The gamma models are extensively used for applied research in the areas of econometrics, engineering and biomedical sciences; but there are few goodness-of-fit tests available to test if the data indeed come from a gamma population. In this paper we employ Hwang and Hu's [1999. On a characterization of the gamma distribution: the independence of the sample mean and the sample coefficient of variation. Ann. Inst. Statist. Math. 51, 749–753] characterization of the gamma population in terms of the independence of sample mean and coefficient of variation for developing such a test. The asymptotic null distribution of the proposed test statistic is obtained and empirically refined for use with samples of moderate size.     

Exact likelihood inference based on an unified hybrid censored sample from the exponential distribution

Recently, exact confidence bounds and exact likelihood inference have been developed based on hybrid censored samples by Chen and Bhattacharyya [Chen, S. and Bhattacharyya, G.K. (1998). Exact confidence bounds for an exponential parameter under hybrid censoring. Communications in Statistics—Theory and Methods, 17, 1857–1870.], Childs et al. [Childs, A., Chandrasekar, B., Balakrishnan, N. and Kundu, D. (2003). Exact likelihood inference based on Type-I and Type-II hybrid censored samples from the exponential distribution. Annals of the Institute of Statistical Mathematics, 55, 319–330.], and Chandrasekar et al. [Chandrasekar, B., Childs, A. and Balakrishnan, N. (2004). Exact likelihood inference for the exponential distribution under generalized Type-I and Type-II hybrid censoring. Naval Research Logistics, 51, 994–1004.] for the case of the exponential distribution. In this article, we propose an unified hybrid censoring scheme (HCS) which includes many cases considered earlier as special cases. We then derive the exact distribution of the maximum likelihood estimator as well as exact confidence intervals for the mean of the exponential distribution under this general unified HCS. Finally, we present some examples to illustrate all the methods of inference developed here.     

Assessing the lifetime performance index with Lomax distribution based on progressive type I interval censored sample

In manufacturing industry, the lifetime performance index CL is applied to evaluate the larger-the-better quality features of products. It can quickly show whether the lifetime performance of products meets the desired level. In this article, first we obtain the maximum likelihood estimator of CL with two unknown parameters in the Lomax distribution on the basis of progressive type I interval censored sample. With the MLE we proposed, some asymptotic confidence intervals of CL are discussed by using the delta method. Furthermore, the MLE of CL is used to establish the hypothesis test procedure under a given lower specification limit L. In addition, we also conduct a hypothesis test procedure when the scale parameter in the Lomax distribution is given. Finally, we illustrate the proposed inspection procedures through a real example. The testing procedure algorithms presented in this paper are efficient and easy to implement.     

Bayesian estimation and prediction based on generalized Type-II hybrid censored sample

ABSTRACT

In this paper, we consider a general form for the underlying distribution and a general conjugate prior, and develop a general procedure for deriving the maximum likelihood and Bayesian estimators based on an observed generalized Type-II hybrid censored sample. The problems of predicting the future order statistics from the same sample and that from a future sample are also discussed from a Bayesian viewpoint. For the illustration of the developed results, the exponential and Pareto distributions are used as examples. Finally, two numerical examples are presented for illustrating all the inferential procedures developed here.     

Bayesian estimation and prediction based on generalized Type-I hybrid censored sample

In this paper, we consider an exponential form for the underlying distributionand a conjugate prior, and develop a procedure for deriving the maximum likelihood and Bayesian estimators based on an observed generalized Type-I hybrid censored sample. The problems of predicting the future order statistics from the same sample and that from a future sample are also discussed from a Bayesian viewpoint. For the illustration of the developed results, the exponential and Pareto distributions are used as examples. Finally, two numerical examples are presented for illustrating all the inferential procedures developed here.     

Interval estimation for the Pareto distribution based on the progressive Type II censored sample

For the complete sample and the right Type II censored sample, Chen [Joint confidence region for the parameters of Pareto distribution. Metrika 44 (1996), pp. 191–197] proposed the interval estimation of the parameter θ and the joint confidence region of the two parameters of Pareto distribution. This paper proposed two methods to construct the confidence region of the two parameters of the Pareto distribution for the progressive Type II censored sample. A simulation study comparing the performance of the two methods is done and concludes that Method 1 is superior to Method 2 by obtaining a smaller confidence area. The interval estimation of parameter ν is also given under progressive Type II censoring. In addition, the predictive intervals of the future observation and the ratio of the two future consecutive failure times based on the progressive Type II censored sample are also proposed. Finally, one example is given to illustrate all interval estimations in this paper.     

Approximate maximum likelihood estimation of the mean and standard deviation of the normal distribution based on type ii censored samples

It is known that the maximum likelihood methods does not provide explicit estimators for the mean and standard deviation of the normal distribution based on Type II censored samples. In this paper we present a simple method of deriving explicit estimators by approximating the likelihood equations appropriately. We obtain the variances and covariance of these estimators. We also show that these estimators are almost as eficient as the maximum likelihood (ML) estimators and just as eficient as the best linear unbiased (BLU), and the modified maximum likelihood (MML) estimators. Finally, we illustrate this method of estimation by applying it to Gupta's and Darwin's data.     

Efficiency estimation of type-I censored sample from the Weibull distribution based on sup-entropy

On the basis of Awad sup-entropy, the efficiency function for type-I censored sample from the Weibull distribution is numerically introduced. The properties of the derived efficiency are discussed. Furthermore, for a given efficiency, the termination time of the experiment, and the maximum likelihood estimates for the Weibull parameters, are proposed. Simulation results are tabulated and discussed. Censored and complete samples are compared for a wide range of the efficiency. The comparisons show the quality of the developed algorithms and the effectiveness of using censoring in estimating with the Weibull distribution.