Marginal information for expectation parameters

The authors consider a special case of inference in the presence of nuisance parameters. They show that when the orthogonalized score function is a function of a statistic S, no Fisher information for the interest parameter is lost by using the marginal distribution of S rather than the full distribution of the observations. Therefore, no information for the interest parameter is recovered by conditioning on an ancillary statistic, and information will be lost by conditioning on an approximate ancillary statistic. This is the case for regular multivariate exponential families when the interest parameter is a subvector of the expectation parameter and the statistic is the maximum likelihood estimate of the interest parameter. Several examples are considered, including the 2 × 2 table.     

A new method for interval estimation of the mean of the Gamma distribution

The two parameter Gamma distribution is widely used for modeling lifetime distributions in reliability theory. There is much literature on the inference on the individual parameters of the Gamma distribution, namely the shape parameter k and the scale parameter θ when the other parameter is known. However, usually the reliability professionals have a major interest in making statistical inference about the mean lifetime μ, which equals the product θk for the Gamma distribution. The problem of inference on the mean μ when both parameters θ and k are unknown has been less attended in the literature for the Gamma distribution. In this paper we review the existing methods for interval estimation of μ. A comparative study in this paper indicates that the existing methods are either too approximate and yield less reliable confidence intervals or are computationally quite complicated and need advanced computing facilities. We propose a new simple method for interval estimation of the Gamma mean and compare its performance with the existing methods. The comparative study showed that the newly proposed computationally simple optimum power normal approximation method works best even for small sample sizes.     

Modified p-Value of Two-Sided Test for Normal Distribution with Restricted Parameter Space

This article proposes a modified p-value for the two-sided test of the location of the normal distribution when the parameter space is restricted. A commonly used test for the two-sided test of the normal distribution is the uniformly most powerful unbiased (UMPU) test, which is also the likelihood ratio test. The p-value of the test is used as evidence against the null hypothesis. Note that the usual p-value does not depend on the parameter space but only on the observation and the assumption of the null hypothesis. When the parameter space is known to be restricted, the usual p-value cannot sufficiently utilize this information to make a more accurate decision. In this paper, a modified p-value (also called the rp-value) dependent on the parameter space is proposed, and the test derived from the modified p-value is also shown to be the UMPU test.     

Graphical Tests for the Assumption of Gamma and Inverse Gaussian Frailty Distributions

The common choices of frailty distribution in lifetime data models include the Gamma and Inverse Gaussian distributions. We present diagnostic plots for these distributions when frailty operates in a proportional hazards framework. Firstly, we present plots based on the form of the unconditional survival function when the baseline hazard is assumed to be Weibull. Secondly, we base a plot on a closure property that applies for any baseline hazard, namely, that the frailty distribution among survivors at time t has the same form as the original distribution, with the same shape parameter but different scale parameter. We estimate the shape parameter at different values of t and examine whether it is constant, that is, whether plotted values form a straight line parallel to the time axis. We provide simulation results assuming Weibull baseline hazard and an example to illustrate the methods.     

On the validation of fiducial techniques

A structured model is essentially a family of random vectors X_θ defined on a probability space with values in a sample space. If, for a given sample value x and for each ω in the probability space, there is at most one parameter value θ for which X_θ(ω) is equal to x, then the model is called additive at x. When a certain conditional distribution exists, a frequency interpretation specific to additive structured models holds, and is summarized in a unique structured distribution for the parameter. Many of the techniques used by Fisher in deriving and handling his fiducial probability distribution are shown to be valid when dealing with a structured distribution.     

Analysis of survival data by an exponential-generalized Poisson distribution

In this paper, we consider the distribution of life length of a series system with random number of components, say Z. Considering the distribution of Z as generalized Poisson, an exponential-generalized Poisson (EGP) distribution is developed. The generalized Poisson distribution is a generalization of the Poisson distribution having one extra parameter. The structural properties of the resulting distribution are presented and the maximum likelihood estimation of the parameters is investigated. Extensive simulation studies are carried out to study the performance of the estimates. The score test is developed to test the importance of the extra parameter. For illustration, two real data sets are examined and it is shown that the EGP model, presented here, fits better than the exponential–Poisson distribution.     

On a Relation Between a Family of Distributions Attaining the Bhattacharyya Bound and That of Linear Combinations of the Distributions from an Exponential Family

Abstract

An unbiased estimation problem of a function g(θ) of a real parameter is considered. A relation between a family of distributions for which an unbiased estimator of a function g(θ) attains the general order Bhattacharyya lower bound and that of linear combinations of the distributions from an exponential family is discussed. An example on a family of distributions involving an exponential and a double exponential distributions with a scale parameter is given. An example on a normal distribution with a location parameter is also given.     

The two-sided power-binomial distribution for modeling binary count data

The two-sided power (TSP) distribution is a flexible two-parameter distribution having uniform, power function and triangular as sub-distributions, and it is a reasonable alternative to beta distribution in some cases. In this work, we introduce the TSP-binomial model which is defined as a mixture of binomial distributions, with the binomial parameter p having a TSP distribution. We study its distributional properties and demonstrate its use on some data. It is shown that the newly defined model is a useful candidate for overdispersed binomial data.     

Improved estimation of scale parameters of morgenstern type bivariate uniform distribution using ranked set sampling

ABSTRACT

In this article we suggest some improved version of estimators of scale parameter of Morgenstern-type bivariate uniform distribution (MTBUD) based on the observations made on the units of the ranked set sampling regarding the study variable Y which is correlated with the auxiliary variable X, when (X, Y) follows a MTBUD. We also suggest some linear shrinkage estimators of scale parameter of Morgenstern type bivariate uniform distribution (MTBUD). Efficiency comparisons are also made in this work.     

On the q-Weibull Distribution and Its Applications

The q-Weibull distribution is a stretched model for Weibull distribution, obtained by introducing a new pathway parameter q, which facilitates a slow transition to the Weibull as q → 1. In this article, we make a detailed study of the properties of the q-Weibull distribution and we apply it to a data on cancer remission times for which this distribution is a better fit than Weibull. Results relating to reliability properties, estimation of parameters, and applications in stress-strength analysis are also obtained.     

Estimation of a Gamma Scale Parameter Under Asymmetric Squared Log Error Loss

Abstract

Estimation of scale parameter under the squared log error loss function is considered with restriction to the principle of invariance and risk unbiasedness. An explicit form of minimum risk scale-equivariant estimator under this loss is obtained. The admissibility and inadmissibility of a class of linear estimators of the form (cT + d) are considered, where T follows a gamma distribution with an unknown scale parameter η and a known shape parameter ν. This includes the admissibility of the minimum risk equivariant estimator on η (MRE).     

The Likelihood Ratio Test for the Presence of Immunes in a Censored Sample

Models for populations with immune or cured individuals but with others subject to failure are important in many areas, such as medical statistics and criminology. One method of analysis of data from such populations involves estimating an immune proportion 1 ? p and the parameter(s) of a failure distribution for those individuals subject to failure. We use the exponential distribution with parameter λ for the latter and a mixture of this distribution with a mass 1 ? p at infinity to model the complete data. This paper develops the asymptotic theory of a test for whether an immune proportion is indeed present in the population, i.e., for H ₀:p = 1. This involves testing at the boundary of the parameter space for p. We use a likelihood ratio test for H ₀. and prove that minus twice the logarithm of the likelihood ratio has as an asymptotic distribution, not the chi-square distribution, but a 50–50 mixture of a chi-square distribution with 1 degree of freedom, and a point mass at 0. The result is proved under an independent censoring assumption with very mild restrictions.     

α-stable laws for noncoding regions in DNA sequences

In this work, we analyze the long-range dependence parameter for a nucleotide sequence in several different transformations. The long-range dependence parameter is estimated by the approximated maximum likelihood method, by a novel estimator based on the spectral envelope theory, by a regression method based on the periodogram function, and also by the detrended fluctuation analysis method. We study the length distribution of coding and noncoding regions for all Homo sapiens chromosomes available from the European Bioinformatics Institute. The parameter of the tail rate decay is estimated by the Hill estimator ?α. We show that the tail rate decay is greater than 2 for coding regions, while for almost all noncoding regions it is less than 2.     

On grouping sets of exponential populations

Suppose we have k random samples each of size n from a two parameter exponential distribution with location parameters μ _i i=1,…,k, and where each item has the same, unknown scale parameter. A multistage procedure is developed to determine t≦k groups such that in any one group the distributions have μ_i's that are not appreciably different. The method yields a unique grouping and extends the approach of the Kumar and Pate1 test.The emphasis is on the development of a procedure based on the null sampling distribution of the maximum gap of the ordered first order statistics from exponential distributions. The procedure is based on complete ordered samples or censored (of any or of all) samples.     

A bivariate F distribution with marginals on arbitrary numerator and denominator degrees of freedom, and related bivariate beta and t distributions

The classical bivariate F distribution arises from ratios of chi-squared random variables with common denominators. A consequent disadvantage is that its univariate F marginal distributions have one degree of freedom parameter in common. In this paper, we add a further independent chi-squared random variable to the denominator of one of the ratios and explore the extended bivariate F distribution, with marginals on arbitrary degrees of freedom, that results. Transformations linking F, beta and skew t distributions are then applied componentwise to produce bivariate beta and skew t distributions which also afford marginal (beta and skew t) distributions with arbitrary parameter values. We explore a variety of properties of these distributions and give an example of a potential application of the bivariate beta distribution in Bayesian analysis.     

Inference for the scale parameter of lifetime distribution of k-unit parallel system based on progressively censored data

In this paper, inference for the scale parameter of lifetime distribution of a k-unit parallel system is provided. Lifetime distribution of each unit of the system is assumed to be a member of a scale family of distributions. Maximum likelihood estimator (MLE) and confidence intervals for the scale parameter based on progressively Type-II censored sample are obtained. A β-expectation tolerance interval for the lifetime of the system is obtained. As a member of the scale family, half-logistic distribution is considered and the performance of the MLE, confidence intervals and tolerance intervals are studied using simulation.     

Computational comparison of the general weighted moments estimators of the scale parameter of a Pareto distribution with a known shape parameter with other estimators based on a multiply type II-censored sample

Wu et al. [Computational comparison for weighted moments estimators and BLUE of the scale parameter of a Pareto distribution with known shape parameter under type II multiply censored sample, Appl. Math. Comput. 181 (2006), pp. 1462–1470] proposed the weighted moments estimators (WMEs) of the scale parameter of a Pareto distribution with known shape parameter on a multiply type II-censored sample. They claimed that some WMEs are better than the best linear unbiased estimator (BLUE) based on the exact mean-squared error (MSE). In this paper, the general WME (GWME) is proposed and the computational comparison of the proposed estimator with the WMEs and BLUE is done on the basis of the exact MSE for given sample sizes and different censoring schemes. As a result, the GWME is performing better than the best estimator among 12 WMEs and BLUE for all cases. Therefore, GWME is recommended for use. At last, one example is given to demonstrate the proposed GWME.     

Power and Sample Size Determination When the Alternative Hypotheses are Given in Quantiles

The power of normal-theory tests about means depends on a noncentrality parameter which is a function of the unknown parameter σ. In order to calculate power and to solve sample-size problems based on power, differences between hypothesized and alternative values of the means are frequently selected as a multiple of σ, a choice which eliminates σ from the noncentrality parameter and permits a solution. Perhaps a more natural (but equivalent) way to express alternatives is to give one or more means as the quantile of order p (say Q_p ) of a distribution with another mean. As we will demonstrate, this kind of alternative also eliminates σ from the problem.     

Goodness-of-fit test for uniform stochastic ordering among several distributions

The likelihood-ratio test (LRT) is considered as a goodness-of-fit test for the null hypothesis that several distribution functions are uniformly stochastically ordered. Under the null hypothesis, H₁ : F₁ ? F₂ ?···? F_N, the asymptotic distribution of the LRT statistic is a convolution of several chi-bar-square distributions each of which depends upon the location parameter. The least-favourable parameter configuration for the LRT is not unique. It can be two different types and depends on the number of distributions, the number of intervals and the significance level α. This testing method is illustrated with a data set of survival times of five groups of male fruit flies.     

A flexible parametric survival model which allows a bathtub-shaped hazard rate function

A new parametric (three-parameter) survival distribution, the lognormal–power function distribution, with flexible behaviour is introduced. Its hazard rate function can be either unimodal, monotonically decreasing or can exhibit a bathtub shape. Special cases include the lognormal distribution and the power function distribution, with finite support. Regions of parameter space where the various forms of the hazard-rate function prevail are established analytically. The distribution lends itself readily to accelerated life regression modelling. Applications to five data sets taken from the literature are given. Also it is shown how the distribution can behave like a Weibull distribution (with negative aging) for certain parameter values.