Exact and approximate runs distributions

Let R = R_n denote the total (and unconditional) number of runs of successes or failures in a sequence of n Bernoulll (p) trials, where p is assumed to be known throughout. The exact distribution of R is related to a convolution of two negative binomial random variables with parameters p and q (=1-p). Using the representation of R as the sum of 1 - dependent indicators, a Berry - Esséen theorem is derived; the obtained rate of sup norm convergence is O(n^-½). This yields an unconditional version of the classical result of Wald and Wolfowitz (1940). The Stein - Chen method for m - dependent random variables is used, together with a suitable coupling, to prove a Poisson limit theorem for R. but with the limiting support set being the set of odd integers, Total variation error bounds (of order O(p) are found for the last result. Applications are indicated.     

Reference priors for a product of normal means when variances are unknown

The reference priors of Berger and Bernardo (1992) are derived for normal populations with unknown variances when the product of means is of interest. The priors are also shown to be Tibshirani's (1989) matching priors.     

Noninformative priors for linear combinations of the normal means

In this paper, we develop noninformative priors for linear combinations of the means under the normal populations. It turns out that among the reference priors the one-at-a-time reference prior satisfies a second order probability matching criterion. Moreover, the second order probability matching priors match alternative coverage probabilities up to the second order and are also HPD matching priors. Our simulation study indicates that the one-at-a-time reference prior performs better than the other reference priors in terms of matching the target coverage probabilities in a frequentist sense.     

Exact and approximate distributions of the chi-square statistic for equiprobability

The distribution of the chi-square goodness-of-fit statistic is studied in the equiprobable case. Tables of exact critical values are given for a = .1, .05, .01, .005; k = 2(1)4, N = 26(1)50; k = 5, N = 26(1)40; k = 6(1)10, N = 26(1)30, where a is the desired significance level, k is the number of cells and N is the sample size. Methods of fitting the true distribution are compared. If k> 3, it is found that a simple additive adjustment to the asymptotic chi-square fit leads to high accuracy even for N between 10 and 20. For k = 2, the Yates corrected chi-square statistic is very accurately fitted by the usual chi-square distribution.     

Comparing variances and means when distributions have non-identical shapes

The paper compares several methods for computing robust 1-α confidence intervals for σ ₁ ²-σ ₂ ², or σ ₁ ²/σ ₂ ², where σ ₁ ² and σ ₂ ² are the population variances corresponding to two independent treatment groups. The emphasis is on a Box-Scheffe approach when distributions have different shapes, and so the results reported here have implications about comparing means. The main result is that for unequal sample sizes, a Box-Scheffe approach can be considerably less robust than indicated by past investigations. Several other procedures for comparing variances, not based on a Box-Scheffe approach, were also examined and found to be highly unsatisfactory although previously published papers found them to be robust when the distributions have identical shapes. Included is a new result on why the procedures examined here are not robust, and an illustration that increasing σ ₁ ²-σ ₂ ² can reduce power in certain situations. Constants needed to apply Dunnett’s robust comparison of means are included.     

Exact and approximate distributions for the product of inverted Dirichlet components

It is well known that X/(X+Y) has the beta distribution when X and Y follow the inverted Dirichlet distribution. In this paper, we derive the exact distribution of the product P=XY (involving the Gauss hypergeometric function) and the corresponding moment properties. We also propose two approximations and show evidence of their goodness of fit. The work is motivated by real-life examples.     

Bayesian confidence intervals for means and variances of lognormal and bivariate lognormal distributions

The lognormal distribution is currently used extensively to describe the distribution of positive random variables. This is especially the case with data pertaining to occupational health and other biological data. One particular application of the data is statistical inference with regards to the mean of the data. Other authors, namely Zou et al. (2009), have proposed procedures involving the so-called “method of variance estimates recovery” (MOVER), while an alternative approach based on simulation is the so-called generalized confidence interval, discussed by Krishnamoorthy and Mathew (2003). In this paper we compare the performance of the MOVER-based confidence interval estimates and the generalized confidence interval procedure to coverage of credibility intervals obtained using Bayesian methodology using a variety of different prior distributions to estimate the appropriateness of each. An extensive simulation study is conducted to evaluate the coverage accuracy and interval width of the proposed methods. For the Bayesian approach both the equal-tail and highest posterior density (HPD) credibility intervals are presented. Various prior distributions (Independence Jeffreys' prior, Jeffreys'-Rule prior, namely, the square root of the determinant of the Fisher Information matrix, reference and probability-matching priors) are evaluated and compared to determine which give the best coverage with the most efficient interval width. The simulation studies show that the constructed Bayesian confidence intervals have satisfying coverage probabilities and in some cases outperform the MOVER and generalized confidence interval results. The Bayesian inference procedures (hypothesis tests and confidence intervals) are also extended to the difference between two lognormal means as well as to the case of zero-valued observations and confidence intervals for the lognormal variance. In the last section of this paper the bivariate lognormal distribution is discussed and Bayesian confidence intervals are obtained for the difference between two correlated lognormal means as well as for the ratio of lognormal variances, using nine different priors.     

Noninformative priors for linear combinations of normal means with unequal variances

For normal populations with unequal variances, we develop matching priors and reference priors for a linear combination of the means. Here, we find three second-order matching priors: a highest posterior density (HPD) matching prior, a cumulative distribution function (CDF) matching prior, and a likelihood ratio (LR) matching prior. Furthermore, we show that the reference priors are all first-order matching priors, but that they do not satisfy the second-order matching criterion that establishes the symmetry and the unimodality of the posterior under the developed priors. The results of a simulation indicate that the second-order matching prior outperforms the reference priors in terms of matching the target coverage probabilities, in a frequentist sense. Finally, we compare the Bayesian credible intervals based on the developed priors with the confidence intervals derived from real data.     

Improved estimators for the common mean and ordered means of two normal distributions with ordered variances

We first consider the problem of estimating the common mean of two normal distributions with unknown ordered variances. We give a broad class of estimators which includes the estimators proposed by Nair (1982) and Elfessi et al. (1992) and show that the estimators stochastically dominate the estimators which do not take into account the order restriction on variances, including the one given by Graybill and Deal (1959). Then we propose a broad class of individual estimators of two ordered means when unknown variances are ordered. We show that in estimating the mean with larger variance, estimators which do not take into account the order restriction on variances are stochastically dominated by the proposed class of estimators which take into account both order restrictions. However, in estimating the mean with smaller variance, similar improvement is not possible even in terms of mean squared error. We also show a domination result in the simultaneous estimation problem of two ordered means. Further, improving upon the unbiased estimators of the two means is discussed.     

A stein two-sample procedure for the general linear model with unequal error variances

The pronerties of the tests and confidence regions for the parameters in the classical general linear model depend upon the equality of the variances of the error terms. The level and power of tests and the confidence coefficients associated with confidence regions are vitiated when the assumption of equality is not true. Even when the error variances are equal the power of tests and the size of confidence regions depend upon the unknown common variance and hence are uncontrollable. This paper presents a two-stage procedure which yields tests and confidence regions which are completely independent of the variances of the errors and hence tests with controllable power and confidence regions of fixed controllable size are obtained.     

Bayesian analysis of multivariate t linear mixed models with missing responses at random

The multivariate t linear mixed model (MtLMM) has been recently proposed as a robust tool for analysing multivariate longitudinal data with atypical observations. Missing outcomes frequently occur in longitudinal research even in well controlled situations. As a powerful alternative to the traditional expectation maximization based algorithm employing single imputation, we consider a Bayesian analysis of the MtLMM to account for the uncertainties of model parameters and missing outcomes through multiple imputation. An inverse Bayes formulas sampler coupled with Metropolis-within-Gibbs scheme is used to effectively draw the posterior distributions of latent data and model parameters. The techniques for multiple imputation of missing values, estimation of random effects, prediction of future responses, and diagnostics of potential outliers are investigated as well. The proposed methodology is illustrated through a simulation study and an application to AIDS/HIV data.     

A modification of the kruskal-wallis statistic for the generalized behrens-fisher problem

The problem of testing the equality of the medians of several populations is considered. Standard distribution-free procedures for this problem require that the populations have the same shape in order to maintain their nominal significance level, ever asymptotically, under the null hypothesis of equal medians , A modification of the Kruskal-Wallis test statistic is proposed which is exactly distribution-free under the usual nonparanetric asswnption that the continuous populations are identical with any shape. It is asymptotically distribution-free when the Continuous populations are asswned to be syrmnetric with equal medians.     

Lad for Multiway Tables

It is shown that transformations based on differencing can be helpful in simplifying the problem of fitting to multiway tables. This is illustrated for LAD fitting, and the computational problems surveyed briefly.     

Testing the equality of several means when the population variances are unequal

A comparative study is made of three tests, developed by James (1951), Welch (1951) and Brown & Forsythe (1974). James presented two methods of which only one is considered in this paper. It is shown that this method gives better control over the size than the other two tests. None of these methods is uniformly more powerful than the other two. In some cases the tests of James and Welch reject a false null hypothesis more often than the test of Brown & Forsythe, but there are also situations in which it is the other way around.

We conclude that for implementation in a statistical software package the very complicated test of James is the most attractive. A practical disadvantage of this method can be overcome by a minor modification.     

Exact linear inference and prediction for exponential distributions based on general progressively type-ii censored samples

In this paper, we make use of an algorithm of Huffer and Lin (2001) in order to develop exact interval estimation for the location and scale parameters of an exponential distribution based on general progressively Type-II censored samples. The exact prediction intervals for failure times of the items censored at the last observation are also presented for one-parameter and two-parameter exponential distributions. Finally, we give two examples to illustrate the methods of inference developed here.     

Robust statistics for testing equality of means or variances

For testing the equality of means (location parameters) of two populations, Tiku (1980a) defined a statistic Tc (based on symmetrically censored samples) and showed that this statistic is robust to underlying populations and is also remarkably powerful. In this paper, we define a similar statistic T (based on samples s with observations censored only on one side) and show that this stat is tic is more powerful than T and nonparametric statistics, C for skew populations. We also provide a modification of this statistic for testing the equality of two population variances.     

Exact and approximate sum representations for the Dirichlet process

The Dirichlet process can be regarded as a random probability measure for which the authors examine various sum representations. They consider in particular the gamma process construction of Ferguson (1973) and the “stick‐breaking” construction of Sethuraman (1994). They propose a Dirichlet finite sum representation that strongly approximates the Dirichlet process. They assess the accuracy of this approximation and characterize the posterior that this new prior leads to in the context of Bayesian nonpara‐metric hierarchical models.     

On the linear combinations of the means of inverse gaussian distributions

Consider the two parameter Inverse Gaussian distribution with mean μ and scale parameter λ.

Suppose one is interested in testing a problem on a linear combination for the means of Inverse Gaussian distributions. For this problem a test and confidence intervals are proposed when: (1) λ’s are known and; (2) λ’s are unknown.

Finally an application of the procedures is illustrated with a data set of failure times of high-speed turbine bearings.     

A two sample ordered alternative test for means and variances

A two sample test of likelihood ratio type is proposed, assuming normal distribution theory, for testing the hypothesis that two samples come from identical normal populations versus the alternative that the populations are normal but vary in mean value and variance with one population having a smaller mean and smaller variance than the other. The small sample and large sample distribution of the proposed statistic are derived assuming normality. Some computations are presented which show the speed of convergence of small sample critical values to their asymptotic counterparts. Comparisons of local power of the proposed test are made with several potential competing tests. Asymptotics for the test statistic are derived when underlying distributions are not necessarily normal.