Shortest and longest length of success runs in binary sequences

The shortest and the longest length of success runs statistics in binary sequences are considered. The sequences are arranged on a line or on a circle. Exact probabilities of these statistics are derived, both in closed formulae via combinatorial analysis, as well as recursively. Furthermore, their joint probability distribution function and cumulative distribution function are obtained. The results are developed first for Bernoulli trials (i.i.d. binary sequences), and then they are generalized to the Polya–Eggenberger sampling scheme. For the latter case, the length of the longest success run is related to other success runs statistics and to reliability of consecutive systems.     

Estimating the distribution of one-dimensional discrete scan statistics viewed as extremes of 1-dependent stationary sequences

Discrete one-dimensional scan statistics can be viewed as extremes of 1-dependent stationary sequences. A result of Haiman [1999. First passage time for some stationary processes. Stochastic Process. Appl. 80, 231–248] provides approximations of the distributions of extremes of 1-dependent stationary sequences together with sharp bounds for the corresponding errors. We apply this result to scan statistics generated by Bernoulli r.v.'s and to the charge problem.     

Testing the difference of two exponential means using generalized p-values

There are no exact fixed-level tests for testing the null hypothesis that the difference of two exponential means is less than or equal to a prespecified value θ₀. For this testing problem, there are several approximate testing procedures available in the literature. Using an extended definition of p-values, Tsui and Weerahandi (1989) gave an exact significance test for this testing problem. In this paper, the performance of that procedure is investigated and is compared with approximate procedures. A size and power comparison is carried out using a simulation study. Its findings show that the test based on the generalized p-value guarantees the intended size and that it is either as good as or outperforms approximate procedures available in the literature, both in power and in size.     

Near-exact distributions for the sphericity likelihood ratio test statistic

In this paper three near-exact distributions are developed for the sphericity test statistic. The exact probability density function of this statistic is usually represented through the use of the Meijer G function, which renders the computation of quantiles impossible even for a moderately large number of variables. The main purpose of this paper is to obtain near-exact distributions that lie closer to the exact distribution than the asymptotic distributions while, at the same time, correspond to density and cumulative distribution functions practical to use, allowing for an easy determination of quantiles. In addition to this, two asymptotic distributions that lie closer to the exact distribution than the existing ones were developed. Two measures are considered to evaluate the proximity between the exact and the asymptotic and near-exact distributions developed. As a reference we use the saddlepoint approximations developed by Butler et al. [1993. Saddlepoint approximations for tests of block independence, sphericity and equal variances and covariances. J. Roy. Statist. Soc., Ser. B 55, 171–183] as well as the asymptotic distribution proposed by Box.     

Approximation of the Bivariate Renewal Function

Recently, Kambo and his co-researchers (2012) proposed a method of approximation for evaluating the one-dimensional renewal function based on the first three moments. Their method is simple and elegant, which gives exact values for well-known distributions. In this article, we propose an analogous method for the evaluation of bivariate renewal function based on the first two moments of the variables and their joint moment. The proposed method yields exact results for certain widely used bivariate distributions like bivariate exponential distribution, bivariate Weibull distributions, and bivariate Pareto distributions. An illustrative example in the form of a two-dimensional warranty problem is considered and comparisons of our method are made with the results of other models.     

An L-statistic approach to a test of exponentiality against IFR alternatives

In this note we consider the problem of testing exponentiality against IFR alternatives. A measure of deviation from exponentiality is developed and a test statistic constructed on the basis of this measure. It is shown that the test statistic is an L-statistic. The asymptotic as well as the exact distribution of the test statistic is obtained and the test is shown to be consistent.     

Relationships between confidence intervals for the ratio of two variances of independent normal distributions

Burk at al (1984) gave a results concerning the comparison of the length of the two different confidence intervals for variance ratio, when the construction of the intervals was based on the principle of “equal tails”11. The purpose of this paper is to be solve the similar problem in case of the principle of “minimal length”.     

Distributions of Runs and Scans on Higher-Order Markov Trees

In this article, we consider the distributions of the number of success runs of specified length and scans on a higher-order Markov tree under three different enumeration schemes (the “non overlapping”, the “at least”, and the “overlapping” scheme). Recursive formulae for the evaluation of their probability generating functions are established. We provide a proper framework for extending the exact distribution theory of runs and scans from based on sequences to based on directed trees. Some numerical results for the run and scan statistics are given in order to illustrate the computational aspects and the feasibility of our theoretical results. Finally, two special reliability systems are considered, which are closely related to our general results.     

Calculation of the Prokhorov distance by optimal quantization and maximum flow

Calculating exact values of the Prokhorov metric for the set of probability distributions on a metric space is a challenging problem. In this paper probability distributions are approximated by finite-support distributions through optimal or quasi-optimal quantization, in such a way that exact calculation of the Prokhorov distance between a distribution and a quantizer can be performed. The exact value of the Prokhorov distance between two quantizers is obtained by solving an optimization problem through the Simplex method. This last value is used to approximate the Prokhorov distance between the two initial distributions, and the accuracy of the approximation is measured. We illustrate the method on various univariate and bivariate probability distributions. Approximation of bivariate standard normal distributions by quasi-optimal quantizers is also considered.     

The exact distribution of the maximum,minimum and the range of Multinomial/Dirichlet and Multivariate Hypergeometric frequencies

The exact distribution of the maximum and minimum frequencies of Multinomial/Dirichlet and Multivariate Hypergeometric distributions of n balls in m urns is compactly represented as a product of stochastic matrices. This representation does not require equal urn probabilities, is invariant to urn order, and permits rapid calculation of exact probabilities. The exact distribution of the range is also obtained. These algorithms satisfy a long-standing need for routines to compute exact Multinomial/Dirichlet and Multivariate Hypergeometric maximum, minimum, and range probabilities in statistical computation libraries and software packages.     

A non-linear congruential pseudo random number generator

A non-linear congruential pseudo random number generator is introduced. This generator does not have the lattice structure in the distribution of tuples of consecutive pseudo random numbers which appears in the case of linear congruential generators. A theorem on the period length of sequences produced by this type of generators is proved. This theorem justifies an algorithm to determine the period length. Finally a simulation problem is described where a linear congruential generator produces completely useless results whereas good results are obtained if a non-linear congruential generator of about the same period length is applied.     

Testing the equality of several covariance matrices

Using a new approach based on Meijer G-functions and computer simulation, we numerically compute the exact null distribution of the modified-likelihood ratio statistic used to test the hypothesis that several covariances matrices of normal distributions are equal. Small samples of different sizes are considered, and for the case of two matrices, we introduce a new test based on determinants, with the null distribution of its criterion also fully computable. Comparisons with published results show the accuracy of our approach, which is proved to be more flexible and adaptable to different cases.     

On convergence of properly weighted samples to the target distribution

We consider importance sampling as well as other properly weighted samples with respect to a target distribution π$\pi$ from a different point of view. By considering the associated weights as sojourn times until the next jump, we define appropriate jump processes. When the original sample sequence forms an ergodic Markov chain, the associated jump process is an ergodic semi-Markov process with stationary distribution π$\pi$. In this respect, properly weighted samples behave very similarly to standard Markov chain Monte Carlo (MCMC) schemes in that they exhibit convergence to the target distribution as well. Indeed, some standard MCMC procedures like the Metropolis–Hastings algorithm are included in this context. Moreover, when the samples are independent and the mean weight is bounded above, we describe a slight modification in order to achieve exact (weighted) samples from the target distribution.     

The modified slash Lindley–Weibull distribution with applications to nutrition data

This work presents an extension of the slash Lindley–Weibull distribution, of which it can be considered a modification. The new family is obtained by using the quotient of two independent random variables: a two-parameter Lindley–Weibull distribution divided by a power of the exponential distribution with parameter equal to 2. We present the pdf and cdf of the new distribution, analyzing their risk functions. Some statistical properties are studied and the moments and coefficients of asymmetry and kurtosis are shown. The parameter estimation problem is carried out by the maximum likelihood method. The method is assessed by a Monte Carlo simulation study. We use nutrition data, which are characterized by high kurtosis, to illustrate the usefulness of the proposed model.     

Almost sure central limit theorem for the maxima and sums of stationary Gaussian sequences

In this paper, we proved an almost sure central limit theorem for the maxima (after centered at the sample mean) and the partial sums of standardized stationary Gaussian sequences under some conditions related to the convergence rate of covariance functions, which extended the existing results.     

Some nonparametric precedence-type tests based on progressively censored samples and evaluation of power

In this paper, we consider the problem of testing the equality of two distributions when both samples are progressively Type-II censored. We discuss the following two statistics: one based on the Wilcoxon-type rank-sum precedence test, and the second based on the Kaplan–Meier estimator of the cumulative distribution function. The exact null distributions of these test statistics are derived and are then used to generate critical values and the corresponding exact levels of significance for different combinations of sample sizes and progressive censoring schemes. We also discuss their non-null distributions under Lehmann alternatives. A power study of the proposed tests is carried out under Lehmann alternatives as well as under location-shift alternatives through Monte Carlo simulations. Through this power study, it is shown that the Wilcoxon-type rank-sum precedence test performs the best.     

Distributions of simple patterns in some kinds of exchangeable sequences

In this article, we consider the distributions of simple patterns in some types of sequences of infinite exchangeable multi-state trials. The distributions on exchangeable multi-state trials are considered in terms of an extension of de Finetti's theorem. As an application of partially exchangeable sequences, distributions on a Markov exchangeable sequence are studied. Furthermore, we propose a new type of partially exchangeable sequence and examine its properties. In addition, we discuss the distribution theory in the case of the finite exchangeable sequences. The results presented here provide a wide framework for developing the exact distribution theory of simple patterns. Finally, some examples are given in order to illustrate our theoretical results.     

A generalized multivariate beta distribution: control charting when the measurements are from an exponential distribution

In Statistical Process Control (SPC) there exists a need to model the run-length distribution of a Q-chart that monitors the process mean when measurements are from an exponential distribution with an unknown parameter. To develop exact expressions for the probabilities of run-lengths the joint distribution of the charting statistics is needed. This gives rise to a new distribution that can be regarded as a generalized multivariate beta distribution. An overview of the problem statement as identified in the field of SPC is given and the newly developed generalized multivariate beta distribution is proposed. Statistical properties of this distribution are studied and the effect of the parameters of this generalized multivariate beta distribution on the correlation between two variables is also discussed.     

Discriminating membrane proteins using the joint distribution of length sums of success and failure runs

Discriminating integral membrane proteins from water-soluble ones, has been over the past decades an important goal for computational molecular biology. A major drawback of methods appeared in the literature, is that most of the authors tried to solve the problem using machine learning techniques. Specifically, most of the proposed methods require an appropriate dataset for training, and consequently the results depend heavily on the suitability of the dataset, itself. Motivated by these facts, in this paper we develop a formal discrimination procedure that is based on appropriate theoretical observations on the sequence of hydrophobic and polar residues along the protein sequence and on the exact distribution of a two dimensional runs-related statistic defined on the same sequence. Specifically, for setting up our discrimination procedure, we study thoroughly the exact distribution of a bivariate random variable, which accumulates the exact lengths of both success and failure runs of at least a specific length in a sequence of Bernoulli trials. To investigate the properties of this bivariate random variable, we use the Markov chain embedding technique. Finally, we apply the new procedure to a well-defined dataset of proteins.     

Normal approximation to the hypergeometric distribution in nonstandard cases and a sub-Gaussian Berry–Esseen theorem

In this paper, we consider simple random sampling without replacement from a dichotomous finite population. We investigate accuracy of the Normal approximation to the Hypergeometric probabilities for a wide range of parameter values, including the nonstandard cases where the sampling fraction tends to one and where the proportion of the objects of interest in the population tends to the boundary values, zero and one. We establish a non-uniform Berry–Esseen theorem for the Hypergeometric distribution which shows that in the nonstandard cases, the rate of Normal approximation to the Hypergeometric distribution can be considerably slower than the rate of Normal approximation to the Binomial distribution. We also report results from a moderately large numerical study and provide some guidelines for using the Normal approximation to the Hypergeometric distribution in finite samples.