On the distribution and expectation of success runs in nonhomogeneous Markov dependent trials

The number of success runs for nonhomogeneous markov dependent trials are represented as the sum of Bernoulli trials and the expected value of runs are obtained by using this representation. The distribution and bounds for the distribution of the longest run are derived for markov dependent trials.     

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.     

Circular success and failure runs in a sequence of exchangeable binary trials

Let X₁,…,X_n be an exchangeable sequence of binary trials arranged on a circle with possible values “1” (success) or “0” (failure). In an exchangeable sequence, the joint distribution of X₁,X₂,…,X_n is invariant under the permutation of its arguments. For the circular sequence, general expressions for the joint distributions of run statistics based on the joint distribution of success and failure run lengths are obtained. As a special case, we present our results for Bernoulli trials. The results presented consist of combinatorial terms and therefore provide easier calculations. For illustration purposes, some numerical examples are given and the reliability of the circular combined k-out-of-n:G and consecutive k_c-out-of-n:G system under stress–strength setup is evaluated.     

An identity for the joint distribution of order statistics and its applications

It is shown that the joint distribution function of several order statistics can be expressed in terms of the distribution functions of the maximum (or the minimum) order statistics of suitable subsamples. The result is used to derive explicit expressions for the expectations of functions of order statistics from certain families of distributions which include the exponential distribution and the power function distribution. The results generalize earlier work by the authors for a single order statistic.     

On the matrix-variate generalized hyperbolic distribution and its Bayesian applications

In the first part of the paper, we introduce the matrix-variate generalized hyperbolic distribution by mixing the matrix normal distribution with the matrix generalized inverse Gaussian density. The p-dimensional generalized hyperbolic distribution of [Barndorff-Nielsen, O. (1978). Hyperbolic distributions and distributions on hyperbolae. Scand. J. Stat., 5, 151–157], the matrix-T distribution and many well-known distributions are shown to be special cases of the new distribution. Some properties of the distribution are also studied. The second part of the paper deals with the application of the distribution in the Bayesian analysis of the normal multivariate linear model.     

On binomial and circular binomial distributions of orderk forl-overlapping success runs of lengthk

The number ofl-overlapping success runs of lengthk inn trials, which was introduced and studied recently, is presently reconsidered in the Bernoulli case and two exact formulas are derived for its probability distribution function in terms of multinomial and binomial coefficients respectively. A recurrence relation concerning this distribution, as well as its mean, is also obtained. Furthermore, the number ofl-overlapping success runs of lengthk inn Bernoulli trials arranged on a circle is presently considered for the first time and its probability distribution function and mean are derived. Finally, the latter distribution is related to the first, two open problems regarding limiting distributions are stated, and numerical illustrations are given in two tables. All results are new and they unify and extend several results of various authors on binomial and circular binomial distributions of orderk.     

On the joint distribution of placement statistics under progressive censoring and applications to precedence test

In reliability and survival analysis, comparison of two or more populations is an important problem. For example, while comparing a treatment group with a control group, one may be interested in determining whether the observations in the treatment group have a longer lifetime than those from the control group, that is, whether the treatment is effective or not. In such a study, it would be extremely valuable to make a decision based on early failures. In this paper, we consider independent progressively Type-II censored random samples from two populations with cumulative distribution function's (cdf) F(·)$F(·)$ and G(·)$G(·)$ respectively, and discuss a precedence test for testing the equality of the two distributions based on placements. For this purpose, we derive the joint distribution of the first l   placement statistics from the progressively censored sample from F(·)$F(·)$. We then derive the exact null distribution of the precedence test statistic which is simply the sum of the placements. We provide the rejection regions for fixed levels of significance and various sample sizes and different progressive censoring schemes.     

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.     

On some aspects of a general class of Yule distribution and its applications

Abstract

A wide class of Yule distribution is introduced here as a generalization of the extended Yule distribution of Martinez-Rodríguez et al. and the generalized Yule distribution of Mishra and investigate some of its properties. Various methods of estimation are employed for estimating the parameters of the distribution and generalized likelihood ratio test procedures are suggested for testing the significance of the parameters of the distribution. All these procedures are illustrated with the help of real data sets.     

A property of the yule distribution and its applications

The relationship Y = RX between two random variables X and Y, where R is distributed independently of X in (0, l), is known to have important consequences in different fields such as income distribution analysis, Inventory decision models, etc.

In this paper it is shown that when X and Y are discrete random variables, relationships of similar nature lead to Yule-type distributions. The implications of the results are studied in connection with problems of income underreporting and inventory decision making.     

Asymptotic joint distribution of sample quantiles and sample mean with applications

Based on a sample from an absolutely continuous distribution F with density f, and with the aid of the Bahadur (Ann. Math. Statist. 37( 1966 ), 577-580) representation of sample quantiles, the asymptotic joint distribution of three statistics, the sample p^th and q^th quantiles (0 < p < q < l) and the sample mean, is obtained. Using the Cramer-Wold device, asymptotic distributions of functions of the three statistics can be derived. In particular, the asymptotic joint distribution of the ratio of sample p^th quantile to sample mean and the ratio of sample q^th quantile to sample mean is presented. Finally, consistent estimators are proposed for the variances and covariances of these limiting distributions.     

Generalized Log-Lindley distribution and its applications in stochastic comparison

In this paper, a new distribution, generalized Log-Lindley distribution, which supports on (0, 1) is introduced as an alternative to the beta distribution. We show the new proposed distribution is also an extension of Log-Lindley distribution. Stochastic comparison for two parallel systems with generalized Log-Lindley distributed components is considered. Our results extend some existing results in the literature.     

On blocks and runs estimators of the extremal index

Given a sample from a stationary sequence of random variables, we study the blocks and runs estimators of the extremal index. Conditions are given for consistency and asymptotic normality of these estimators. We show that moment restrictions assumed by Hsing (Stochast. Process. Appl. 37(1), 117–139; Ann. Statist. 21(4), 2043-2021) may be relaxed if a stronger mixing condition holds. The CLT for the runs estimator seems to be proven for the first time.     

A bootstrap approximation to the joint distribution of sum and maximum of a stationary sequence

This paper establishes the asymptotic validity for the moving block bootstrap as an approximation to the joint distribution of the sum and the maximum of a stationary sequence. An application is made to statistical inference for a positive time series where an extreme value statistic and sample mean provide the maximum likelihood estimates for the model parameters. A simulation study illustrates small sample size behavior of the bootstrap approximation.     

On the joint distribution of the random variables ‘number of inversions’ and ‘number of outstanding variables’ in a randomly arranged sequence

The bivariate probability distribution of the random variables [number of inversions] and [number of outstanding variables] in a sequence of n i.i.d. random variables is derived. As an application, the null covariance between the test statistics proposed by Mann and Brunk, respectively, for the ‘trend in location’ problem is obtained. It is shown that these test statistics are asymptotically uncorrelated under the null hypothesis.     

A stochastic representation for the lp-norm symmetric distribution and its applications

The family of l_p-norm symmetric distributions was proposed by Yue and Ma and is a natural generalization to the family of l₁-norm symmetric distributions studied by Fang et al. In this article, we propose a stochastic representation for the l_p-norm symmetric distribution for any constant p > 0. The stochastic representation is expressed through independent and identically distributed uniform U(0, 1) random variables. It is illustrated that the stochastic representation can be applied to statistical simulation and uniform experimental design.     

On the distribution of quadratic forms and applications in the analysis of repeated measurements

In this paper a unified approach is given to the distribution of scalar quadratic forms for dependent variables. Necessary and sufficient conditions are found for the sums of squares of the various hierarchical layers in ANOVA to be distributed as multiples of chi-square variables. Results concerning the usual univariate F-tests in ANOVA of repeated measurements are derived as a special case.     

On constructing sequences estimating the mixing distribution with applications

This paper shows that by minimizing a Chebychev norm a mixing distribution can be constructed which converges weakly to the true mixing distribution with probability one. Deely and Kruse (1968) established a similar result for the supremum norm. For both norms the constructed mixing distribution is computed by solving a linear programming problem, but this problem is considerably smaller when the Chebychev norm is used. Thus a suitable mixing distribution can be constructed from solving a linear programming problem with considerably less computational work than was previously known. To illustrate the application of this simpler procedure it is applied to derive nonparametric empirical Bayes estimates in a simulation study. Some density estimates are also illustrated.