On k-th record times,record values and their moments

We give new formula for moments of k-th record values in terms of Stirling numbers of the first kind. In particular, the formulae allow to derive the explicit formulae for moments of k-th lower record values from exponential distribution which have not been known yet. Moreover, some interesting identities involving harmonic numbers are also obtained as corollaries to presented results.     

Upper and lower bounds on the variances of linear combinations of the kth records

We describe a method of determining upper bounds on the variances of linear combinations of the kth records values from i.i.d. sequences, expressed in terms of variances of parent distributions. We also present conditions for which the bounds are sharp, and those for which the respective lower ones are equal to zero. A special attention is paid to the case of the kth record spacings, i.e. the differences of consecutive kth record values.     

Projection Mean–Variance Bounds on Expectations of kth Record Values from Restricted Families

We present sharp mean–variance bounds for expectations of kth record values based on distributions coming from restricted families of distributions. These families are defined in terms of convex or star ordering with respect to generalized Pareto distribution. The bounds for expectations of kth record values from DD, DFR, DDA, and DFRA families are special cases of our results. The bounds are derived by application of the projection method.     

Nonparametric model checking for k-out-of-n systems

It is an important problem in reliability analysis to decide whether for a given k-out-of-n system the static or the sequential k-out-of-n model is appropriate. Often components are redundantly added to a system to protect against failure of the system. If the failure of any component of the system induces a higher rate of failure of the remaining components due to increased load, the sequential k-out-of-n model is appropriate. The increase of the failure rate of the remaining components after a failure of some component implies that the effects of the component redundancy are diminished. On the other hand, if all the components have the same failure distribution and whenever a failure occurs, the remaining components are not affected, the static k-out-of-n model is adequate. In this paper, we consider nonparametric hypothesis tests to make a decision between these two models. We analyze test statistics based on the profile score process as well as test statistics based on a multivariate intensity ratio and derive their asymptotic distribution. Finally, we compare the different test statistics.     

SHARP BOUNDS FOR EXPECTATIONS OF kTH RECORD INCREMENTS

This paper presents sharp bounds for expectations of non‐adjacent increments of kth record statistics, measured in various scale units, for a sequence of independent identically distributed random variables with continuous cumulative distribution function. The results for kth record spacings are considered as special cases. The paper also characterizes probability distributions for which the bounds are attained.     

An optimal response adaptive biased coin design with k heteroscedastic treatments

For comparing treatments in clinical trials, Atkinson (1982) introduced optimal biased coins for balancing patients across treatment assignments by using D-optimality under the assumption of homoscedastic responses of different treatments. However, this assumption can be violated in many real applications. In this paper, we relax the homoscedasticity assumption in the k treatments setting with k>2. A general family of optimal response adaptive biased coin designs are proposed following Atkinson's procedure. Asymptotic properties of the proposed designs are obtained. Some advantages of the proposed design are discussed.     

On perturbations of Stein operator

In this article, we obtain a Stein operator for the sum of n independent random variables (rvs) which is shown as the perturbation of the negative binomial (NB) operator. Comparing the operator with NB operator, we derive the error bounds for total variation distance by matching parameters. Also, three-parameter approximation for such a sum is considered and is shown to improve the existing bounds in the literature. Finally, an application of our results to a function of waiting time for (k₁, k₂)-events is given.     

Circular consecutive k-out-of-n systems with exchangeable dependent components

In this paper we study circular consecutive k-out-of-n systems consisting of exchangeable components. We derive explicit expressions for both unconditional and conditional survival functions for 2k+1≥n, while signature based mixture representations for general k are obtained. The applications and computational results concerned with mean residual life function and stochastic ordering are presented.     

Classification performance resulting from a 2-means

The k-means procedure is probably one of the most common nonhierachical clustering techniques. From a theoretical point of view, it is related to the search for the k principal points of the underlying distribution. In this paper, the classification resulting from that procedure for k=2 is shown to be optimal under a balanced mixture of two spherically symmetric and homoscedastic distributions. Then, the classification efficiency of the 2-means rule is assessed using the second order influence function and compared to the classification efficiencies of Fisher and Logistic discriminations. Influence functions are also considered here to compare the robustness to infinitesimal contamination of the 2-means method w.r.t. the generalized 2-means technique.     

Discrete q-distributions on Bernoulli trials with a geometrically varying success probability

Consider a sequence of independent Bernoulli trials and assume that the odds of success (or failure) or the probability of success (or failure) at the ith trial varies (increases or decreases) geometrically with rate (proportion) q, for increasing i=1,2,…. Introducing the notion of a geometric sequence of trials as a sequence of Bernoulli trials, with constant probability, that is terminated with the occurrence of the first success, a useful stochastic model is constructed. Specifically, consider a sequence of independent geometric sequences of trials and assume that the probability of success at the jth geometric sequence varies (increases or decreases) geometrically with rate (proportion) q, for increasing j=1,2,…. On both models, let X_n be the number of successes up the nth trial and T_k (or W_k) be the number of trials (or failures) until the occurrence of the kth success. The distributions of these random variables turned out to be q-analogues of the binomial and Pascal (or negative binomial) distributions. The distributions of X_n, for n→∞$n\to \infty$, and the distributions of W_k, for k→∞$k\to \infty$, can be approximated by a q  -Poisson distribution. Also, as k→0$k\to 0$, a zero truncated negative q  -binomial distribution _Uk=_Wk|_Wk>0$U_k=W_k|W_k>0$ can be approximated by a q-logarithmic distribution. These discrete q-distributions and their applications are reviewed, with critical comments and additions. Finally, consider a sequence of independent Bernoulli trials and assume that the probability of success (or failure) is a product of two sequences of probabilities with one of these sequences depending only the number of trials and the other depending only on the number of successes (or failures). The q-distributions of the number X_n of successes up to the nth trial and the number T_k of trials until the occurrence of the kth success are similarly reviewed.     

On the quality of k-means clustering based on grouped data

Let us have a probability distribution P   (possibly empirical) on the real line R$\mathbb{R}$. Consider the problem of finding the k-mean of P, i.e. a set A of at most k points that minimizes given loss-function. It is known that the k-mean can be found using an iterative algorithm by Lloyd [1982. Least squares quantization in PCM. IEEE Transactions on Information Theory 28, 129–136]. However, depending on the complexity of the distribution P, the application of this algorithm can be quite resource-consuming. One possibility to overcome the problem is to group the original data and calculate the k-mean on the basis of the grouped data. As a result, the new k-mean will be biased, and our aim is to measure the loss of the quality of approximation caused by such approach.     

A note on the mean residual life function of a coherent system with exchangeable or nonidentical components

This article investigates some properties of the mean residual life function of (n−k+1)-out-of-n systems, when the lifetimes of the system components are independent random variables but not necessarily identically distributed and when the joint distribution of the component lifetimes is exchangeable, extending the results of Asadi and Goliforushani (2008) [On the mean residual life function of coherent systems. IEEE Transactions on Reliability 57 (4) 574-580] for the case of independent and identically distributed components. The extension to a coherent system with exchangeable components is also given.     

A new characterization of Elfving's method for high dimensional computation

We give a new characterization of Elfving's (1952) method for computing c-optimal designs in k dimensions which gives explicit formulae for the k unknown optimal weights and k unknown signs in Elfving's characterization. This eliminates the need to search over these parameters to compute c-optimal designs, and thus reduces the computational burden from solving a family of optimization problems to solving a single optimization problem for the optimal finite support set. We give two illustrative examples: a high dimensional polynomial regression model and a logistic regression model, the latter showing that the method can be used for locally optimal designs in nonlinear models as well.     

RECURRENCE RELATIONS FOR SINGLE AND PRODUCT MOMENTS OF GENERALIZED ORDER STATISTICS FROM PARETO,GENERALIZED PARETO,AND BURR DISTRIBUTIONS

We give recurrence relations for single and product moments of generalized order statistics under the concept of Kamps from Pareto, generalized Pareto and Burr distributions. The results include as particular cases the above relations for moments of k–th record values.     

On a system of nonhomogeneous components sharing a common frailty

The components of a reliability system subjected to a common random environment usually have dependent lifetimes. This paper studies the stochastic properties of such a system with lifetimes of the components following multivariate frailty models and multivariate mixed proportional reversed hazard rate (PRHR) models, respectively. Through doing stochastic comparison, we devote to throwing a new light on how the random environment affects the number of working components of a reliability system and on assessing the performance of a k-out-of-n system.     

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.     

Comparison of Records and Maxima

We establish best upper bounds on the expected differences of records and sample maxima, and kth records and kth maxima based on sequences of independent random variables with identical continuous distribution and finite variance. The bounds are expressed in terms of the standard deviation units of the parent distribution. We also provide conditions for attaining the bounds.     

Optimal k-circulant supersaturated designs

Optimal k-circulant supersaturated designs have been constructed in literature using computer intensive methods. A systematic method of construction for multi-level experiments based on balanced incomplete block designs is presented in this paper. The method is also applicable to two-level experiments. Illustrative examples are also given.