Discrete time cold standby repairable system: Combinatorial analysis

ABSTRACT

In this article, we obtain exact expression for the distribution of the time to failure of discrete time cold standby repairable system under the classical assumptions that both working time and repair time of components are geometric. Our method is based on alternative representation of lifetime as a waiting time random variable on a binary sequence, and combinatorial arguments. Such an exact expression for the time to failure distribution is new in the literature. Furthermore, we obtain the probability generating function and the first two moments of the lifetime random variable.     

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.     

The Distribution of the Record Position and its Applications

This is mainly an expository article on the positions of records in sequences of ordered elements. Such sequences are obtained, for example, when observing and ordering continuous iid random variables. In practice, records are of interest, for example, in meteorology and sports. A k-record is obtained when a new element is placed at position k counted from the top. The sequences of time points, when new k-records occur, are studied by elementary random walk methods. In the last section, it is shown that the time scale can be changed so that the time points of the k-records follow, approximately, a Poisson process.     

On the exact distribution of the sum of the largest n−k out of n normal random variables with differing mean values

Motivated by an application in Electrical Engineering, we derive the exact distribution of the sum of the largest n?k out of n normally distributed random variables, with differing mean values. Comparisons are made with two normal approximations to this distribution—one arising from the asymptotic negligibility of the omitted order statistics and one from the theory of L-statistics. The latter approximation is found to be in excellent agreement with the exact distribution.     

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.     

OPTIMAL SEMIPARAMETRIC INFERENCE FOR THE TAIL INDEX BASED ON RATIOS OF THE LARGEST EXTREMES

The k largest order statistics in a random sample from a common heavy‐tailed parent distribution with a regularly varying tail can be characterized as Fréchet extremes. This paper establishes that consecutive ratios of such Fréchet extremes are mutually independent and distributed as functions of beta random variables. The maximum likelihood estimator of the tail index based on these ratios is derived, and the exact distribution of the maximum likelihood estimator is determined for fixed k, and the asymptotic distribution as k →∞ . Inferential procedures based upon the maximum likelihood estimator are shown to be optimal. The Fréchet extremes are not directly observable, but a feasible version of the maximum likelihood estimator is equivalent to Hill's statistic. A simple diagnostic is presented that can be used to decide on the largest value of k for which an assumption of Fréchet extremes is sustainable. The results are illustrated using data on commercial insurance claims arising from fires and explosions, and from hurricanes.     

Lower bounds on the expectations of upper record values

For infinite sequences of independent random variables with identical continuous distributions, we establish optimal lower bounds on the deviations of the expectations of record values from population means in units generated by the central absolute moments of various orders. The bounds are non-negative for the classic record values, and non-positive for the other kth records with k?2. We also provide analogous bounds for the record increments.     

Asymptotically optimal Berry-Esseen-type bounds for distributions with an absolutely continuous part

Recursive and closed form upper bounds are offered for the Kolmogorov and the total variation distance between the standard normal distribution and the distribution of a standardized sum of n independent and identically distributed random variables. The method employed is a modification of the method of compositions along with Zolotarev's ideal metric. The approximation error in the CLT obtained vanishes at a rate O(n^−k/2+1), provided that the common distribution of the summands possesses an absolutely continuous part, and shares the same k−1 (k?3) first moments with the standard normal distribution. Moreover, for the first time, these new uniform Berry-Esseen-type bounds are asymptotically optimal, that is, the ratio of the true distance to the respective bound converges to unity for a large class of distributions of the summands. Thus, apart from the correct rate, the proposed error estimates incorporate an optimal asymptotic constant (factor). Finally, three illustrative examples are presented along with numerical comparisons revealing that the new bounds are sharp enough even to be used in practical statistical applications.     

Simultaneous arrival of customers to two different queues and modeling dependence via copula approach

ABSTRACT

In classical queueing systems, a customer is allowed to wait only in one queue to receive the service. In practice, when there exist a number of queues rendering the same service, some customers may tend to simultaneously take turn in more than one queue with the aim to receive the service sooner and thus reduce their waiting time. In this article, we introduce such a model and put forward a methodology to deal with the situation. In this regard, we consider two queues and assume that if a customer, who has turn in both queues, receives the service from one of the queues, the other turn is automatically withdrawn. This circumstance for the model brings about some abandonment in each queue as some customers receive the service from the other one. We study the customer’s waiting time in the mentioned model, which is defined as the minimum of waiting times in both queues and obtain probability density function of this random variable. Our approach to obtain probability density function of each of the waiting time random variables is to rely on the existing results for the abandonment case. We examine the situation for the cases of independence and dependence of the waiting time random variables. The latter is treated via a copula approach.     

Randomly weighted sums of linearly wide quadrant-dependent random variables with heavy tails

This paper investigates tail behavior of the randomly weighted sum ∑ⁿ_{k = 1}θ_kX_k and reaches an asymptotic formula, where X_k, 1 ? k ? n, are real-valued linearly wide quadrant-dependent (LWQD) random variables with a common heavy-tailed distribution, and θ_k, 1 ? k ? n, independent of X_k, 1 ? k ? n, are n non-negative random variables without any dependence assumptions. The LWQD structure includes the linearly negative quadrant-dependent structure, the negatively associated structure, and hence the independence structure. On the other hand, it also includes some positively dependent random variables and some other random variables. The obtained result coincides with the existing ones.     

Equal-bin-width histogram versus equal-bin-count histogram

The histogram has all its bin widths equal to some non-random number arbitrary set by an analyst (EBWH). In the result, particular bin counts are random variables. This paper presents also a histogram that is constructed in a converse manner. Bin counts are all equal to some non-random number arbitrary set by an analyst (EBCH). In the result, particular bin widths are random variables. The first goal of the paper is a choose of constant bin width (of bin numbers k) in the EBWH, which maximize the similarity measure in the Monte Carlo simulation. The second goal is a choose of constant bin count in the EBCH, which maximize the similarity measure in the Monte Carlo simulation. The third goal is to present similarity measures between empirical and theoretical data. The fourth goal is the comparative analysis of two histogram methods by means of the frequency formula. The first additional goal is a tip how to proceed in EBCH when modulo(n,k)≠0. The second additional goal is the software in the form of a Mathcad file with the implementation of EBWH and EBCH.Subject classification codes: 62G30, 62G07KEYWORDS: Empirical density function, histogram, random number generator, Monte Carlo simulation, similarity measure     

On exact strong laws of large numbers for ratios of random variables and their applications

Abstract

We study the almost sure convergence of weighted sums of ratios of independent random variables satisfying some general, mild conditions. The obtained results are applied to exact laws for order statistics. An exact law for independent random variables which are nonidentically distributed is also proved and applied to ratios of adjacent order statistics for a sample of uniformly distributed random variables.     

Competing risks model with random censoring on the leff

We consider a model involvingk competing risks when the random variables of interest (risks) are censored from the left with the unobservable random variable. The nonparametrical estimators for survival functions of risks are presented and the estimators are strongly approximated with the best rates by appropriate Gaussian processes.     

UPPER BOUNDS FOR THE HARMONIC MEAN, WITH AN APPLICATION TO EXPERIMENTAL DESIGN

For positive-valued random variables, the paper provides a sequence of upper bounds for the harmonic mean, the ith of these bounds being exact if and only if the random variable is essentially i-valued. Sufficient conditions for the convergence of the bounds to the harmonic mean are given. The bounds have a number of applications, particularly in experimental design where they may be used to check how close a given design is to A-optimality     

Asymptotic tail behavior of a random sum with conditionally dependent subexponential summands

In this paper, we obtain some results for the asymptotic behavior of the tail probability of a random sum S_τ = ∑^τ_{k = 1}X_k, where the summands X_k, k = 1, 2, …, are conditionally dependent random variables with a common subexponential distribution F, and the random number τ is a non negative integer-valued random variable, independent of {X_k: k ? 1}.     

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 exact computation of the density and of the quantiles of linear combinations of t and F random variables

The inversion formula for evaluation of the distribution of a linear combination of independent t and F random variables, respectively, is suggested. The method is applied to computing the exact confidence intervals for the common mean of several normal populations. This method is compared with the known approximate methods.     

Optimal selection and ordering of columns in supersaturated designs

Two methods to select columns for assigning factors to work on supersaturated designs are proposed. The focus of interest is the degree of non-orthogonality between the selected columns. One method is the exhaustive enumeration of selections of p columns from all k columns to find the exact optimality, while the other is intended to find an approximate solution by applying techniques used in the corresponding analysis, aiming for ease of use as well as a reduction in the large computing time required for large k with the first method. Numerical illustrations for several typical design matrices reveal that the resulting “approximately” optimal assignments of factors to their columns are exactly optimal for any p. Ordering the columns in E(s²)-optimal designs results in promising new findings including a large number of E(s²)-optimal designs.     

Density estimation for samples satisfying a certain absolute regularity condition

Let {ξ_i} be an absolutely regular sequence of identically distributed random variables having common density function f(x). Let H_k(x,y) (k=1, 2,…) be a sequence of Borel-measurable functions and f_n(x)=n^?1(H_n(x,ξ₁)+…+H_n(x,ξ_n)) the empirical density function. In this paper, the asymptotic property of the probability P(sup_x|f_n(x)?f(x)|>ε) (n→∞) is studied.     

On almost sure convergence for weighted sums of pairwise negatively quadrant dependent random variables

Let {X _n , n ≥ 1} be a sequence of pairwise negatively quadrant dependent (NQD) random variables. In this study, we prove almost sure limit theorems for weighted sums of the random variables. From these results, we obtain a version of the Glivenko–Cantelli lemma for pairwise NQD random variables under some fragile conditions. Moreover, a simulation study is done to compare the convergence rates with those of Azarnoosh (Pak J Statist 19(1):15–23, 2003) and Li et al. (Bull Inst Math 1:281–305, 2006).