Replacement policies for age and working numbers with random life cycle

It has been modeled for several replacement policies in literatures that the whole life cycle or operating interval of an operating unit should be finite rather than infinite as is done with the traditional method. However, it is more natural to consider the case in which the finite life cycle is a fluctuated parameter that could be used to estimate replacement times, which will be taken up in this article. For this, we first formulate a general model in which the unit is replaced at random age U, random time Y for the first working number, random life cycle S, or at failure X, whichever occurs first. The following models included in the general model, such that replacement done at age T when variable U is a degenerate distribution, and replacement done at working numbers N summed by number N of variable Y, are optimized. We obtain the total expected cost until replacement and the expected replacement cost rate for each model. Optimal age T, working number N, and a pair of (T, N) are discussed analytically and computed numerically.     

A simple generalization of the Box–Muller method for obtaining a pair of correlated standard normal variables

In the simple and widely used method of Box–Muller [G. Box and M. Muller, A note on the generation of random normal deviates, Ann. Math. Statist. 29 (1958), pp. 610–611], from a pair of uniform and independent random variables in (0,1), a pair of standard and independent normal variables is obtained. In this article, we present a very simple and elegant generalization of this method to obtain a pair of correlated standard normal variables with a given coefficient of correlation. This generalized method, which is computationally very easy, is interpreted in geometric terms, considering a translation of the uniform interval (0,1) and a rotation of a defined angle, both related to the coefficient of correlation. Some numerical results are simulated and statistically analysed, proving that the generalization is extremely simple and powerful.     

On the Sum of Symmetric Random Variables

Let U and V be two symmetric (about zero) random variables with U + V symmetric about C; here C is a constant. It is easy to see that if U and V are mutually independent, or if both U and V satisfy the weak law of large numbers, then C = 0. So, intuitively, we would suspect that C = 0 in general. However, we show that there exist two random variables U and V symmetric about 0 with U + V symmetric about C ≠ 0 The example given is closely related to one given by Alejandro D. De Acosta in another context.     

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}.     

On An Intriguing Distributional Identity

For a continuous random variable X with support equal to (a, b), with c.d.f. F, and g: Ω₁ → Ω₂ a continuous, strictly increasing function, such that Ω₁∩Ω₂?(a, b), but otherwise arbitrary, we establish that the random variables F(X) ? F(g(X)) and F(g^{? 1}(X)) ? F(X) have the same distribution. Further developments, accompanied by illustrations and observations, address as well the equidistribution identity U ? ψ(U) = ^dψ^{? 1}(U) ? U for U ～ U(0, 1), where ψ is a continuous, strictly increasing and onto function, but otherwise arbitrary. Finally, we expand on applications with connections to variance reduction techniques, the discrepancy between distributions, and a risk identity in predictive density estimation.     

Characterizations based on regression assumptions of order statistics

Let X₍₁₎≤X₍₂₎≤···≤X_(n) be the order statistics from independent and identically distributed random variables {X_i, 1≤i≤n} with a common absolutely continuous distribution function. In this work, first a new characterization of distributions based on order statistics is presented. Next, we review some conditional expectation properties of order statistics, which can be used to establish some equivalent forms for conditional expectations for sum of random variables based on order statistics. Using these equivalent forms, some known results can be extended immediately.     

ON CHARACTERIZATIONS THROUGH MIXED SUMS

Let X, X‘ and X“ be independent and have the same law, and let U be uniform on [0,1] and independent of X’ and XI”. A problem which has received considerable attention is to identify the laws of X for which X =c U(X‘+X“), the equality being in law. If X is positive it has an exponential law, and mixture solutions exist without the sign restriction. The problem has been extended to allow more summands on either side and to let U have a special beta law. In the positive case X then has a gamma law. This problem is tackled in full generality by considering an integral equation for the characteristic function of X. Uniqueness results and necessary conditions are given. The general case is reduced to that where X is positive, and the solution of the corresponding integral equation is constructed under the necessary conditions. This is shown to characterize all possible laws of X when there is a single summand on the left and U has an arbitrary law on [0,1]. Examples are given and known particular results are sharpened.     

ON A SHARED ALLELE TEST OF RANDOM MATING

A simple random sample is observed from a population with a large number‘K’ of alleles, to test for random mating. Of n couples, n_ijkl have female genotype ij and male genotype kl (i, j, k, l{1,…, A‘}). The large contingency table is collapsed into three counts, n₀, n₁ and n₂ where n_p is the number of couples with s alleles in common (s = 0,1, 2). The counts are estimated by np?_o where n₀, is the estimated probability of a couple having s alleles in common under the hypothesis of random mating. The usual chi-square goodness of fit statistic X² compares observed (n_s) with expected (np?) over the three categories, s = 0,1,2. An empirical observation has suggested that X² is close to having a chi-square distribution with two degrees of freedom (X) despite a large number of parameters implicitly estimated in _e. This paper gives two theorems which show that x is indeed the approximate distribution of X² for large n and K₁“, provided that no allele type over-dominates the others.     

The Joint Distribution of the Sum and the Maximum of IID Exponential Random Variables

The authors establish the joint distribution of the sum X and the maximum Y of IID exponential random variables. They derive exact formuli describing the random vector (X, Y), including its joint PDF, CDF, and other characteristics; marginal and conditional distributions; moments and related parameters; and stochastic representations leading to further properties of infinite divisibility and self-decomposability. The authors also discuss parameter estimation and include an example from climatology that illustrates the modeling potential of this new bivariate model.     

Classes of power semicircle laws that are randomly weighted average distributions

We give an affirmative answer to the conjecture raised in Soltani and Roozegar [On distribution of randomly ordered uniform incremental weighted averages: divided difference approach. Statist Probab Lett. 2012;82(5):1012–1020] that a certain class of power semicircle distributions, parameterized by n, gives the distributions of the average of n independent and identically Arcsine random variables weighted by the cuts of (0,1) by the order statistics of a uniform (0, 1) sample of size n?1, for each n. Then we establish the central limit theorem for this class of distributions. We also use the Demni [On generalized Cauchy–Stieltjes transforms of some beta distributions. Comm Stoch Anal. 2009;3:197–210] results on the connection between the ordinary and generalized Cauchy or Stieltjes transforms, and introduce new classes of randomly weighted average distributions.     

CHARACTERIZATIONS OF SHIFT-INVARIANT DISTRIBUTIONS BASED ON SUMMATION MODULO ONE

Let X₁Y_1,…, Y_n be independent random variables. We characterize the distributions of X and Y_j satisfying the equation {X+Y₁++Y_n}=_dX, where {Z} denotes the fractional part of a random variable Z. In the case of full generality, either X is uniformly distributed on [0,1), or Y_j has.a shifted lattice distribution and X is shift-invariant. We also give a characterization of shift-invariant distributions. Finally, we consider some special cases of this equation.     

Tolerance Limits Based on the Multivariate MaxMin Chart

In a previous paper, we have showed how to obtain sequences of number proved random. With this aim, we used sequences of noises y_n such that the conditional probabilities have Lipschitz coefficients not too large. We transformed them using Fibonacci congruences. Then, we obtained sequences x_n which admit the IID model for correct model. This method consisted to value the work of Marsaglia in order to build his CD-ROM. But we did not use Rap Music (as Marsaglia), but texts files. This method also uses an extractor and at the same time the notion of correct models. In this paper, we apply this method to numbers provided by machines or chips. Unfortunately, it is less sure than they have Lipschtiz coefficient not too large. But we can solve this problem: it suffices to use the Central Limit Theorem. We do it modulo 1. In this case, we use a new limit theorem, the XOR Limit theorem : asymptotic distribution of sum of random vectors modulo 1 are asymptotically independent. Then Lipschtiz coefficient of associated sequences are not too large and we can obtain IID sequences by using Fibonacci congruences.     

The Beta Product Distribution with Complex Parameters

The family consisting of the distributions of products of two independent beta variables is extended to include cases where some of the parameters are not positive but negative or complex. This “beta product” distribution is expressible as a Meijer G function. An example (from risk theory) where such a distribution arises is given: an infinite sum of products of independent random variables is shown to have a distribution that is the product convolution of a complex-parameter beta product and an independent exponential. The distribution of the infinite sum is a new explicit solution of the stochastic equation X = (in law) B(X + C). Characterizations of some G distributions are also proved.     

On the Ghoudi,Khoudraji, and Rivest test for extreme‐value dependence

Ghoudi, Khoudraji & Rivest [The Canadian Journal of Statistics 1998;26:187–197] showed how to test whether the dependence structure of a pair of continuous random variables is characterized by an extreme‐value copula. The test is based on a U‐statistic whose finite‐ and large‐sample variance are determined by the present authors. They propose estimates of this variance which they compare to the jackknife estimate of Ghoudi, Khoudraji & Rivest ( 1998 ) through simulations. They study the finite‐sample and asymptotic power of the test under various alternatives. They illustrate their approach using financial and geological data. The Canadian Journal of Statistics © 2009 Statistical Society of Canada     

Finding the PDF of the hypoexponential random variable using the Kad matrix similar to the general Vandermonde matrix

ABSTRACT

The sum of independent exponential random variables – the hypoexponential random variables – plays an important role of modeling in many domains. Khuong and Kong in (2006) Khuong, H.V., Kong, H.Y. (2006). General expression for pdf of a sum of independent exponential random variables. IEEE Commun. Lett. 10: 159–161.[Crossref], [Web of Science ®] , [Google Scholar] were concerned in evaluating the performance of some diversity scheme, which deals with the problem of finding the probability density function of this hypoexponential random variable. They considered a particular case of m independent exponential random variable, when l random variables have the same mean and m ? l remaining random variables of different means and they found a closed expression of its probability density function. In this paper, we consider the general case of the hypoexponential random variable when the means do not have to be distinct. We find a more simple and general closed expression of its probability density function than that of Khuong and Kong. This expression is obtained using a new defined matrix called the Kad matrix, which is similar to the general Vandermonde matrix. Eventually, we present an application illustrating our work.     

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.     

Three-stage chain sampling plans

This article outlines the structure of a generalized family of three-stage chain sampling plans, extending the concept of two-stage chain sampling plans of Dodge and Stephens (1966) which is an extension of the original work of Dodge (1955). Expressions are derived for the OC curves for several sets of general sets of three-stage chain sampling plans, with cumulative acceptance numbers of (c₁,c₂,c₃) = (0,1,2), (0,1,3), (0,2,3), (1,2,3),(0,1,4),(0,2,4), (0,3,4), (1,2,4), (1,3,4) and (2,3,4). In the original work of Dodge (1955) only acceptance numbers of 0,1 were used and in the extension work of Dodge and Stephens (1966) acceptance numbers of (c₁,c₂) = (0,1) ,(0,2), (1,2), (0,3), (1,3), (0,4) and (1,4) were used with selected sets of values of k₁,and k₂ (the number of lots considered for cumulation in the first and second stage respectively). In this paper the 0C curves are derived more generally for any k₁, k₂and k₃(the number of lots considered for cumulation in the first, second and third stages respectively) combination and 0C curves of a number of plans are given and comparisons are made with some single sampling, two-stage chain sampling and multiple sampling plans.     

Minimum distance lack-of-fit tests for fixed design

This paper discusses a class of tests of lack-of-fit of a parametric regression model when design is non-random and uniform on [0,1]. These tests are based on certain minimized distances between a nonparametric regression function estimator and the parametric model being fitted. We investigate asymptotic null distributions of the proposed tests, their consistency and asymptotic power against a large class of fixed and sequences of local nonparametric alternatives, respectively. The best fitted parameter estimate is seen to be n^1/2-consistent and asymptotically normal. A crucial result needed for proving these results is a central limit lemma for weighted degenerate U statistics where the weights are arrays of some non-random real numbers. This result is of an independent interest and an extension of a result of Hall for non-weighted degenerate U statistics.     

Extreme behaviour for bivariate elliptical distributions

The authors examine the asymptotic behaviour of conditional threshold exceedance probabilities for an elliptically distributed pair (X, Y) of random variables. More precisely, they investigate the limiting behaviour of the conditional distribution of Y given that X becomes extreme. They show that this behaviour differs between regularly and rapidly varying tails.