The distribution of sums, products and ratios for Lawrance and Lewis's bivariate exponential random variables

We derive the exact distributions of R=X+Y, P=XY and W=X/(X+Y) and the corresponding moment properties when X and Y follow Lawrence and Lewis's bivariate exponential distribution. The expressions turn out to involve special functions. We also provide extensive tabulations of the percentage points associated with the distributions. These tables—obtained using intensive computing power—will be of use to practitioners of the bivariate exponential distribution.     

A convenient way of generating gamma random variables using generalized exponential distribution

In this paper we propose a very convenient way to generate gamma random variables using generalized exponential distribution, when the shape parameter lies between 0 and 1. The new method is compared with the most popular Ahrens and Dieter method and the method proposed by Best. Like Ahrens and Dieter and Best methods our method also uses the acceptance-rejection principle. But it is observed that our method has greater acceptance proportion than Ahrens and Dieter or Best methods.     

Efficient simulation of a bivariate exponential conditionals distribution

The bivariate distribution with exponential conditionals (BEC) is introduced by Arnold and Strauss [1988. Bivariate distributions with exponential conditionals, J. Amer. Statist. Assoc. 83, 522-527]. This work presents a simple and fast algorithm for simulating random variates from this density.     

Large deviations for distributions of sums of random variables: Markov chain method

Let {ξ _k } _k=0^∞ be a sequence of i.i.d. real-valued random variables, and let g(x) be a continuous positive function. Under rather general conditions, we prove results on sharp asymptotics of the probabilities $ P\left\{ {\frac{1} {n}\sum\limits_{k = 0}^{n - 1} {g\left( {\xi _k } \right) < d} } \right\} $ P\left\{ {\frac{1} {n}\sum\limits_{k = 0}^{n - 1} {g\left( {\xi _k } \right) < d} } \right\} , n → ∞, and also of their conditional versions. The results are obtained using a new method developed in the paper, namely, the Laplace method for sojourn times of discrete-time Markov chains. We consider two examples: standard Gaussian random variables with g(x) = |x| ^p , p > 0, and exponential random variables with g(x) = x for x ≥ 0.     

Approximation of exponential functions of random variables by statistical linearization

The approximation of a nonlinear function of a random variable by a linear function is considered. An exponential function, which arises in problems such as random sampling and random parameter linear systems, is treated in detail. The values of the two parameters which define the linear function are given for various approximation criteria. The results are then applied to the Gaussian random variable case, with explicit results and numerical comparisons.     

Testing for a class of bivariate exponential distributions

Bivariate and multivariate exponential distributions are widely applied in several areas such as reliability, queueing systems or hydrology. A frequently used bivariate exponential distribution is the Moran–Downton distribution. Because of this reason, this paper proposes a goodness-of-fit test for this distribution. The test statistic exploits the analytically convenient formula of its characteristic function. Large sample properties of the proposed test such as consistency against fixed and local alternatives are studied. The finite sample performance is numerically studied. Finally, an application of this distribution to hydrological data is presented.     

Expected warranty cost of two-attribute free-replacement warranties based on a bivariate exponential distribution

Many warranty policies offered by manufacturers are limited by two factors, typically time since purchase, and usage of the product. Such policies are referred to as two-attribute (or two-dimensional) warranty policies and can be described in terms of regions in a plane with one axis representing time and the other axis the usage. In this paper, we consider two-attribute warranty policies for non-repairable items. The item failures are described in terms of a bivariate exponential distribution. Analytical expressions are derived for the resulting two-dimensional renewal function and the cost of warranty. An illustrative numerical example is presented to study the effect of correlation between the warranty variables on the cost of warranty.     

An EM algorithm for estimating the parameters of bivariate Weibull distribution under random censoring

We consider the problem of estimation of the parameters of the Marshall-Olkin Bivariate Weibull distribution in the presence of random censoring. Since the maximum likelihood estimators of the parameters cannot be expressed in a closed form, we suggest an EM algorithm to compute the same. Extensive simulations are carried out to conclude that the estimators perform efficiently under random censoring.     

Implementing lifetime performance index of products with two-parameter exponential distribution

The manufacturing industry has prioritised enhancing the quality, lifetime and conforming rate of products. Process capability indices (PCIs) are used to measure process potential and performance. The process capability is evaluated with product survival time and a longer lifetime implies a better process capability and a higher reliability. In order to save experimental time and cost, a censored sample arises in practice. In the case of product possessing a two-parameter exponential distribution, this study constructs a uniformly minimum variance unbiased estimator (UMVUE) of the lifetime performance index based on the type II right-censored sample. Then the UMVUE of the lifetime performance index is utilised to develop the new hypothesis testing procedure in the condition of known lower specification limit. Finally, two practical examples are illustrated to employ the testing procedure to determine whether the product is reliable.     

Chebyshev approximation of the distribution function of nonnegative random variables

The research described in this publication was made possible in part by grant No. UAL200 from the Joint Fund of the Government of Ukraine and International Science Foundation.     

The central limit theorems for fuzzy random variables

The new concept of the central limit theorem for fuzzy random variables is discussed in this paper by proposing the convergence in distribution for fuzzy random variables. We first consider the limit properties of fuzzy numbers by invoking the Hausdorff metric and then we extend it to the weak and strong convergence of fuzzy distribution functions. We provide a notion of fuzzy normal distribution. Then the central limit theorem for fuzzy random variables follows naturally.     

An efficient iterative algorithm for designing an asymptotically optimal modified unrestricted uniform polar quantization of bivariate Gaussian random variables

In this paper, a more efficient and a more accurate algorithm is developed for designing asymptotically optimal unrestricted uniform polar quantization (UUPQ) of bivariate Gaussian random variables compared to the existing algorithms on this subject. The proposed algorithm is an iterative one defining the analytical model of asymptotically optimal UUPQ in only a few iterations. The UUPQ model is also improved via optimization of the last magnitude reconstruction level so that the mean squared error (MSE) is minimal. Moreover, for the straightforward performance assessment of our analytical UUPQ model an asymptotic formula for signal to quantization noise ratio (SQNR) is derived, which is reasonably accurate for any rate (R) greater than or equal to 2.5 bits/sample. It is demonstrated empirically that our asymptotically optimal UUPQ model outperforms the previous UUPQ models in terms of SQNR. Eventually, the transition from the analytical to the practically designed UUPQ model, as an important aspect in quantizer design, is considered in the paper and, as a result, a novel method to achieve this is provided. The proposed method is applicable to the practical design of any unrestricted polar quantization.     

Exact distribution of the linear combination of p Gumbel random variables

The distribution of linear combinations of independent Gumbel random variables arises explicitly in many applied areas, including economics, electrical and electronic engineering, and hydrological sciences. However, the exact distribution has not been known in its most general form. In this paper, the exact distribution of the linear combination Z=α₁ X ₁+···+α _p X _p is derived when X _j are independent Gumbel random variables. A computer program is provided for the associated percentile points.     

Strong convergence for weighted sums of fuzzy random sets

In this paper, we establish some results on strong convergence for sums and weighted sums of uniformly integrable fuzzy random sets taking values in the space of upper-semicontinuous fuzzy sets in R^p.     

FIM for Arnold and Strauss's bivariate gamma distribution

The Fisher information matrix (FIM) as well as tools for its numerical computation is derived for Arnold and Strauss’ [1988. Bivariate distributions with exponential conditionals, J. Amer. Statist. Assoc. 83 (1988) 522-527] bivariate gamma distribution. Numerical tabulations of the matrix are also provided for practical purposes.     

Triangular fuzzification of random variables and power of distribution tests: Empirical discussion

A fuzzifying process of finitely valued random variables by means of triangular fuzzy sets is analyzed. Empirical studies show that if the random variable takes on a small number of different values, the one-sample test about the (fuzzy) mean of the fuzzified random variable is frequently more powerful than the classical test about the mean of the original random variable. This empirical conclusion is theoretically supported as follows: whenever the number of different values of a random variable X is up to 4, the mean of the fuzzified random variable captures the whole information on its distribution. As a consequence, the statistical test about the mean of the fuzzified random variable can be considered in fact as a goodness-of-fit test for the original random variable and, analogously, the J-sample test becomes a test for the equality of J distributions. Comparative simulation studies of these procedures with respect to other well-known methods are carried out. A real-life example illustrates the introduced methodology.     

The computer generation of poisson random variables

An exact method for the generation of Poisson random variables on a computer is presented. The average time required per random variate decreases as the Poisson parameter tends to infinity.     

Applications of dimensionality reduction and exponential sums to graph automorphism

Given a connected undirected graph, we associate a simplex with it such that two graphs are isomorphic if and only if their corresponding simplices are congruent under an isometric map. In the first part of the paper, we study the effectiveness of a dimensionality reduction approach to Graph Automorphism. More precisely, we show that orthogonal projections of the simplex onto a lower dimensional space preserves an automorphism if and only if the space is an invariant subspace of the automorphism. This insight motivates the study of invariant subspaces of an automorphism. We show the existence of some interesting (possibly lower dimensional) invariant subspaces of an automorphism. As an application of the correspondence between a graph and its simplex, we show that there are roughly a quadratic number of invariants that uniquely characterize a connected undirected graph up to isomorphism.In the second part, we present an exponential sum formula for counting the number of automorphisms of a graph and study the computation of this formula. As an application, we show that for a fixed prime p and any graph G, we can count, modulo p, the number of permutations that violate a multiple of p edges in G in polynomial time.