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.     

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.     

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.     

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.     

Chi-squared components for tests of fit and improved models for the grouped exponential distribution

We consider testing for an exponential distribution with unspecified rate parameter when it is only possible to observe the counts in groups with boundaries specified before sighting the data. On the basis of a size and power study we recommend that tests of fit for the exponential distribution be based on the Anderson-Darling statistic and the SW2 statistic recommended by Gulati and Neus [(2001). Goodness-of-fit statistics for the exponential distribution when the data are grouped. In: Huber-Carol, C., Balakrishnan, N., Nikulin, M.S., Mesbah, M., (Eds.), Goodness-of-Fit Tests and Validity of Models. Birkhauser, Boston, pp. 113-123 (Chapter 9)]. We also suggest that inference based on one of these be complemented by examination of the components of the X² statistic. We illustrate how to use these components to give improved models.     

Existence and uniqueness of solution for fuzzy random differential equations with non-Lipschitz coefficients

In this paper, a class of fuzzy random differential equations with non-Lipschitz coefficients is studied. The existence and uniqueness of solutions for fuzzy random differential equations with non-Lipschitz coefficients is first proved. Then the dependence of fuzzy random differential equations on initial values is discussed. Finally the non-confluence property of the solution for fuzzy random differential equation is investigated. Our main tool is the Gronwall lemma.     

The complementary exponential geometric distribution: Model, properties, and a comparison with its counterpart

In this paper, we proposed a new two-parameter lifetime distribution with increasing failure rate, the complementary exponential geometric distribution, which is complementary to the exponential geometric model proposed by Adamidis and Loukas (1998). The new distribution arises on a latent complementary risks scenario, in which the lifetime associated with a particular risk is not observable; rather, we observe only the maximum lifetime value among all risks. The properties of the proposed distribution are discussed, including a formal proof of its probability density function and explicit algebraic formulas for its reliability and failure rate functions, moments, including the mean and variance, variation coefficient, and modal value. The parameter estimation is based on the usual maximum likelihood approach. We report the results of a misspecification simulation study performed in order to assess the extent of misspecification errors when testing the exponential geometric distribution against our complementary one in the presence of different sample size and censoring percentage. The methodology is illustrated on four real datasets; we also make a comparison between both modeling approaches.     

逆映照仿真技术确定主控变量DCS控制限

以对减四线生产数据处理为例,介绍了自主开发的确定主控变量DCS控制限的逆映照仿真技术。该技术通过模式分类,映照、逆映照,样本变量分布统计,仿真,以及生产检测,可最终给出主控变量的DCS控制限,实施生产优化控制。提出的方法紧密结合生产实际,技术处理严谨,方案切实可行。     

Toward a definition and understanding of correlation for variables constrained by random relations

Random relations are random sets defined on a two-dimensional space (or higher). After defining the correlation for two variables constrained by a random relation as an interval, the effect of imprecision was studied by using a multi-valued mapping, whose domain is a space of joint random variables. This perspective led to the notions of consistent and non-consistent marginals, which parallel those of epistemic independence, and unknown interaction and epistemic independence for random sets, respectively. The calculation of the correlation bounds entails solving two optimisation problems that are NP-hard. When the entire random relation is available, it is shown that the hypothesis of non-consistent marginals leads to correlation bounds that are much larger (four orders of magnitude in some cases) than those obtained under the hypothesis of consistent marginals; this hierarchy parallels the hierarchy between probability bounds for unknown interaction and strong independence, respectively. Solutions of the optimisation problems were found at the extremes of their feasible intervals in 80–100% of the cases when non-consistent marginals were assumed, but this range became 75–84% when consistent marginals were assumed. When only the marginals are available, there is a complete loss of knowledge in the correlation, and the correlation interval is nearly vacuous or vacuous (i.e. [ ? 1,1]) even if the measurements are sufficiently accurate in which their narrowed intervals do not overlap. Solutions to the optimisation problems were found at the extremes of their feasible intervals 50% or less of the times.     

The essential order of approximation for nearly exponential type neural networks

1 Introduction Artificial neural networks have been extensively applied in various fields of science and engineering. Why is so is mainly because the feedforward neural networks (FNNs) have the universal approximation capability[1-9]. A typical example of…     

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.     

Sums of exchangeable Bernoulli random variables for family and litter frequency data

We describe new families of discrete distributions that are used to model sums of exchangeable Bernoulli random variables. These discrete distributions can be parameterized in terms of their range, mean, variance, and shape parameters. These models are fitted to an example involving mortality rates for children in a survey of families in Brazil. The methods illustrate that mortality rates in this survey increase with family size and that the correlation of within-family mortality status also depends on the family size. These methods are also applied to a laboratory study of birth defects in mice.     

A small sample comparison of maximum likelihood, moments and L-moments methods for the asymmetric exponential power distribution

Three methods of estimation, namely maximum likelihood, moments and L-moments, when data come from an asymmetric exponential power distribution are considered. This is a very flexible four-parameter family exhibiting variety of tail and shape behaviours. The analytical expression of the first four L-moments of these distributions are derived, allowing for the use of L-moments estimators. A simulation study compares the three estimation methods in small samples.     

Semi-supervised PLVR models for process monitoring with unequal sample sizes of process variables and quality variables

As the key indicators of chemical processes, the quality variables, unlike process variables, are often difficult to obtain at the high frequency. Obtaining the data of quality variables is expensive, so the data are only collected as a small portion of the whole dataset. It is common to see in both continuous and batch processes that the sample sizes of process variables and quality variables are unequal. To effectively integrate two different observation sources, including quality variables collected at a low frequency and process variables sampled at a high rate, a semi-supervised probabilistic latent variable regression model (SSPLVR) is proposed in this article. It enhances the performance monitoring of the variations of process variables and quality variables. The proposed semi-supervised model is applied to continuous and batch processes respectively. The SSPLVR model calibrated by the expectation-maximization algorithm is derived and the corresponding statistics is also systematically developed for the fault detection. Finally, two simulated case studies, TE benchmark for a continuous process problem and the penicillin fermentation for a batch process problem, are presented to illustrate the effectiveness of the proposed method.     

A priori determination of bounds for the optimal value of variables in integer linear programming problems

The paper outlines a simple technique for determining a narrower interval containing the coordinates of the optimal solution to integer linear programming problems. Computational experiments show that the lengths of the given ranges of variables decreases by 30–90% on average. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 2, pp. 86–93, March–April 2006.     

An algorithm for uniform random sampling of points in and on a hypersphere

A new algorithm, based on a set of generalized polar coordinates, is given for uniform random sampling of points in the interior or on the surface of an N-dimensional hypersphere. Computation times are compared for a variety of algorithms on several high-speed computers.     

Parameter estimation for generalized Dirichlet distributions from the sample estimates of the first and the second moments of random variables

Generalized Dirichlet distributions have a more flexible covariance structure than Dirichlet distributions, and the computation for the moments of a generalized Dirichlet distribution is still tractable. For situations under which Dirichlet distributions are inappropriate for data analysis, generalized Dirichlet distributions will generally be an applicable alternative. When the expected values and the covariance matrix of random variables can be estimated from available data, this study introduces ways to estimate the parameters of a generalized Dirichlet distribution for analyzing compositional data. Under the assumption that the sample mean of every variable must be considered for parameter estimation, we present methods for choosing the statistics from a sample covariance matrix to construct a generalized Dirichlet distribution. Some rules for removing inappropriate statistics from a sample covariance matrix to speed up the estimation process are also established. An example for Taiwan’s car market is introduced to demonstrate the applicability of the parameter estimation methods.     

Delay-dependent exponential state estimators for stochastic neural networks of neutral type with both discrete and distributed delays

This paper considered the state estimation for stochastic neural networks of neutral type with discrete and distributed delays. By using available output measurements, the state estimator can approximate the neuron states, and the asymptotic property of the state error is mean square exponential stable and also almost surely exponential stable in the presence of discrete and distributed delays. Under the Lipschitz assumptions for the activation functions and the measurement nonlinearity, a delay-dependent linear matrix inequality (LMI) criterion is proposed to guarantee the existence of the desired estimators by constructing an appropriate Lyapunov-Krasovskii function. It is shown that the existence conditions and the explicit expression of the state estimator can be parameterised in terms of the solution to a LMI. Finally, two numerical examples are presented to demonstrate the validity of the theoretical results and show that the theorem can provide less conservative conditions.