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.     

Asymptotics of the Moments of Extreme-Value Related Distribution Functions

The asymptotic cost of many algorithms and combinatorial structures is related to the extreme-value Gumbel distribution exp(-exp(-x)). The following list is not exhaustive: Trie, Digital Search Tree, Leader Election, Adaptive Sampling, Counting Algorithms, trees related to the Register Function, Composition of Integers, some structures represented by Markov chains (Column-Convex Polyominoes, Carlitz Compositions), Runs and number of distinct values of some multiplicity in sequences of geometrically distributed random variables. Sometimes we can start from an exact (discrete) probability distribution, sometimes from an asymptotic analysis of the discrete objects (e.g., urn models) before establishing the relationship with the Gumbel distribution function. Also some Markov chains are either exactly and directly given by the structure itself, or as a limiting Markov process. The main motivation of the paper is to compute the asymptotic distribution and the moments of the random variables in question. The moments are usually given by a dominant part and a small fluctuating part. We use Laplace and Mellin transforms and singularity analysis and aim for a unified treatment in all cases. Furthermore, our goal is a purely mechanical computation of dominant and fluctuating components, with the help of a computer algebra system. We provide each time the first three moments, but the treatment is (almost) completely automatic. We need some real analysis for the approximations and apart from that only easy complex analysis; simple poles and a few special functions.     

Extended Rama Distribution: Properties and Applications

In this paper, the Rama distribution (RD) is considered, and a new model called extended Rama distribution (ERD) is suggested. The new model involves the sum of two independent Rama distributed random variables. The probability density function (pdf) and cumulative distribution function (cdf) are obtained and analyzed. It is found that the new model is skewed to the right. Several mathematical and statistical properties are derived and proved. The properties studied include moments, coefficient of variation, coefficient of skewness, coefficient of kurtosis and moment generating function. Some simulations are undertaken to illustrate the behavior of these properties. In addition, the reliability analysis of the distribution is investigated through the hazard rate function, reversed hazard rate function and odds function. The parameter of the distribution is estimated based on the maximum likelihood method. The distributions of order statistics for ERD are also presented. The performance of the suggested model is compared with several other lifetime distributions based on some goodness of fit tests on a real dataset. It turns out that the suggested model is more flexible than its competitors considered in this study, for modeling real lifetime data.     

High-resolution models of motion of macromolecules in cell membranes

The path of a macromolecule on a cell membrane is modeled by a sum of independent identically distributed random variables. Random variables with simple discrete distribution functions capture some important aspects of the jump or hop diffusion reported from single particle tracking experiments that measure the motion of single molecules on a cell membrane. The detail provided by the distribution function for the random variables is critical for accurate simulations of the motion and interactions of macromolecules on the cell membrane. Additionally, the probability distribution for the random variables is easily estimated from single-particle tracking data. The diffusion constant is given by the second moment of the probability distribution, which agrees with the diffusion constant estimated from the mean-square displacement, and thus represents far less information than the distribution function.     

Optimal facility location with random throughput costs

In this paper we consider the optimal location and size of facilities where the throughput costs for each facility are random. Given a set of origins and a set of destinations, we want to determine the optimal location and size of a set of intermediate facilities in order to minimize the expected total generalized transportation cost. The generalized transportation cost of a freight unit from an origin to a destination passing through a facility is the sum of two terms: the transportation cost from the origin to the destination through the facility and the throughput cost of the facility. While the first term is deterministic, the second one is stochastic with a Gumbel probability distribution. Looking for the expected value of the optimal solution, a mixed deterministic nonlinear problem for the optimal location of the facilities is derived. Two heuristics, which give very good approximations to the optimum, are proposed.     

Superposition properties and performance bounds of stochastictimed-event graphs

This paper addresses the performance evaluation of stochastic timed-event graphs. The transition firing times are random variables with general distribution. We first consider a stochastic timed-event graph in which the firing times are generated by time superposition (or addition) of two sets of random variable sequences. Properties of this system are established. Chiefly, we prove that the average cycle time is subadditive, i.e., it is smaller than the sum of the average cycle times of the two stochastic timed-event graphs in which the firing times are generated by one of the two sets of random variable sequences, respectively. Based on these superposition properties, we derive various upper bounds of the average cycle time of a general stochastic timed-event graph. In particular, we obtain upper bounds which converge to the exact average cycle time as the standard deviations of the firing times decrease. Finally, we derive performance bounds for stochastic timed-event graphs with bounded firing times     

Exact algorithms for computing p-values of statistics-linear combination of 3-nomial variables

Algorithms for computing exact distribution values of statistics-linear combination of some multinomial variables called 3-nomial variables are derived. In the general case, there are no explicit formulae of the cumulative distribution function of these statistics that allow the direct computation of p-values. The exact distribution in the independent case and in some dependent ones is studied. For each of the two cases of dependence, an algorithm based on recursive relationships is proposed, giving the number of favorable cases and combinatorics to minimize computation time. For each of these NFC algorithms, concurrent PGF algorithms and some realized comparisons are presented.     

Approximations of bayes and minimax risks and the least favorable distribution

In this paper we present a method for approximating the risks and Bayes risk associated with a Bayes decision procedure. Additionally, our method leads to approximating the least favorable distribution and the risk associated with the minimax decision procedure. We assume two states of nature (or classes of patterns) and multivariate probability density functions. Taylor series expansions are used, and an nth-order polynomial equation derived from such expansions provides an approximation to one of the least favorable probabilities. An application to a normally distributed random vector of observables is presented with numerical comparisons. The method can be generalized to cases having more than two states of nature by using Taylor series expansions in several variables.     

广义高斯分布随机变量生成算法

以几个在［０,１］区间上相互独立的均匀分布随机变量为基础,综合运用了反函数法、变换抽样法、舍选抽样法以及综合法产生了广义高斯分布的随机变量,并给出了严格的理论证明以及详细的实现算法和实验结果。算法的推导和实现虽然是针对方差σ＝ １０,形状参数ｖ＝ ０．７ 的广义高斯分布完成的,然而通过相应参数的调整,该算法可以产生具有任意形状参数和任意方差的广义高斯分布随机变量。     

A new replacement warranty policy indexed by the product’s correlated failure and usage time

We study a new warranty policy for non-repairable products which is indexed by two correlated random variables, age and usage and covers all failures in (0, t]. Two different warranty costs for the replacement of the failed product are considered, according to its usage being greater or less than a pre-specified level s > 0. A bivariate probability distribution function is applied to incorporate the correlation effect of the two variables. Analytical expressions of the probability density function of the total warranty cost and its expected value, the probability distribution functions of the number of the failed products with usage greater or less than s and their corresponding expected values and costs are derived. Limit results are also obtained. The results obtained are useful measures in establishing the compensation policy and the evaluation of its performance under the proposed warranty. Illustrative numerical examples of the expected cost for Paulson, Pareto and Beta Stacy bivariate distributions are presented and discussed. In particular for Paulson’s bivariate probability distribution, closed form expressions for the expected costs are obtained.     

Exact prediction intervals for future exponential lifetime based on random generalized order statistics

In this paper we develop two pivotal quantities to construct exact predication intervals for future exponential lifetime based on a random number of lower generalized order statistics. The distribution functions of the two pivotal quantities, when the sample size is assumed to be integer-valued random variable, are derived. Three important special cases for the random sample size are presented. A simulation study is conducted for illustrative purposes.     

Stochastic Optimal Design for Unknown Linear Discrete‐Time System Zero‐Sum Games in Input‐Output form Under Communication Constraints

In this paper, stochastic optimal strategy for unknown linear discrete‐time system quadratic zero‐sum games in input‐output form with communication imperfections such as network‐induced delays and packet losses, otherwise referred to as networked control system (NCS) zero‐sum games, relating to the H_∞ optimal control problem is solved in a forward‐in‐time manner. First, the linear discrete‐time zero sum state space representation is transformed into a linear NCS in the state space form after incorporating random delays and packet losses and then into the input‐output form. Subsequently, the stochastic optimal approach, referred to as adaptive dynamic programming (ADP), is introduced which estimates the cost or value function to solve the infinite horizon optimal regulation of unknown linear NCS quadratic zero‐sum games in the presence of communication imperfections. The optimal control and worst case disturbance inputs are derived based on the estimated value function in the absence of state measurements. An update law for tuning the unknown parameters of the value function estimator is derived and Lyapunov theory is used to show that all signals are asymptotically stable (AS) and that the estimated control and disturbance signals converge to optimal control and worst case disturbances, respectively. Simulation results are included to verify the theoretical claims.     

An improved quantum-behaved particle swarm optimization and its application to medical image registration

This paper investigates the quantum-behaved particle swarm optimization (QPSO) algorithm from the perspective of estimation of distribution algorithm (EDA) which reveals the reason of QPSO's superiority. A revised QPSO (RQPSO) technique with a novel iterative equation is also proposed. The modified technique is deduced from the distribution function of the sum of two random variables with exponential and normal distribution, respectively. We present a diversity-controlled RQPSO (DRQPSO) algorithm, which helps prevent the evolutionary algorithms’ tendency to be easily trapped into local optima as a result of rapid decline in diversity. Both the RQPSO and DRQPSO are tested on three benchmark functions, as well as in medical image registration for performance comparison with the particle swarm optimization and QPSO.     

Generating inverse Gaussian random variates by approximation

The inverse Gaussian distribution is a useful distribution with important applications. But there is less discussion in the literature on sampling of this distribution. The method given in [Atkinson, A.C., 1982. The simulation of generalized inverse Gaussian and hyperbolic random variables. SIAM Journal on Scientific and Statistical Computing 3(4), 502-515] is based on rejection method where some (uniform) random numbers from the sample are discarded. This feature makes it difficult to take advantage of the low discrepancy sequences which have important applications. In [Michael, J., Schucany, W., Haas, R., 1976. Generating random variates using transformations with multiple roots. The American Statistician 30(2), 88-90], Michael et al. give a method to generate random variables with inverse Gaussian distribution. In their method, two pseudorandom numbers uniformly distributed on (0, 1) are needed in order to generate one inverse Gaussian random variable. In this short paper, we present a new method, based on direct approximate inversion, to generate the inverse Gaussian random variables. In this method, only one pseudorandom number is needed in generating one inverse Gaussian random variate. This method enables us to make use of the better convergence of low discrepancy sequence than the pseudorandom sequence. Numerical results show the superiority of low discrepancy sequence than the pseudorandom sequence in simulating the mean of the inverse Gaussian distribution by using our sampling method. Further application of this method in exotic option pricing under the normal inverse Gaussian model is under investigation.     

Some properties of fuzzy random renewal processes

Fuzzy random variable is a measure function from a probability space to a collection of fuzzy variables. Based on the fuzzy random theory, this paper addresses some properties of fuzzy random renewal processes generated by a sequence of independent and identically distributed (iid) fuzzy random interarrival times. The relationship between the expected value of the fuzzy random renewal variable and the distribution functions of the /spl alpha/-pessimistic values and /spl alpha/-optimistic values of the interarrival times is discussed. Furthermore, the fuzzy random style of renewal equation is provided. Finally, fuzzy random Blackwell's renewal theorem and Smith's key renewal theorem are also given.     

Process Simulation Using Randomized Markov Chain and Truncated Marginal Distribution

Generating pseudo random objects is one of the key issues in computer simulation of complex systems. Most earlier simulation systems include procedures for the generation of independent and identically distributed random variables or some classical random processes, such as white noise. In this paper we propose a new approach to the generation of wide ranges of processes that are characterized by marginal distribution and autocorrelation function that are significant in many cases. The proposed algorithm is based on the use of truncated distribution that gives more simplicity and efficiency in comparison with the previous one. The effectiveness of the proposed algorithm is verified using computer simulation of various real examples.     

Using Pearson Correlation to Improve Envelopes around the Distributions of Functions

Given two random variables whose dependency relationship is unknown, if a new random variable is defined whose samples are some function of samples of the given random variables, the distribution of this function is not fully determined. However, envelopes can be computed that bound the space through which its cumulative distribution function must pass. If those envelopes could be made to bound a smaller space, the cumulative distribution, while still not fully determined, would at least be more constrained. We show how information about the correlation between values of given random variables can lead to better envelopes around the cumulative distribution of a function of their values.     

Delay-distribution-dependent stability of stochastic discrete-time neural networks with randomly mixed time-varying delays

In this paper, the stability analysis problem for a new class of discrete-time neural networks with randomly discrete and distributed time-varying delays has been investigated. Compared with the previous work, the distributed delay is assumed to be time-varying. Moreover, the effects of both variation range and probability distribution of mixed time-delays are taken into consideration in the proposed approach. The distributed time-varying delays and coupling term in complex networks are considered by introducing two Bernoulli stochastic variables. By using some novel analysis techniques and Lyapunov–Krasovskii function, some delay-distribution-dependent conditions are derived to ensure that the discrete-time complex network with randomly coupling term and distributed time-varying delay is synchronized in mean square. A numerical example is provided to demonstrate the effectiveness and the applicability of the proposed method.     

Analysis of two-unit stand-by system with switch failure

A two-unit warm stand-by redundant system is analysed under the assumptions that the switching device (s.d.) is imperfect and the repair facility is not available (dead time) after each repair completion. The repair time of the units, the s.d. and the dead time are arbitrarily distributed random variables. The lifetimes of the unit while on-line and while standing by and that of the s.d. are exponentially distributed random variables. The reliability characteristics of interest to system designers have been derived. Some results obtained earlier are shown as particular cases.     

On Estimation in M/G/c/c Queues

We derive the minimum variance unbiased estimator (MVUE) and the maximum likelihood estimator (MLE) of the stationary probability function (pf) of the number of customers in a collection of independent M / G / c / c subsystems. It is assumed that the offered load and number of servers in each subsystem are unknown. We assume that observations of the total number of customers in the system are utilized because it may be impractical or impossible to observe individual server occupancies. Both estimators depend on the R distribution (the distribution of the sum of independent right truncated Poisson random variables) and R numbers.