The asymptotic sufficiency of sparse order statistics in tests of fit with nuisance parameters

In an earlier paper, it was shown that for the problem of testing that a sample comes from a completely specified distribution, a relatively small number of order statistics is asymptotically sufficient, and for all asymptotic probability calculations the joint distribution of these order statistics can be assumed to be normal. In the present paper, these results are extended to certain cases where the problem is to test the hypothesis that a sample comes from a distribution which is a member of a specified parametric family of distributions, with the parameters unspecified.     

On the bivariate negative binomial distribution of mitchell and paulson

The bivariate negative binomial distribution of Mitchell and Paulson [17] for the case b = c = 0 is shown to be equivalent to the accident proneness model of Edwards and Gurland [4] and Subrahmaniam [19,20]. The diagonal series expansion of its joint probability function is then derived. Two other formulations of this distribution are also considered: (i) as a mixture model, which showed how it arises as the discrete analogue to the Wicksell-Kibble bivariate gamma distribution, and (ii) as a consequence of the linear birth-and-death process with immigration.     

Asymptotic joint normality of an increasing number of multivariate order statistics and associated cell frequencies

Take n independent identically distributed (IID) observations from a continuous r-variate population, and choose some order statistics from each of the r variates. These order statistics are used to construct a grid in r-dimensional space. Under certain conditions, it is shown that as n increases we can choose an increasing number of order statistics in such a way that the asymptotic joint distribution of the chosen order statistics and of the frequencies of sample points falling in the cells of the grid can be assumed to be a normal distribution. An application to testing independence of random variables is given.     

Some bayes tests and their asymptotic properties for the multivariate,multisample goodness-of-fit problem

Independent samples are taken from C multivariate populations with continuous but unknown cumulative distribution function c.d.f.). The problem is to test the hypothesis that the C population c.d.f's are identical to a specified c.d.f. We approach this problem by first transforming the data so that the hypothesis being tested is that the common distribution is uniform over a unit hypercube. We then construct some Bayes tests and investigate their asymptotic properties. These tests are based on the asymptotic normality of the number of observations falling in the “asymptotically sufficient groupings”.     

Optimal and suboptimal procedures in group sequential sampling

This paper investigates the problem of choosing between two simple hypothesis, H₀ and H₁, in terms of independent, identically distributed random variables, when observations can be taken in groups. At any stage in the decision process it must be decided whether to stop and take action now or to continue, in which case the size of the next group of observations must be decided upon. The problem is to find an optimal procedure incorporating a stopping, group size (batch) and terminal action rule. It is proven, in general, that the optimal stopping and terminal action rule is of the sequential probability ratio type (SPRT). Fixed stopping rules of the SPRT type are studied and an iterative procedure of the policy improvement type, both with and without a value determination step, is developed. It is shown, for the general situation, that both the average risk and scheduling rule converge to the optima. Also, six suboptimal scheduling rules are considered with respect to the average risks they achieve. Numerical results are presented to illustrate the effectiveness of the procedures.     

一类多目标模糊系数线性规划问题

讨论了一类所有系数均为模糊数的多目标线性规划问题 .通过对模糊数的比较 ,将模糊多目标线性规划模型转化为清晰的多目标模型 ,并应用一种基于线性隶属函数的模糊规划算法求其协调解 .最后给出了一个数值例子 .     

基于随机拟合的悬浮式深弹弹阵特征建模＊

通过分析悬浮式深弹发射后在空中的弹道特性,建立了深弹弹着点坐标的计算模型。采取蒙特卡洛方法,分别对舰艇六自由度状态下,单管和六联装火箭深弹的弹着点坐标进行了仿真计算,对弹着点的分布规律进行了研究,并得出弹着点的联合密度函数。结果表明,在发射参数存在正态扰动时,弹着点散布区域均呈椭圆形分布,弹着点坐标均仍服从正态分布。     

基于随机拟合的悬浮式深弹弹阵特征建模

通过分析悬浮式深弹发射后在空中的弹道特性,建立了深弹弹着点坐标的计算模型。采取蒙特卡洛方法,分别对舰艇六自由度状态下,单管和六联装火箭深弹的弹着点坐标进行了仿真计算,对弹着点的分布规律进行了研究,并得出弹着点的联合密度函数。结果表明,在发射参数存在正态扰动下,弹着点散布区域均呈椭圆形分布,弹着点坐标均仍服从正态分布。     

Nonstationary stochastic gold-mining: A time-sequential tactical-allocation problem

This paper presents an extension of gold-mining problems formulated in earlier work by R. Bellman and J. Kadane. Bellman assumes there are two gold mines labeled A and B, respectively, each with a known initial amount of gold. There is one delicate gold-mining machine which can be used to excavate one mine per day. Associated with mine A is a known constant return rate and a known constant probability of breakdown. There is also a return rate and probability of breakdown for mine B. Bellman solves the problem of finding a sequential decision procedure to maximize the expected amount of gold obtained before breakdown of the machine. Kadane extends the problem by assuming that there are several mines and that there are sequences of constants such that the jth constant for each mine represents the return rate for the jth excavation of that mine. He also assumes that the probability of breakdown during the jth excavation of a mine depends on j. We extend these results by assuming that the return rates are random variables with known joint distribution and by allowing the probability of breakdown to be a function of previous observations on the return rates. We show that under certain regularity conditions on the joint distributions of the random variables, the optimal policy is: at each stage always select a mine which has maximal conditional expected return per unit risk. This gold-mining problem is also a formulation of the problem of time-sequential tactical allocation of bombers to targets. Several examples illustrating these results are presented.     

Note on sequential search

Optimal distribution of search effort problems for certain kinds of moving targets are considered. It is shown that they can be formulated by the use of appropriate Markovian decision models.     

Estimation of the survival function of a repairable standby series system

This article deals with the statistical analysis of an N-component series system supported by an active standby and one repair facility. Assuming that the life and repair times of the components are independent exponential random variables, the probability distribution of the first passage to the system failure time is shown to be a convolution of two independent exponential distributions. Three observation schemes are considered to obtain the maximum likelihood estimates of the survival function. Information matrices are supplied. Numerical results based on Monte Carlo simulation are presented. It is noted that component level information (failure rate, repair rate) is not necessary for estimating the survival function of the system.     

Simplified estimates of the parameters of the double exponential distribution based on optimum order statistics from a middle-censored sample

In an ordered sample from a given population, a few of the consecutive observations from somewhere in the middle may be missing Further, we may be constrained to use a few, and not all, of the remaining observations for purposes of estimation of population parameters. In this paper, such a situation is considered for the double exponential distribution and best linear unbiased estimates are obtained for its parameters, based on a choice of an optimum set of order statistics when the number of observations in the set are prefixed.     

Warranty policies and burn-in

Warranty is an important factor for consumer durable products in the marketplace. However, the warranty cost may drastically reduce profitability. Burn in is a common procedure to improve the quality of products after they have been produced, but it is also costly. By taking both the burn-in procedure and warranty policy into consideration, several cost functions can be formulated and optimized. Assuming that the failure-rate function of the product has a bathtub shape, it is shown that the optimal burn-in times that minimize the considered cost functions never exceed the first change point of the failure-rate function. The continuous dependence of the optimal burn-in times on the model parameters and the underlying distribution is also established. © 1997 John Wiley & Sons, Inc. Naval Research Logistics 44: 199–209, 1997     

On the existence of joint production functions

Within a general framework of production correspondences satisfying a set of weak axioms necessary and sufficient conditions for the existence of a joint production function are given. Without enforcing the strong disposability of inputs or outputs it is shown that a joint production function exists if and only if both input and output correspondences are strictly increasing along rays.     

An application of the MTP2 property on bounds on system reliability

This paper is concerned with the joint prior distribution of the dependent reliabilities of the components of a binary system. When this distribution is MTP₂ (Multivariate Totally Positive of Order 2), it is shown in general that this actually makes the machinery of Natvig and Eide [7] available to arrive at the posterior distribution of the system's reliability, based on data both at the component and system level. As an illustration in a common environmental stress case, the joint prior distribution of the reliabilities is shown to have the MTP₂ property. We also show, similarly to Gåsemyr and Natvig [3], for the case of independent components given component reliabilities how this joint prior distribution may be based on the combination of expert opinions. A specific system is finally treated numerically. © 1997 John Wiley & Sons, Inc. Naval Research Logistics 44: 741–755, 1997     

Optimal lot sizing and inspection policy for an EMQ model with imperfect inspections

An EMQ model with a production process subject to random deterioration is considered. The process can be monitored through inspections, and both the lot size and the inspection schedule are subject to control. The “in-control” periods are assumed to be generally distributed and the inspections are imperfect, i.e., the true state of the process is not necessarily revealed through an inspection. The objective is the joint determination of the lot size and the inspection schedule, minimizing the long-run expected average cost per unit time. Both discrete and continuous cases are examined. A dynamic programming formulation is considered in the case where the inspections can be performed only at discrete times, which is typical for the parts industry. In the continuous case, an optimum inspection schedule is obtained for a given production time and given number of inspections by solving a nonlinear programming problem. A two-dimensional search procedure can be used to find the optimal policy. In the exponential case, the structure of the optimal inspection policy is established using Lagrange's method, and it is shown that the optimal inspection times can be found by solving a nonlinear equation. Numerical studies indicate that the optimal policy performs much better than the optimal policy with periodic inspections considered previously in the literature. The case of perfect inspections is discussed, and an extension of the results obtained previously in the literature is presented. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 165–186, 1998     

宽带目标回波模型的物理解释及小波变换表示

在点目标回波模型基础上,通过对理想运动体目标的分析,引入了距离-速度联合分布密度和时延-时间伸缩联合分布密度函数的概念,详细推导了延展目标的宽带回波模型,解释了各密度函数和宽带扩展函数的物理意义及相互关系,最后给出了宽带回波模型的小波变换表示方法.     

The joint signature of coherent systems

We investigate the joint signature of $m$ coherent systems, under the assumption that the components have independent and identically distributed lifetimes. The joint signature, for a particular ordering of failure times, is an $m$‐dimensional matrix depending solely on the composition of the systems and independent of the underlying distribution function of the component lifetimes. The elements of the $m$‐dimensional matrix are formulated based on the joint signatures of numerous series of parallel systems. The number of the joint signatures involved is an exponential function of the number of the minimal cut sets of each original system and may, therefore, be significantly large. We prove that although this number is typically large, a great number of the joint signatures are repeated, or removed by negative signs. We determine the maximum number of different joint signatures based on the number of systems and components. It is independent of the number of the minimal cut sets of each system and is polynomial in the number of components. Moreover, we consider all permutations of failure times and demonstrate that the results for one permutation can be of use for the others. Our theorems are applied to various examples. The main conclusion is that the joint signature can be computed much faster than expected.     

A cost analysis of sampling inspection under Military Standard 105D

Military Standard 105D has been almost universally adopted by government and private consumers for the lot-by-lot sampling inspection of product which may be inspected on a dichotomoun basis The plan specifies, for each lot size, a random sample size and set of acceptance numbers (maximum allowable number of defectives in each sample). The acceptance numbers are based upon the binomial distribution and depend upon the quality required by the purchaser. Where several consecutive lots are submitted, a shift to less severe (“reduced”) inspection or more severe (“tightened”) inspection is specified when the ongoing quality is very high or low. Further experience permits a return to normal sampling from either of these states This paper examines the long range costs of such a sampling scheme. The three inspection types are considered as three distinct Markov chains, with periodic transitions from chain to chain. The expected sample size and the expected proportion of rejected product are determined as a function of the two parameters under control of the manufacturer, lot size and product quality. Some numerical examples are given which illustrate how to compute the overall cost of sampling inspection. Suggestions are made concerning the choice of parameters to minimize this cost.     

Inventory control of by-products

The system to be controlled produces n products simultaneously in fixed proportions every time it is activated. Demands for the products in any period are components of an n dimensional vector random variable with known distribution function. Cases of excess demands backlogged and excess demands lost are considered. In the former the notion of k convexity can be generalized to guarantee relatively simple form for the optimal policy in an n decision problem. In the latter, this generalization was not successful although when there is no setup cost, a convexity argument can be used to show that the optimal policy has a simple form.