On the characterization of the exponential distribution by the independence of its integer and fractional parts

It is well–known that the fractional part and integer part of an exponentially distributed random variable are independent. Under some regularity conditions, the independence of the fractional part and integer part of a positive random variable characterizes the exponential distribution.     

Random variables with independent integer and fractional parts

For random variables with independent integer and fractional parts: a canonical form is given for those with positive differentiable densities, and a condition ensuring exponentially is made less restrictive.     

On maximum likelihood prediction based on Type II doubly censored exponential data

In this paper, the maximum likelihood predictor (MLP) of the kth ordered observation, t _k, in a sample of size n from a two-parameter exponential distribution as well as the predictive maximum likelihood estimators (PMLE's) of the location and scale parameters, θ and β, based on the observed values t _r, …, t _s (1≤r≤s<k≤n), are obtained in closed forms, contrary to the belief they cannot be so expressed. When θ is known, however, the PMLE of β and MLP of t _k do not admit explicit expressions. It is shown here that they exist and are unique; sharp lower and upper bounds are also provided. The derived predictors and estimators are reasonable and also have good asymptotic properties. As applications, the total duration time in a life test and the failure time of a k-out-of-n system may be predicted. Finally, an illustrative example is included. Received: August 1999     

Linearity of regression for non-adjacent order statistics

Let X ₁,X ₂,…,X _n be a random sample from a continuous distribution with the corresponding order statistics X _1:n≤X _2:n≤…≤X _n:n. All the distributions for which E(X _{k+r: n}|X _k:n)=a X _k:n+b are identified, which solves the problem stated in Ferguson (1967). Received February 1998     

Conditional distributions of generalized order statistics and some characterizations

Generalized order statistics have been introduced in Kamps (1995a). They enable a unified approach to several models of ordered random variables, e.g. (ordinary) order statistics, record values, sequential order statistics, record values from non-identical distributions. The purpose of this paper is to develop conditional distributions of one generalized order statistic given another and to characterize the underlying continuous distribution by different conditional expectations. Well-known results for ordinary order statistics and record values are extended to generalized order statistics. Received: July 1997     

On the asymptotic behaviour of the Stringer bound1

The Stringer bound is a widely used nonparametric 100(1 -α)% upper confidence bound for the fraction of errors in an accounting population. This bound has been found in practice to be rather conservative, but no rigorous mathematical proof of the correctness of the Stringer bound as an upper confidence bound is known and also no counterexamples are available. In a pioneering paper Bickel (1992) has given some fixed sample support to the bound's conservatism together with an asymptotic expansion in probability of the Stringer bound, which has led to his claim of the asymptotic conservatism of the Stringer bound. In the present paper we obtain expansions of arbitrary order of the coefficients in the Stringer bound. As a consequence we obtain Bickel's asymptotic expansion with probability 1 and we show that the asymptotic conservatism holds for confidence levels 1 -α, with α∈ (0,1/2]. It means that in general also in a finite sampling situation the Stringer bound does not necessarily have the right confidence level. Based on our expansions we propose a modified Stringer bound which has asymptotically precisely the right nominal confidence level. Finally, we discuss other consequences of the expansions of the Stringer bound such as a central limit theorem, a law of the iterated logarithm and the functional versions of them.     

Fisher information in record data

We present some general results on Fisher information (FI) contained in upper (or lower) record values and associated record times generated from a sequence of i.i.d. continuous variables. For the record data obtained from a random sample of fixed size, we establish an interesting relationship between its FI content and the FI in the data consisting of sequential maxima. We also consider the record data from an inverse sampling plan (Samaniego and Whitaker, 1986). We apply the general results to evaluate the FI in upper as well as lower records data from the exponential distribution for both sampling plans. Further, we discuss the implication of our results to statistical inference from these record data. Received: December 2001 Acknowledgements. This research was supported by Fondo Nacional de Desarrollo Cientifico y Tecnologico (FONDECYT) grants 7990089 and 1010222 of Chile. We would like to thank the Department of Statistics at the University of Concepción for its hospitality during the stay of H. N. Nagaraja in Chile in March of 2000, when the initial work was done. We are grateful to the two referees for various comments that let to improvements in the paper.     

On testing for independence against right tail increasing in bivariate models

A random variableY is right tail increasing (RTI) inX if the failure rate of the conditional distribution ofX givenY>y is uniformly smaller than that of the marginal distribution ofX for everyy0. This concept of positive dependence is not symmetric inX andY and is stronger than the notion of positive quadrant dependence. In this paper we consider the problem of testing for independence against the alternative thatY is RTI inX. We propose two distribution-free tests and obtain their limiting null distributions. The proposed tests are compared to Kendall's and Spearman's tests in terms of Pitman asymptotic relative efficiency. We have also conducted a Monte Carlo study to compare the powers of these tests.Research supported by an NSERC Canada operating grant at the University of Alberta.Part of this research was done while visiting the University of Alberta supported by the NSERC Canada grant of the first author.     

Ordering results of the smallest and largest order statistics from independent heterogeneous exponentiated gamma random variables

In this paper, we discuss stochastic comparisons of lifetimes of series and parallel systems with heterogeneous exponentiated gamma components. The results established here are developed in two directions. First, when a system possibly has different shape and scale parameters and the matrix of those different parameters following the chain majorization order, we study the reversed hazard rate order of parallel systems and the usual stochastic order of series systems. Next, by using the concept of vector majorization, we establish the usual stochastic order of series systems and the reversed hazard rate order of parallel systems.     

贯穿概率论的n重贝努里概型

以n重贝努里概型为基础,利用极限分布,讨论了概率论中最常见的二项分布、泊松分布、指数分布、正态分布之间的关系,从而得到n重贝努里概型贯穿概率论教程的结论。     

On the standarized empirical process

In this note a new proof of a theorem of CSAKI (1975, 1982) concerning the almost sure behaviour of the standardized empirical process is established.     

Sharp exponential and entropy bounds on expectations of generalized order statistics

Sharp lower and upper bounds on expected values of generalized order statistics are proven by the use of Moriguti's inequality combined with the Young inequality. The bounds are expressed in terms of exponential moments or entropy. They are attainable providing new characterizations of some nontrivial distributions. Received October 2001/Revised May 2002     

On characterizing distributions by conditional expectations of functions of order statistics

In the present paper, we give some general theorems on characterizations based on conditional expectations of the functions of order statistics. In addition, we indicate special forms of the theorems for the familiar probability distributions.     

Monitoring a Process of Exponentially Distributed Characteristics through Minimizing the Sum of the Squared Differences

This study presents an optimal exponent for transforming the exponentials for statistical process control (SPC) applications. The optimal exponent, 3.5454, is determined by minimizing the sum of the squared differences between two distinct cumulative probability functions. The normal distribution closely approximates the transformed distribution. The study investigates an interval of indifference for the exponent using two criteria, the square root of sum of the squared differences ( and the Kullback-Leibler distance (K-L). This interval is [3.4, 3.77], which implies that exponents falling in this interval have very similar results in transforming the exponentials. The study explores an example of flash memory wafer. The individual chart using the transformed data eliminating location the parameter has better detection power than that without eliminating the location parameter. Moreover, the control chart is also easier for practitioners to interpret and implement than probabilistic control limits.     

On a study of the exponential and Poisson characteristics of the Poisson process

Based on the exponential and Poisson characteristics of the Poisson process, in this work we present some characterizations of the Poisson process as a renewal process. More precisely, let γ_t be the residual life at time t of the renewal process A={A(t),t≥0 }, under suitable condition, we prove that if Var(γ_t)=E ² (γ_t),∀t≥0, then A is a Poisson process. Secondly, we show that if Var (A(t)) is proportional to E (A(t)), then A is a Poisson process also, and Var (A(t))=E (A(t)). Received: August 1999     

Recurrence relations for moment and conditional moment generating functions of generalized order statistics

In this paper, we obtain recurrence relations for moment and conditional moment generating functions of generalized order statistics (gos) based on random samples drawn from a population whose distribution is a member of a doubly truncated class of distributions denoted by . Members of the class are characterized in Section (2) based on recurrence relations for moment generating functions (moments) of gos. In Section (3), we shall characterize members of the class based on recurrence relations for conditional moment generating functions (conditional moments) of gos. These results are specialized to the left, right and non-truncated cases. Ordinary order statistics and ordinary record values are also obtained as special cases of the gos. Characterizations of some members of class such as the Weibull, compound Weibull, Pareto, power function (beta is a special case), Gompertz and compound Gompertz distributions are given as illustrative examples.     

独立董事的独立性有待提高

文章从独立董事的产生机制、薪酬机制、职责履行机制、考核机制等方面作了全面的分析,探讨了在现有制度条件下,独立董事能否真正实现其独立性的问题。     

Functional laws of the iterated logarithm for small increments of empirical processes

Let F , denote the uniform empirical distribution based on the first n ≥ 1 observations from an i.i.d. sequence of uniform (0, 1) random variables. We describe the almost sure limiting behavior of the sets of increment functions {F_n(t + h_n.) - F_n(t): 0 ≤ t ≤ 1 - h_n}, when {h_n: n ≥ 1) is a nonincreasing sequence of constants such that nh_n /log n ← 0.     

论统计数据联网直报的应用与发展

利用联网直报采集统计数据,以科学合理的指标体系为基本需求框架,建立统计部门数据共享,有计划地对全市政府统计部门、政府各职能部门以及国民经济行业主管部门需要搜集的社会经济数据资源进行持续地收集、整合、存储和管理。