An optimum t-test for the scale parameter of an extreme-value distribution

A Student's t-test proposed by Ogawa is considered for the hypothesis Ho: σ=σo against the alternative hypothesis H₁: σ ≠ σo, where σ is the scale parameter of the Extremevalue distribution of smallest values with known location parameter μ. The test is based on a few sample quantiles chosen from a large sample so as to give asymptotically maximum power to the test when the number of sample quantiles is fixed. A table which facilitates the computation of the test statistic is given. Several schemes for determining the ranks of the sample quantiles by the optimal spacings are compared and the effect of the bias of the estimate of σ on the test is investigated through a Monte Carlo study.     

Testing for positive quadrant dependence in ordinal contingency table

Two new randomization tests are introduced for ordinal contingency tables for testing independence against strictly positive quadrant dependence, i.e., P(X > x,Y > y) ≥ P(X > x)P(Y > y) for all x,y with strict inequality for some x and y. For a number of cases, simulation is used to compare the estimated power of these tests versus those standard tests based on Kendall's T, Spearman's p, Pearson's X², the usual likelihood ratio test, and a test based upon the log-odds ratio. In these cases, subsets of the alternative region are identified where each of the testing statistics is superior. The new tests are found to be more powerful than the standard tests over a broad range of the alternative regions for these cases.     

Optimum quantiles for the linear estimation of the parameters of the extreme value distribution in complete and censored samples

The present study is concerned with the determination of a few observations from a sufficiently large complete or censored sample from the extreme value distribution with location and scale parameters μ and σ, respectively, such that the asymptotically best linear unbiased estimators (ABLUE) of the parameters in Ref. [24] yield high efficiencies among other choices of the same number of observations. (All efficiencies considered are relative to the Cramér-Rao lower bounds for regular unbiased estimators.) The study is on the asymptotic theory and under Type II censoring scheme. For the estimation of μ when σ is known, it has been proved that there exists a unique optimum spacing whether the sample is complete, right censored, left censored, or doubly censored. Several tables are prepared to aid in the numerical computation of the estimates as well as to furnish their efficiencies. For the estimation of σ when μ is known, it has been observed that there does not exist a unique optimum spacing. Accordingly we have obtained a spacing based on a complete sample which yields high efficiency. A similar table as above is prepared. When both μ and σ are unknown, we have considered four different spacings based on a complete sample and chosen the one yielding highest efficiency. A table of the efficiencies is also prepared. Finally we apply the above results for the estimation of the scale and/or shape parameters of the Weibull distribution.     

Testing guaranteed exponentiality

Among distributions that are used in survival analysis, of interest to us here is the Guaranteed exponential (for details about the distribution, see Gross and Clark 7). In this article, we present a new approach to testing guaranteed exponentiality. Using engineering data on the number of load cycles until a crack appears in the titanium and steel sheet in aircraft, medical data on the number of weeks tumor patients survived, data on modulus of rupture, a measurement on the breaking strength of lumber, and original I.Q. data on 112 children attending kindergarten classes in San Jose and San Mateo Counties, CA, we illustrate our method. The first two data sets are known to follow exponential distribution, while the last data set is known to follow normal distribution. Our method confirms these and nonexponentiality for the third data set. A discussion on the merits and disadvantages of our approach is included. Based on simulation, a comparison is made between the power of our test and EDF test statistic A². Our test has excellent power for many nonexponential alternatives.     

A note on testing for exponentiality

In this note some extensions are made to previous work by a number of authors on the development of tests for exponentiality. The most recent example is due to Fercho and Ringer in which they compare the small sample powers of a few well-known test statistics for the hypothesis of a constant failure rate. It is the primary intent of this current work to extend Gnedenko's F test to situations with hypercensoring and to provide guidance for its use, particularly when a log-normal distribution is the alternative.     

Randomization tests for initialization bias in simulation output

A new approach for detecting initialization bias in the mean of a simulation output series is presented. The problem is cast in a hypothesis-testing framework using randomization (permutation) tests. The approach is extremely flexible in that it enables the use of a wide variety of test statistics. Moreover, no assumptions are needed concerning the distribution of the sample of observations. The test is computationally intensive; however, the required computing power does not exceed the capabilities of a personal computer. © 1993 John Wiley & Sons, Inc.     

On estimating population characteristics from record-breaking observations. i. parametric results

Consider an experiment in which only record-breaking values (e.g., values smaller than all previous ones) are observed. The data available may be represented as X₁,K₁,X₂,K₂, …, where X₁,X₂, … are successive minima and K₁,K₂, … are the numbers of trials needed to obtain new records. We treat the problem of estimating the mean of an underlying exponential distribution, and we consider both fixed sample size problems and inverse sampling schemes. Under inverse sampling, we demonstrate certain global optimality properties of an estimator based on the “total time on test” statistic. Under random sampling, it is shown than an analogous estimator is consistent, but can be improved for any fixed sample size.     

Tests of independence of continuous random variables which guard against special alternatives

For the problem of testing the independence of a set of continuous random variables, where the test is to be based on n independent and identically distributed observations, tests are constructed that guard against all alternatives but are particularly effective against special alternatives of interest. The tests are based on the numbers of observations falling in the cells of a grid constructed using order statistics.     

On the number of games played in a best of (2n – 1) series

Let p(⩾0.5) denote the probability that team A beats B in a single game. The series continues until either A or B wins n games. Assuming that these games are independent replications, we study some features of the distribution of X_n, the number of games played in the series. It is shown that X_n is unimodal, has an IFRA distribution, and is stochastically decreasing in p. Close approximations to its mode, mean, and variance are given. Finally, it is shown that the maximum-likelihood estimator of p based on X_n is unique.     

Estimating a survival curve when new is better than used with respect to a distinguished set

Hollander, Park, and Proschan define a survival function S of a positive random variable X to be new better than used at age t₀ (NBU-{t₀}) if S satisfies $ \begin{array}{*{20}c} {\frac{{S(x + t_0)}}{{S\left({t_0} \right)}} \le S\left(x \right),} & {{\rm for}\,{\rm all}\,x\, \ge \,0,} \\ \end{array}$ where S(x) = P(X > x). The NBU-{t₀} class is a special case of the NBU-A family of survival distributions, where A is a subset of [0, ∞). These families introduce a variety of modeling possibilities for use in reliability studies. We treat problems of nonparametric estimation of survival functions from these classes by estimators which are themselves members of the classes of interest. For a number of such classes, a recursive estimation technique is shown to produce closed-form estimators which are strongly consistent and converge to the true survival distribution at optimal rates. For other classes, additional assumptions are required to guarantee the consistency of recursive estimators. As an example of the latter case, we demonstrate the consistency of a recursive estimator for S ∈ NBU-[t₀, ∞) based on lifetime data from items surviving a preliminary “burn-in” test. The relative precision of the empirical survival curve and several recursive estimators of S are investigated via simulation; the results provide support for the claim that recursive estimators are superior to the empirical survival curve in restricted nonparametric estimation problems of the type studied here.     

Testing goodness of fit to the increasing failure rate family

One branch of the reliability literature is concerned with devising statistical procedures with various nonparametric “restricted family” model assumptions because of the potential improved operating characteristics of such procedures over totally nonparametric ones. In the single-sample problem with unknown increasing failure rate (IFR) distribution F, (1) maximum-likelihood estimators of F have been calculated, (2) upper or lower tolerance limits for F have been determined, and (3) tests of the null hypothesis that F is exponential have been constructed. Barlow and Campo proposed graphical methods for assessing goodness of fit to the IFR model when the validity of this assumption is unknown. This article proposes several analytic tests of the IFR null hypothesis based on the maximum distance and area between the cumulative hazard function and its greatest convex minorant (GCM), and the maximum distance and area between the total time on test statistic and its GCM. A table of critical points is provided to implement a specific test having good overall power properties.     

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.     

Estimating the accuracy of the coverages of outer β-content tolerance intervals,which control both tails of the normal distribution

Tolerance limits which control both tails of the normal distribution so that there is no more than a proportion β₁ in one tail and no more than β₂ in the other tail with probability γ may be computed for any size sample. They are computed from X? - k₁S and X? - k₂S, where X? and S are the usual sample mean and standard deviation and k₁ and k₂ are constants previously tabulated in Odeh and Owen [3]. The question addressed is, “Just how accurate are the coverages of these intervals (– Infin;, X? – k₁S) and (X? + k₂S, ∞) for various size samples?” The question is answered in terms of how widely the coverage of each tail interval differs from the corresponding required content with a given confidence γ′.     

寿命分布的参数Bootstrap拟合优度检验方法

拟合优度检验在统计和可靠性等领域具有非常重要的地位,基于参数Bootstrap重采样的思想,对未知参数的常用寿命分布进行拟合优度检验。数值仿真结果表明,相对于传统的经验分布函数检验,这种基于参数Bootstrap的拟合优度检验具有更高的功效,特别是在小样本的情况下,优势明显。     

On estimating the scale parameter of the exponential distribution with a known linear relation between the location and the scale parameter

In this article we study the estimation of the average excess life θ in a two-parameter exponential distribution with a known linear relationship between α (the minimum life) and θ of the form α = aθ, where a is known and positive. A comparison of the efficiencies of estimators which are linear combinations of the smallest sample value and the sample sum of deviations from the smallest sample value and the maximum likelihood estimators is made for various sample sizes and different values of a. It is shown that these estimators are dominated in the risk by the minimum-risk scale equivariant estimator based on sufficient statistics. A class of Bayes estimators for inverted gamma priors is constructed and shown to include a minimum-risk scale equivariant estimator in it. All the members of this class can be computed easily.     

Estimation of a hidden service distribution of an M/G/∞ system

The maximum likelihood estimator of the service distribution function of an M/G/∞ service system is obtained based on output time observations. This estimator is useful when observation of the service time of each customer could introduce bias or may be impossible. The maximum likelihood estimator is compared to the estimator proposed by Mark Brown, [2]. Relative to each other, Brown's estimator is useful in light traffic while the maximum likelihood estimator is applicble in heavy trafic. Both estimators are compared to the empirical distribution function based on a sample of service times and are found to have drawbacks although each estimator may have applications in special circumstances.     

Estimating value in a uniform auction

Consider an auction in which increasing bids are made in sequence on an object whose value θ is known to each bidder. Suppose n bids are received, and the distribution of each bid is conditionally uniform. More specifically, suppose the first bid X₁ is uniformly distributed on [0, θ], and the i^th bid is uniformly distributed on [X_i?1, θ] for i = 2, …?, n. A scenario in which this auction model is appropriate is described. We assume that the value θ is un known to the statistician and must be esimated from the sample X₁, X₂, …?, X_n. The best linear unbiased estimate of θ is derived. The invariance of the estimation problem under scale transformations in noted, and the best invariant estimation problem under scale transformations is noted, and the best invariant estimate of θ under loss L(θ, a) = [(a/θ) ? 1]² is derived. It is shown that this best invariant estimate has uniformly smaller mean-squared error than the best linear unbiased estimate, and the ratio of the mean-squared errors is estimated from simulation experiments. A Bayesian formulation of the estimation problem is also considered, and a class of Bayes estimates is explicitly derived.     

Estimating the probability that a simulated system will be the best

Consider a stochastic simulation experiment consisting of v independent vector replications consisting of an observation from each of k independent systems. Typical system comparisons are based on mean (long‐run) performance. However, the probability that a system will actually be the best is sometimes more relevant, and can provide a very different perspective than the systems' means. Empirically, we select one system as the best performer (i.e., it wins) on each replication. Each system has an unknown constant probability of winning on any replication and the numbers of wins for the individual systems follow a multinomial distribution. Procedures exist for selecting the system with the largest probability of being the best. This paper addresses the companion problem of estimating the probability that each system will be the best. The maximum likelihood estimators (MLEs) of the multinomial cell probabilities for a set of v vector replications across k systems are well known. We use these same v vector replications to form v^k unique vectors (termed pseudo‐replications) that contain one observation from each system and develop estimators based on AVC (All Vector Comparisons). In other words, we compare every observation from each system with every combination of observations from the remaining systems and note the best performer in each pseudo‐replication. AVC provides lower variance estimators of the probability that each system will be the best than the MLEs. We also derive confidence intervals for the AVC point estimators, present a portion of an extensive empirical evaluation and provide a realistic example. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 341–358, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10019     

Large deviation probabilities for order statistics

Asymptotic representations are found for the large deviation probabilities that the nα-th order statistic exceeds δ, where δ>α. The probabilities are first expressed in terms of the empirical distribution function, and then the 1960 theorem of Bahadur and Ranga Rao is applied. The result is then shown to be more precise than a logarithmic statement in a 1969 paper of Sievers dealing with the asymptotic relative efficiency of the sample median test.     

Bounds and properties of the expected value of sample information for a project-selection problem

In this article we extend the work of Mehrez and Stulman [5] on the expected value of perfect information (EVPI) to the expected value of sample information (EVSI) for a class of economic problems dealing with the decision to reject or accept an investment project. It is shown that shifting the mean of the underlying a priori distribution of X, the project's monetary value from zero in either direction will decrease the associated EVSI of Y, the random sampled information. A theorem is then presented which gives an upper bound on the EVSI over all distributions of Y, as well as the structure of the posterior mean E[X|Y] for which this upper bound is achieved. Finally, the case where E[X|Y] is linear in Y is discussed and its performance compared with that of the optimal case.