Testing fit with nuisance location and scale parameters

For each n, X₁(n),…, X_n(n) are independent and identically distributed random variables, each with cumulative distribution function F(x) which is known to be absolutely continuous but is otherwise unknown. The problem is to test the hypothesis that \documentclass{article}\pagestyle{empty}\begin{document}$ F(x) = G\left( {{\textstyle{{x - \theta _1 } \over {\theta _2 }}}} \right) $\end{document}, where the cumulative distribution function G_x is completely specified and satisfies certain regularity conditions, and the parameters θ₁, θ₂ are unknown and unspecified, except that the scale parameter θ₂, is positive. Y₁ (n) ≦ Y₂ (n) ≦ … ≦ Y_n (n)are the ordered values of X₁(n),…, X_n(n). A test based on a certain subset of {Y_i(n)} is proposed, is shown to have asymptotically a normal distribution when the hypothesis is true, and is shown to be consistent against all alternatives satisfying a mild regularity condition.     

An algorithm for separable piecewise convex programming problems

We present a branch and bound algorithm to solve mathematical programming problems of the form: Find x =|(x₁,…x_n) to minimize Σ?_i0(x₁) subject to x?G, l≦x≦L and Σ?_i0(x₁)≦0, j=1,…,m. With l=(l₁,…,l_n) and L=(L₁,…,L_n), each ?_ij is assumed to be lower aemicontinuous and piecewise convex on the finite interval [l_i.L_i]. G is assumed to be a closed convex set. The algorithm solves a finite sequence of convex programming problems; these correspond to successive partitions of the set C={x|l ≦ x ≦L} on the bahis of the piecewise convexity of the problem functions ?_ij. Computational considerations are discussed, and an illustrative example is presented.     

Covariance of weibull order statistics

Let X₁ < X₂ <… < X_n denote an ordered sample of size n from a Weibull population with cdf F(x) = 1 - exp (?x^p), x > 0. Formulae for computing Cov (X_i, X_j) are well known, but they are difficult to use in practice. A simple approximation to Cov(X_i, X_j) is presented here, and its accuracy is discussed.     

On estimating the reliability of a component subject to several different stresses (strengths)

This paper is concerned with estimating p = P(X₁ < Y …, X_n < Y) or q =P (X < Y₁, …, X < Y_n) where the X's and Y's are all independent random variables. Applications to estimation of the reliability p from stress-strength relationships are considered where a component is subject to several stresses X₁, X₂, …, X_N whereas its strength, Y, is a single random variable. Similarly, the reliability q is of interest where a component is made of several parts all with their individual strengths Y₁, Y₂ …, Y_N and a single stress X is applied to the component. When the X's and Y's are independent and normal, maximum likelihood estimates of p and q have been obtained. For the case N = 2 and in some special cases, minimum variance unbiased estimates have been given. When the Y's are all exponential and the X is normal with known variance, but unknown mean (or uniform between 0 and θ, θ being unknown) the minimum variance unbiased estimate of q is established in this paper.     

Distribution of sample correlation coefficients

Let (Y, X_l,…, X_K) be a random vector distributed according to a multivariate normal distribution where X_l,…, X_K are considered as predictor variables and y is the predictand. Let r_i, and R_i denote the population and sample correlation coefficients, respectively, between Y and X_i. The population correlation coefficient r_i is a measure of the predictive power of X_i. The author has derived the joint distribution of R_l,…, R_K and its asymptotic property. The given result is useful in the problem of selecting the most important predictor variable corresponding to the largest absolute value of r_i.     

On estimating population characteristics from record-breaking observations II. Nonparametric 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. Such data arise in life testing and stress testing and in industrial quality-control experiments. When only a single sequence of random records are available, efficient estimation of the underlying distribution F is possible only in a parametric framework (see Samaniego and Whitaker [9]). In the present article we study the problem of estimating certain population quantiles nonparametrically from such data. Furthermore, under the assumption that the process of observing random records can be replicated, we derive and study the nonparametric maximum-likelihood estimator F̂ of F. We establish the strong uniform consistency of this estimator as the number of replications grows large, and identify its asymptotic distribution theory. The performance of F̂ is compared to that of two possible competing estimators.     

Estimating a survival curve when new is better than used in expectation

Let X be a positive random variable. The distribution F of X is said to be “new better than used in expectation,” or “NBUE,” if E(X) ⩾ E(X – t|X > t) for all t ⩾ 0. Suppose X₁, …, X_n, is a random sample from an NBUE distribution F. The problem of estimating F by a distribution which is itself NBUE is considered. The estimator G_n, defined as the NBUE distribution supported on the sample which minimizes the (sup norm) distance between the NBUE class and the empirical distribution function, is studied. The strong uniform consistency of G_n, is proven, and a numerical algorithm for obtaining G_n, is given. Our approach is applied to provide an estimate of the distribution of lifetime following the diagnosis of chronic granulocytic leukemia based on data from a National Cancer Institute study.     

Asymptotic inference about a density function at an end of its range

For each n, X₁(n),…X_n(n) are independent and identically distributed random variables, with common probability density function Where c, θ, α, and r(y) are all unknown. It is shown that we can make asymptotic inferences about c, θ, and α, when r(y) satisfies mild conditions.     

On computing the expected discounted return in a markov chain

The discounted return associated with a finite state Markov chain X₁, X₂… is given by g(X₁)+ αg(X₂) + α²g(X₃) + …, where g(x) represents the immediate return from state x. Knowing the transition matrix of the chain, it is desired to compute the expected discounted return (present worth) given the initial state. This type of problem arises in inventory theory, dynamic programming, and elsewhere. Usually the solution is approximated by solving the system of linear equations characterizing the expected return. These equations can be solved by a variety of well-known methods. This paper describes yet another method, which is a slight modification of the classical iterative scheme. The method gives sequences of upper and lower bounds which converge mono-tonely to the solution. Hence, the method is relatively free of error control problems. Computational experiments were conducted which suggest that for problems with a large number of states, the method is quite efficient. The amount of computation required to obtain the solution increases much slower with an increase in the number of states, N, than with the conventional methods. In fact, computational time is more nearly proportional to N², than to N³.     

A note on circular error probabilities

An approximation for P(X² + Y² ≤ K²σ²₁) based on an unpublished result of Kleinecke is derived, where X and Y are independent normal variables having zero means and variances σ²₁ and σ²₂ and σ₁ ≥ σ₂. Also, we provide asymptotic expressions for the probabilities for large values of β = K²(1 - c²)/4c² where c = σ₂/σ₁. These are illustrated by comparing with values tabulated by Harter [6]. Solution of K for specified P and c is also considered. The main point of this note is that simple and easily calculable approximations for P and K can be developed and there is no need for numerical evaluation of integrals.     

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.     

Deterministic and stochastic scheduling with teamwork tasks

We study a class of new scheduling problems which involve types of teamwork tasks. Each teamwork task consists of several components, and requires a team of processors to complete, with each team member to process a particular component of the task. Once the processor completes its work on the task, it will be available immediately to work on the next task regardless of whether the other components of the last task have been completed or not. Thus, the processors in a team neither have to start, nor have to finish, at the same time as they process a task. A task is completed only when all of its components have been processed. The problem is to find an optimal schedule to process all tasks, under a given objective measure. We consider both deterministic and stochastic models. For the deterministic model, we find that the optimal schedule exhibits the pattern that all processors must adopt the same sequence to process the tasks, even under a general objective function GC = F(f₁(C₁), f₂(C₂), … , f_n(C_n)), where f_i(C_i) is a general, nondecreasing function of the completion time C_i of task i. We show that the optimal sequence to minimize the maximum cost MC = max f_i(C_i) can be derived by a simple rule if there exists an order f₁(t) ≤ … ≤ f_n(t) for all t between the functions {f_i(t)}. We further show that the optimal sequence to minimize the total cost TC = ∑ f_i(C_i) can be constructed by a dynamic programming algorithm. For the stochastic model, we study three optimization criteria: (A) almost sure minimization; (B) stochastic ordering; and (C) expected cost minimization. For criterion (A), we show that the results for the corresponding deterministic model can be easily generalized. However, stochastic problems with criteria (B) and (C) become quite difficult. Conditions under which the optimal solutions can be found for these two criteria are derived. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004     

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.     

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.     

Bayesian inference of survival curve from record-breaking observations: Estimation and asymptotic results

In a variety of industrial situations experimental outcomes are only record-breaking observations. The data available may be represented as X₁, K_1., X₂, K₂,…, where X₁, X₂,… are the successive minima and K₁, K₂, … are the number of trials needed to obtain new records. Samaniego and Whitaker [11, 12] discussed the problem of estimating the survival function in both parametric and nonparametric setups when the data consisted of record-breaking observations. In this article we derive nonparametric Bayes and empirical Bayes estimators of the survival function for such data under a Dirichlet process prior and squared error loss. Furthermore, under the assumptions that the process of observing random records can be replicated, the weak convergence of the Bayes estimator is studied as the number of replications grows large. The calculations involved are illustrated by adopting Proschan's [9] data on successive failure times of air conditioning units on Boeing aircraft, for our purpose. The nonparametric maximum likelihood estimators of the survival function for different choices of the prior are displayed for comparison purposes.     

Confidence intervals for ranked means

Suppose that observations from populations π₁, …, π_k (k ≥ 1) are normally distributed with unknown means μ₁., μ_k, respectively, and a common known variance σ². Let μ_[1] μ … ≤ μ_[k] denote the ranked means. We take n independent observations from each population, denote the sample mean of the n observation from π₁ by X _i (i = 1, …, k), and define the ranked sample means X _[1] ≤ … ≤ X _[k]. The problem of confidence interval estimation of μ₍₁₎, …,μ_[k] is stated and related to previous work (Section 1). The following results are obtained (Section 2). For i = 1, …, k and any γ(0 < γ < 1) an upper confidence interval for μ_[i] with minimal probability of coverage γ is (? ∞, X _[i]+ h) with h = (σ/n^1/2) Φ^?1(γ^1/k-i+1), where Φ(·) is the standard normal cdf. A lower confidence interval for μ_[i] with minimal probability of coverage γ is (X _i[i] – g, + ∞) with g = (σ/n^1/2) Φ^?1(γ^1/i). For the upper confidence interval on μ_[i] the maximal probability of coverage is 1– [1 – γ^1/k-i+1]ⁱ, while for the lower confidence interval on μ_[i] the maximal probability of coverage is 1–[1– γ^1/i] ^k-i+¹. Thus the maximal overprotection can always be calculated. The overprotection is tabled for k = 2, 3. These results extend to certain translation parameter families. It is proven that, under a bounded completeness condition, a monotone upper confidence interval h(X ₁, …, X _k) for μ_[i] with probability of coverage γ(0 < γ < 1) for all μ = (μ_[1], …,μ_[k]), does not exist.     

A characterization of the wald distribution

Suppose X is a random variable having an absolutely continuous distribution function F(x). We assume that F(x) has the Wald distribution. A relation between the probability density function of X⁻¹ with that of X is used to characterize the Wald distribution.     

A note on minimizing the expected makespan in flowshops subject to breakdowns

Consider a two machine flow shop and n jobs. The processing time of job j on machine i is equal to the random variable X_ij One of the two machines is subject to breakdown and repair. The objective is to find the schedule that minimizes the expected makespan. Two results are shown. First, ifP(X_2j ≧ X_1j) = 1 for all j and the random variables X₁₁, X₁₂,…, X_1n are likelihood ratio ordered, then the SEPT sequence minimizes the expected makespan when machine 2 is subject to an arbitrary breakdown process; if P(X_1j≧X_2j) = 1 and X₂₁, X₂₂,….,X_2n are likelihood ratio ordered, then the LEPT sequence minimizes the expected makespan when machine 1 is subject to an arbitrary breakdown process. A generalization is presented for flow shops with m machines. Second, consider the case where X_1j and X_2j are i.i.d. exponentially distributed with rate λ_j. The SEPT sequence minimizes the expected makespan when machine 2 is subject to an arbitrary breakdown process and the LEPT sequence is optimal when machine 1 is subject to an arbitrary breakdown process. © 1995 John Wiley & Sons, Inc.     

A note on a prediction interval for a first-order Gauss Markov process

Let X_t, t = 1,2, ?, be a stationary Gaussian Markov process with E(X_t) = μ and Cov(X_t, X_t+k) = σ²ρ^k. We derive a prediction interval for X_2n+1 based on the preceding 2n observations X₁,X₂, ?,X_2n.     

A test for the hypothesis that two extreme-value scale parameters are equal

A statistic is determined for testing the hypothesis of equality for scale parameters from two populations, each of which has the first asymptotic distribution of smallest (extreme) values. The probability distribution is derived for this statistic, and critical values are determined and given in tabular form for a one-sided or two-sided alternative, for censored samples of size n₁ and n₂, n₁ = 2, 3, …. 6, n₂ = 2, 3, …. 6. The power function of the test for certain alternatives is also calculated and listed in each case considered.