Classical and Bayesian estimation of P(Y < X) for Kumaraswamy's distribution

The aim of this paper is to study the estimation of the reliability R=P(Y<X) when X and Y are independent random variables that follow Kumaraswamy's distribution with different parameters. If we assume that the first shape parameter is common and known, the maximum-likelihood estimator (MLE), the exact confidence interval and the uniformly minimum variance unbiased estimator of R are obtained. Moreover, when the first parameter is common but unknown, MLEs, Bayes estimators, asymptotic distributions and confidence intervals for R are derived. Furthermore, Bayes and empirical Bayes estimators for R are obtained when the first parameter is common and known. Finally, when all four parameters are different and unknown, the MLE of R is obtained. Monte Carlo simulations are performed to compare the different proposed methods and conclusions on the findings are given.     

On A strongly consistent nonparametric density estimator for the deconvolution problem

The problem of nonparametric estimation of a probability density function is studied when the sample observations are contaminated with random noise. Previous authors have proposed estimators which use kernel density and deconvolution techniques. The appearance and properties of the previously proposed estimators are affected by constants M_n and h_n which the user may choose. However, the optimal choices of these constants depend on the sample size n, the noise distribution and the unknown distribution which is being estimated. Hence, in practice, M_n and h_n are optimally selected as functions of the data. In this paper it is shown that a class of the proposed estimators are uniformly, strongly consistent when M_n and h_n are allowed to be random variables. Even when M_n and h_n are constants, these results are new findings.     

Comparison of design for quadratic regression on cubes

Designs for quadratic regression are considered when the possible choices of the controllable variable are points x=(x₁,x₂,…,x_q) in the q-dimensional cube of side 2. The designs that are optimum with respect to such criteria as those of D-, A-, and E-optimality are compared in their performance relative to these and other criteria. Some of the results are developed algebraically; others, numerically. The possible supports of E-optimum designs are much more numerous than the D-optimum supports characterized earlier. The A-optimum design appears to be fairly robust in its efficiency, under variation of criterion.     

Reliability Analysis of Unrepairable Systems with k-out-of-m:G Subsystems Subject to Suspended Animation

Two unrepairable series structure systems with k-out-of-m:G subsystems and spares are investigated. The first one consists of a k-out-of-m:G subsystem and a series subsystem while the other consists of two k-out-of-m:G subsystems. The systems have identical components with identical lifetime distributions and the working components are suspended as soon as the systems are down. Two Markov models are proposed for the reliability analysis of such systems and closed form results on the reliability and the mean time to failure (MTTF) are presented. Numerical examples are given to illustrate the impact of several parameters on the reliability of the systems.     

Forecasting Vector ARMA Processes With Systematically Missing Observations

The following two predictors are compared for time series with systematically missing observations: (a) A time series model is fitted to the full series X_t , and forecasts are based on this model, (b) A time series model is fitted to the series with systematically missing observations Y _τ, and forecasts are based on the resulting model. If the data generation processes are known vector autoregressive moving average (ARMA) processes, the first predictor is at least as efficient as the second one in a mean squared error sense. Conditions are given for the two predictors to be identical. If only the ARMA orders of the generation processes are known and the coefficients are estimated, or if the process orders and coefficients are estimated, the first predictor is again, in general, superior. There are, however, exceptions in which the second predictor, using seemingly less information, may be better. These results are discussed, using both asymptotic theory and small sample simulations. Some economic time series are used as illustrative examples.     

A review of optimal designs in survey sampling

Results in five areas of survey sampling dealing with the choice of the sampling design are reviewed. In Section 2, the results and discussions surrounding the purposive selection methods suggested by linear regression superpopulation models are reviewed. In Section 3, similar models to those in the previous section are considered; however, random sampling designs are considered and attention is focused on the optimal choice of π_j. Then in Section 4, systematic sampling methods obtained under autocorrelated superpopulation models are reviewed. The next section examines minimax sampling designs. The work in the final section is based solely on the randomization. In Section 6 methods of sample selection which yield inclusion probabilities π_j = n/N and π_ij = n(n - 1)/N(N - 1), but for which there are fewer than _NC_n possible samples, are mentioned briefly.     

A simulation study of the Bennett test and Miller test for coefficients of variation

In the article, properties of the Bennett test and Miller test are analyzed. Assuming that the sample size is the same for each sample and considering the null hypothesis that the coefficients of variation for k populations are equal against the hypothesis that k ? 1 coefficients of variation are the same but differ from the coefficient of variation for the kth population, the empirical significance level and the power of the test are studied. Moreover, the dependence of the test statistic and the power of the test on the ratio of coefficients of variation are considered. The analyses are performed on simulated data.     

Editorial Collaborators

Abstract

The present note explores sources of misplaced criticisms of P-values, such as conflicting definitions of “significance levels” and “P-values” in authoritative sources, and the consequent misinterpretation of P-values as error probabilities. It then discusses several properties of P-values that have been presented as fatal flaws: That P-values exhibit extreme variation across samples (and thus are “unreliable”), confound effect size with sample size, are sensitive to sample size, and depend on investigator sampling intentions. These properties are often criticized from a likelihood or Bayesian framework, yet they are exactly the properties P-values should exhibit when they are constructed and interpreted correctly within their originating framework. Other common criticisms are that P-values force users to focus on irrelevant hypotheses and overstate evidence against those hypotheses. These problems are not however properties of P-values but are faults of researchers who focus on null hypotheses and overstate evidence based on misperceptions that p?=?0.05 represents enough evidence to reject hypotheses. Those problems are easily seen without use of Bayesian concepts by translating the observed P-value p into the Shannon information (S-value or surprisal) –log₂(p).     

Some new constructions of circular balanced repeated measurements designs

Repeated measurements designs are widely used in medicine, pharmacology, animal sciences, and psychology. In this article, some infinite series are developed to generate the balanced repeated measurements designs for p (periods) even. For p odd, construction procedures are also described. Catalogues of the proposed designs are also presented for p = 5, 7, 9, when v ≤ 100.     

Estimation of P(Y < X) for Weibull Distribution Under Progressive Type-II Censoring

Based on progressively Type II censored samples, we consider the estimation of R = P(Y < X) when X and Y are two independent Weibull distributions with different shape parameters, but having the same scale parameter. The maximum likelihood estimator, approximate maximum likelihood estimator, and Bayes estimator of R are obtained. Based on the asymptotic distribution of R, the confidence interval of R are obtained. Two bootstrap confidence intervals are also proposed. Analysis of a real data set is given for illustrative purposes. Monte Carlo simulations are also performed to compare the different proposed methods.     

Riordan matrices and higher-dimensional lattice walks

An algebraic combinatorial method is used to count higher-dimensional lattice walks in Z^m${\mathbb{Z}}^m$ that are of length n ending at height k. As a consequence of using the method, Sands’ two-dimensional lattice walk counting problem is generalized to higher dimensions. In addition to Sands’ problem, another subclass of higher-dimensional lattice walks is also counted. Catalan type solutions are obtained and the first moments of the walks are computed. The first moments are then used to compute the average heights of the walks. Asymptotic estimates are also given.     

New tests based on Rubin's empirical distribution function

In this paper we derive some new tests for goodness-of-fit based on Rubin's empirical distribution function (EDF). Substituting Rubin's EDF for the classical EDF in the Kolmogorov–Smirnov, Cramér–von Mises, Anderson–Darling statistics, since Rubin's EDF for a given sample is a randomized distribution function, randomized statistics are derived, of which the qth quantile and the expectation are chosen as test statistics. We show that the new tests are consistent under simple hypothesis. Several power comparisons are also performed to show that the new tests are generally more powerful than the classical ones.     

A-optimal designs for heteroscedastic multifactor regression models

Abstract

This paper searches for A-optimal designs for Kronecker product and additive regression models when the errors are heteroscedastic. Sufficient conditions are given so that A-optimal designs for the multifactor models can be built from A-optimal designs for their sub-models with a single factor. The results of an efficiency study carried out to check the adequacy of the products of optimal designs for uni-factor marginal models when these are used to estimate different multi-factor models are also reported.     

Tests of symmetry based on transformed empirical processes

A probability distribution function F is said to be symmetric when 1 ‐ F(x) ‐ F(‐x) = 0 for all x∈ R. Given a sequence of alternatives contiguous to a certain symmetric F₀, the authors are concerned with testing for the null hypothesis of symmetry. The proposed tests are consistent against any nonsymmetric alternative, and their power with respect to the given sequence can easily be optimized. The tests are constructed by means of transformed empirical processes with an adequate selection of the underlying isometry, and the optimum power is obtained by suitably choosing the score functions. The test statistics are very easy to compute and their asymptotic distributions are simple.     

Estimation and testing stationarity for double-autoregressive models

Summary.  The paper considers the double-autoregressive model y _t  =  φ y _{t −1}+ ɛ _t with ɛ _t  =     . Consistency and asymptotic normality of the estimated parameters are proved under the condition E  ln | φ  +√ α η _t |<0, which includes the cases with | φ |=1 or | φ |>1 as well as     . It is well known that all kinds of estimators of φ in these cases are not normal when ɛ _t are independent and identically distributed. Our result is novel and surprising. Two tests are proposed for testing stationarity of the model and their asymptotic distributions are shown to be a function of bivariate Brownian motions. Critical values of the tests are tabulated and some simulation results are reported. An application to the US 90-day treasury bill rate series is given.     

INTERRELATIONSHIPS BETWEEN lP-ISOTROPIC DENSITIES AND lP-ISOTROPIC SURVIVAL FUNCTIONS, AND DE FINETTI REPRESENTATIONS OF SCHUR-CONCAVE SURVIVAL FUNCTIONS

Interrelationships between l_P-isotropic densities and l_P-isotropic survival functions are studied. We obtain representations for densities whose survival functions are l_P-isotropic and for survival functions whose densities are l_P-isotiopic. The former are mixtures of products of Weibull densities and the latter are mixtures of products of Gamma survival functions. Based on a theorem proved by Cooke, we show that de Finetti representations for Schur-concave survival functions that have the same level sets as a product measure can be represented as mixtures of products of increasing failure rate survival functions.     

Developing a variables repetitive group sampling plan based on process capability index C pk with unknown mean and variance

In this paper, variables repetitive group sampling plans are developed based on the process capability index C _pk when the quality characteristic follows a normal distribution with unknown mean and variance. The sampling plan parameters such as the sample size and the acceptance constant are determined to minimize the average sample number. Symmetric and asymmetric cases, in percent defectives due to two specification limits, are dealt with for specified combinations of acceptable quality level and limiting quality level. Tables are provided and examples are given to use proposed plans in practice.     

A discrete probability problem in card shuffling

Abstract

There are n cards serially numbered from 1 to n. The cards are shuffled and placed in a line one after the other on top of a table with faces up. The numbers on the faces are read from left to right. If there are consecutive numbers in increasing order of magnitude the corresponding cards are merged into one. After the merger, the cards are numbered serially from one to whatever the number of cards we now have. The cards are shuffled and placed in a line one after another on top of the table with faces up. The process continues until we have only one card left. In this paper, we develop a probabilistic recurrence relation approach to obtain the mean, variance, and distribution of the number of shuffles needed. A Markov chain formulation and its properties are discussed in the paper as well.     

Reliability Evaluation for a Multi-State System Under Stress-Strength Setup

The two most commonly used reliability models in engineering applications are binary k-out-of-n:G and consecutive k-out-of-n:G systems. Multi-state k-out-of-n:G and multi-state consecutive k-out-of-n:G systems have been proposed as an extension of these systems and they have been found to be more flexible tool for modeling engineering systems. In this article, multi-state systems, in particular, multi-state k-out-of-n:G and multi-state consecutive k-out-of-n:G, are considered in a stress-strength setup. The states of the system are classified considering the number of components whose strengths above (below) the multiple stresses available in an environment. The exact state probabilities are provided and the results are illustrated for various stress-strength distributions. Maximum likelihood estimators of state probabilities are also presented.