Majorization and the spectral radius of starlike trees

A starlike tree is a tree with exactly one vertex of degree greater than two. The spectral radius of a graph G, that is denoted by \(\lambda (G)\), is the largest eigenvalue of G. Let k and \(n_1,\ldots ,n_k\) be some positive integers. Let \(T(n_1,\ldots ,n_k)\) be the tree T (T is a path or a starlike tree) such that T has a vertex v so that \(T{\setminus } v\) is the disjoint union of the paths \(P_{n_1-1},\ldots ,P_{n_k-1}\) where every neighbor of v in T has degree one or two. Let \(P=(p_1,\ldots ,p_k)\) and \(Q=(q_1,\ldots ,q_k)\), where \(p_1\ge \cdots \ge p_k\ge 1\) and \(q_1\ge \cdots \ge q_k\ge 1\) are integer. We say P majorizes Q and let \(P\succeq _M Q\), if for every j, \(1\le j\le k\), \(\sum _{i=1}^{j}p_i\ge \sum _{i=1}^{j}q_i\), with equality if \(j=k\). In this paper we show that if P majorizes Q, that is \((p_1,\ldots ,p_k)\succeq _M(q_1,\ldots ,q_k)\), then \(\lambda (T(q_1,\ldots ,q_k))\ge \lambda (T(p_1,\ldots ,p_k))\).     

Impact of temporal aggregation on stock control performance of intermittent demand estimators: Empirical analysis

Intermittent demand is characterized by occasional demand arrivals interspersed by time intervals during which no demand occurs. These demand patterns pose considerable difficulties in terms of forecasting and stock control due to their compound nature, which implies variability both in terms of demand arrivals and demand sizes. An intuitively appealing strategy to deal with such patterns from a forecasting and stock control perspective is to aggregate demand in lower-frequency ‘time buckets’, thereby reducing the presence of zero observations. In this paper, we investigate the impact of forecasting aggregation on the stock control performance of intermittent demand patterns. The benefit of the forecasting aggregation approach is empirically assessed by means of analysis on a large demand dataset from the Royal Air Force (UK). The results show that the aggregation forecasting approach results in higher achieved service levels as compared to the classical forecasting approach. Moreover, when the combined service-cost performance is considered, the results also show that the former approach is more efficient than the latter, especially for high target service levels.     

Robust allocation of a defensive budget considering an attacker's private information

Attackers' private information is one of the main issues in defensive resource allocation games in homeland security. The outcome of a defense resource allocation decision critically depends on the accuracy of estimations about the attacker's attributes. However, terrorists' goals may be unknown to the defender, necessitating robust decisions by the defender. This article develops a robust-optimization game-theoretical model for identifying optimal defense resource allocation strategies for a rational defender facing a strategic attacker while the attacker's valuation of targets, being the most critical attribute of the attacker, is unknown but belongs to bounded distribution-free intervals. To our best knowledge, no previous research has applied robust optimization in homeland security resource allocation when uncertainty is defined in bounded distribution-free intervals. The key features of our model include (1) modeling uncertainty in attackers' attributes, where uncertainty is characterized by bounded intervals; (2) finding the robust-optimization equilibrium for the defender using concepts dealing with budget of uncertainty and price of robustness; and (3) applying the proposed model to real data.     

On the moments of record values

In this paper some general relations for expectations of functions of record values are established. It is seen that these relations may be used to obtain recurrence relations for moments of record values. Bounds on expectations of record values with numerical computations are presented. Applications to the characterizations of the generalizeed exponential distribution are also given.     

Bivariate Sign Test for One-Sample Bivariate Location Model Using Ranked Set Sample

In this article, we introduce a bivariate sign test for the one-sample bivariate location model using a bivariate ranked set sample (BVRSS). We show that the proposed test is asymptotically more efficient than its counterpart sign test based on a bivariate simple random sample (BVSRS). The asymptotic null distribution and the non centrality parameter are derived. The asymptotic distribution of the vector of sample median as an estimator of the locations of the bivariate model is introduced. Theoretical and numerical comparisons of the asymptotic efficiency of the BVRSS sign test with respect to the BVSRS sign test are also given.     

Estimation of P(Y < X) for the Three-Parameter Generalized Exponential Distribution

This article considers the estimation of R = P(Y < X) when X and Y are distributed as two independent three-parameter generalized exponential (GE) random variables with different shape parameters but having the same location and scale parameters. A modified maximum likelihood method and a Bayesian technique are used to estimate R on the basis of independent complete samples. The Bayes estimator cannot be obtained in explicit form, and therefore it has been determined using an importance sampling procedure. An analysis of a real life data set is presented for illustrative purposes.     

The Effectiveness of Multistage Ranked Set Sampling in Stratifying the Population

The spectral analysis of Gaussian linear time-series processes is usually based on uni-frequential tools because the spectral density functions of degree 2 and higher are identically zero and there is no polyspectrum in this case. In finite samples, such an approach does not allow the resolution of closely adjacent spectral lines, except by using autoregressive models of excessively high order in the method of maximum entropy. In this article, multi-frequential periodograms designed for the analysis of discrete and mixed spectra are defined and studied for their properties in finite samples. For a given vector of frequencies ω, the sum of squares of the corresponding trigonometric regression model fitted to a time series by unweighted least squares defines the multi-frequential periodogram statistic IM(ω). When ω is unknown, it follows from the properties of nonlinear models whose parameters separate (i.e., the frequencies and the cosine and sine coefficients here) that the least-squares estimator of frequencies is obtained by maximizing I ^M(ω). The first-order, second-order and distribution properties of I ^M(ω) are established theoretically in finite samples, and are compared with those of Schuster's uni-frequential periodogram statistic. In the multi-frequential periodogram analysis, the least-squares estimator of frequencies is proved to be theoretically unbiased in finite samples if the number of periodic components of the time series is correctly estimated. Here, this number is estimated at the end of a stepwise procedure based on pseudo-Flikelihood ratio tests. Simulations are used to compare the stepwise procedure involving I ^M(ω) with a stepwise procedure using Schuster's periodogram, to study an approximation of the asymptotic theory for the frequency estimators in finite samples in relation to the proximity and signal-to-noise ratio of the periodic components, and to assess the robustness of I ^M(ω) against autocorrelation in the analysis of mixed spectra. Overall, the results show an improvement of the new method over the classical approach when spectral lines are adjacent. Finally, three examples with real data illustrate specific aspects of the method, and extensions (i.e., unequally spaced observations, trend modeling, replicated time series, periodogram matrices) are outlined.     

Three New Multivariate Process Capability Indices

Different multivariate process capability indices are developed by researchers to evaluate process capability when vectors of quality characteristics are considered in a study. This article presents three indices referred to as NCpM, MCpM, and NMC _PM in order to evaluate process capability in multivariate environment. The performance of the proposed indices is investigated numerically. Simulation results indicate that the proposed indices have descended estimation error and improved performance compared to the existing ones. These results can be important to researchers and practitioners who are interested in evaluating process capability in multivariate domain.     

Shrinkage Estimation Under Multivariate Elliptic Models

The estimation of the location vector of a p-variate elliptically contoured distribution (ECD) is considered using independent random samples from two multivariate elliptically contoured populations when it is apriori suspected that the location vectors of the two populations are equal. For the setting where the covariance structure of the populations is the same, we define the maximum likelihood, Stein-type shrinkage and positive-rule shrinkage estimators. The exact expressions for the bias and quadratic risk functions of the estimators are derived. The comparison of the quadratic risk functions reveals the dominance of the Stein-type estimators if p ≥ 3. A graphical illustration of the risk functions under a “typical” member of the elliptically contoured family of distributions is provided to confirm the analytical results.     

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.