Robustness of R-Chart to Non Normality

This study addresses the appropriate d ₃ values for constructing range control charts (R-charts) when the distributions of the processes are the uniform, triangular, exponential, and Erlang. Comparisons of the range charts are based on Type I error probabilities obtained using simulations. The results reveal that inappropriate use of the d ₃ values strongly affected the performance of the R-charts. Practitioners should be more careful in selecting suitable coefficients when using R-charts methods to process data. The distribution of the processes must be examined before the coefficients are chosen.     

Explicit analytical solutions for ARL of CUSUM chart for a long-memory SARFIMA model

This paper aims to derive explicit analytical solutions for Average Run Length (ARL) of CUSUM chart for the SARFIMA(P,D,Q)_S process with exponential white noise. Measurement of performance was done with the ARL in terms of percentage error and CPU time. The results obtained from the explicit formulas were compared focusing on the performance using the numerical integral equation (NIE) method. Both methods had similarly excellent agreement with the percentage error at less than 0.25%. Meanwhile, the explicit formulas consumed less CPU time than the NIE method. It is clear that the explicit formulas are a good alternative in real applications.     

Tables to Facilitate Fitting Tadikamalla and Johnson's LB Distributions

This paper is concerned with estimating the parameters of Tadikamalla-Johnson's L_B distribution using the first four moments. Tables of the parameters of the L_B distribution are given for selected values of skewness (0.0(0.05) 1.0(0.1)2.0) and corresponding available values of kurtosis at intervals of 0.2. The construction and use of these tables is explained with a numerical example.     

On using percentiles to fit data by a johnson distribution

The well-known Johnson system of distributions was developed by N. L. Johnson (1949). Slifker and Shapiro (1980) presented a criterion for choosing a member from the three distributional classes (S_B,S_L, and S_v) in the Johnson system to fit a set of data. The criterion is based on the value of a quantile ratio which depends on a specified positive z value and the parameters of the distribution. In this paper, we present some properties of the quantile ratio for various distributions and for some selected z values. Some comments are made on using the criterion for selecting a Johnson distribution to fit empirical data.     

Generalized Least Squares Model Averaging

In this article, we propose a method of averaging generalized least squares estimators for linear regression models with heteroskedastic errors. The averaging weights are chosen to minimize Mallows’ C_p-like criterion. We show that the weight vector selected by our method is optimal. It is also shown that this optimality holds even when the variances of the error terms are estimated and the feasible generalized least squares estimators are averaged. The variances can be estimated parametrically or nonparametrically. Monte Carlo simulation results are encouraging. An empirical example illustrates that the proposed method is useful for predicting a measure of firms’ performance.     

Statistical analysis of process capability indices with measurement errors: The case ofC p

Process capability indices (PCIs) have been widely used in manufacturing industries to previde a quantitative measure of process potential and performance. While some efforts have been dedicated in the literature to the statistical properties of PCIs estimators, scarce attention has been given to the evaluation of these properties when sample data are affected by measurement errors. In this work we deal with the problem of measurement errors effects on the performance of PCIs. The analysis is illustrated with reference toC _p , i.e. the simplest and most common measure suggested to evaluate process capability. The authors would like to thank two anonymous referees for their comments and suggestion that were useful in the preparation and improvement of this paper. This work was partially supported by a MURST research grant.     

Standard and robust orthogonal regression

A fast routine for converting regression algorithms into corresponding orthogonal regression (OR) algorithms was introduced in Ammann and Van Ness (1988). The present paper discusses the properties of various ordinary and robust OR procedures created using this routine. OR minimizes the sum of the orthogonal distances from the regression plane to the data points. OR has three types of applications. First, L ₂ OR is the maximum likelihood solution of the Gaussian errors-in-variables (EV) regression problem. This L ₂ solution is unstable, thus the robust OR algorithms created from robust regression algorithms should prove very useful. Secondly, OR is intimately related to principal components analysis. Therefore, the routine can also be used to create L ₁, robust, etc. principal components algorithms. Thirdly, OR treats the x and y variables symmetrically which is important in many modeling problems. Using Monte Carlo studies this paper compares the performance of standard regression, robust regression, OR, and robust OR on Gaussian EV data, contaminated Gaussian EV data, heavy-tailed EV data, and contaminated heavy-tailed EV data.     

Empirical Comparison of Nonparametric Regression Estimates on Real Data

The performance of nine different nonparametric regression estimates is empirically compared on ten different real datasets. The number of data points in the real datasets varies between 7, 900 and 18, 000, where each real dataset contains between 5 and 20 variables. The nonparametric regression estimates include kernel, partitioning, nearest neighbor, additive spline, neural network, penalized smoothing splines, local linear kernel, regression trees, and random forests estimates. The main result is a table containing the empirical L₂ risks of all nine nonparametric regression estimates on the evaluation part of the different datasets. The neural networks and random forests are the two estimates performing best. The datasets are publicly available, so that any new regression estimate can be easily compared with all nine estimates considered in this article by just applying it to the publicly available data and by computing its empirical L₂ risks on the evaluation part of the datasets.     

A program to calculate bootstrap confidence intervals for the process capability index Cpk

Franklin and Wasserman (1991) introduced the use of Bootstrap sampling procedures for deriving nonparametric confidence intervals for the process capability index, C_pk, which are applicable for instances when at least twenty data points are available. This represents a significant reduction in the usually recommended sample requirement of 100 observations (see Gunther 1989). To facilitate and encourage the use of these procedures. a FORTRAN program is provided for computation of confidence intervals for C_pk. Three methods are provided for this calculation including the standard method, the percentile confidence interval, and the biased - corrected percentile confidence interval.     

Estimation of covariance matrix via the sparse Cholesky factor with lasso

In this paper, we discuss a parsimonious approach to estimation of high-dimensional covariance matrices via the modified Cholesky decomposition with lasso. Two different methods are proposed. They are the equi-angular and equi-sparse methods. We use simulation to compare the performance of the proposed methods with others available in the literature, including the sample covariance matrix, the banding method, and the L₁-penalized normal loglikelihood method. We then apply the proposed methods to a portfolio selection problem using 80 series of daily stock returns. To facilitate the use of lasso in high-dimensional time series analysis, we develop the dynamic weighted lasso (DWL) algorithm that extends the LARS-lasso algorithm. In particular, the proposed algorithm can efficiently update the lasso solution as new data become available. It can also add or remove explanatory variables. The entire solution path of the L₁-penalized normal loglikelihood method is also constructed.     

A robust process capability index

The existing process capability indices (PCI's) assume that the distribution of the process being investigated is normal. For non-normal distributions, PCI's become unreliable in that PCI's may indicate the process is capable when in fact it is not. In this paper, we propose a new index which can be applied to any distribution. The proposed indexC_f:, is directly related to the probability of non-conformance of the process. For a given random sample, the estimation of C_f boils down to estimating non-parametrically the tail probabilities of an unknown distribution. The approach discussed in this paper is based on the works by Pickands (1975) and Smith (1987). We also discuss the construction of bootstrap confidence intervals of C_f: based on the so-called accelerated bias correction method (BC _a:). Several simulations are carried out to demonstrate the flexibility and applicability of C_f:. Two real life data sets are analyzed using the proposed index.     

An application of difference sets to a problem concerning graphical codes

By using difference sets, we give an answer to the following problem concerning graphical codes: When is the binary code generated by the complete graph K_n contained in some binary Hamming code? It turns out that this holds if and only if n is one of the numbers 2, 3 and 6.     

Tables to facilitate fitting lu distributions

This paper is concerned with estimating the parameters of Tadikamalla-Johnson's L_Udistributions based on the method of moments. Tables of the parameters of the L_U distribution are given for selected values of skewness (0.0(0.05) 1.0(0.1)2.0) and for twenty values of kurtosis at intervals of 0.2. The construction and use of these tables is explained with a numerical example.     

A generalized p-value approach to inference on the performance measures of an M/Ek/1 queueing system

Abstract

The hypothesis tests of performance measures for an M/E_k/1 queueing system are considered. With pivotal models deduced from sufficient statistics for the unknown parameters, a generalized p-value approach to derive tests about parametric functions are proposed. The focus is on derivation of the p-values of hypothesis testing for five popular performance measures of the system in the steady state. Given a sample T, let p(T) be the p values we developed. We derive a closed form expression to show that, for small samples, the probability P(p(T) ? γ) is approximately equal to γ, for 0 ? γ ? 1.     

Constrained Maximization of the Shapiro–Wilk W Statistic to Estimate Parameters of the Johnson S B Distribution

This article presents a constrained maximization of the Shapiro Wilk W statistic for estimating parameters of the Johnson S _B distribution. The gradient of the W statistic with respect to the minimum and range parameters is used within a quasi-Newton framework to achieve a fit for all four parameters. The method is evaluated with measures of bias and precision using pseudo-random samples from three different S _B populations. The population means were estimated with an average relative bias of less than 0.1% and the population standard deviations with less than 4.0% relative bias. The methodology appears promising as a tool for fitting this sometimes difficult distribution.     

Some series of proper generalized neighbor designs

Neighbor balanced designs are used to remove the neighbor effects but most of these designs require a large number of blocks. To neutralize the neighbor effects in such situations, GN₂-designs are most desirable. This article deals with the (i) refinement of some series of GN₂-designs constructed by Zafaryab et al. (2010) and (ii) construction of some new series of GN₂-designs in circular blocks of equal size.     

Conditions for D A -Maximin Marginal Designs for Generalized Linear Mixed Models to be Uniform

Optimal design theory deals with the assessment of the optimal joint distribution of all independent variables prior to data collection. In many practical situations, however, covariates are involved for which the distribution is not previously determined. The optimal design problem may then be reformulated in terms of finding the optimal marginal distribution for a specific set of variables. In general, the optimal solution may depend on the unknown (conditional) distribution of the covariates. This article discusses the D _A -maximin procedure to account for the uncertain distribution of the covariates. Sufficient conditions will be given under which the uniform design of a subset of independent discrete variables is D _A -maximin. The sufficient conditions are formulated for Generalized Linear Mixed Models with an arbitrary number of quantitative and qualitative independent variables and random effects.     

Robust Coordinate Descent Algorithm Robust Solution Path for High-dimensional Sparse Regression Modeling

The L₁-type regularization provides a useful tool for variable selection in high-dimensional regression modeling. Various algorithms have been proposed to solve optimization problems for L₁-type regularization. Especially the coordinate descent algorithm has been shown to be effective in sparse regression modeling. Although the algorithm shows a remarkable performance to solve optimization problems for L₁-type regularization, it suffers from outliers, since the procedure is based on the inner product of predictor variables and partial residuals obtained from a non-robust manner. To overcome this drawback, we propose a robust coordinate descent algorithm, especially focusing on the high-dimensional regression modeling based on the principal components space. We show that the proposed robust algorithm converges to the minimum value of its objective function. Monte Carlo experiments and real data analysis are conducted to examine the efficiency of the proposed robust algorithm. We observe that our robust coordinate descent algorithm effectively performs for the high-dimensional regression modeling even in the presence of outliers.     

On least absolute values estimation

The resistance of least absolute values (L₁) estimators to outliers and their robustness to heavy-tailed distributions make these estimators useful alternatives to the usual least squares estimators. The recent development of efficient algorithms for L₁ estimation in linear models has permitted their use in practical data analysis. Although in general the L₁ estimators are not unique, there are a number of properties they all share. The set of all L₁ estimators for a given model and data set can be characterized as the convex hull of some extreme estimators. Properties of the extreme estimators and of the L₁-estimate set are considered.     

A special purpose linear programming algorithm for obtaining least absolute value estimators in a linear model with dummy variables

Dummy (0, 1) variables are frequently used in statistical modeling to represent the effect of certain extraneous factors. This paper presents a special purpose linear programming algorithm for obtaining least-absolute-value estimators in a linear model with dummy variables. The algorithm employs a compact basis inverse procedure and incorporates the advanced basis exchange techniques available in specialized algorithms for the general linear least-absolute-value problem. Computational results with a computer code version of the algorithm are given.