Showing papers in "Communications in Statistics-theory and Methods in 2006"

General Multi-Level Modeling with Sampling Weights

Abstract: In this article we study the approximately unbiased multi-level pseudo maximum likelihood (MPML) estimation method for general multi-level modeling with sampling weights. We conduct a simulation study to determine the effect various factors have on the estimation method. The factors we included in this study are scaling method, size of clusters, invariance of selection, informativeness of selection, intraclass correlation, and variability of standardized weights. The scaling method is an indicator of how the weights are normalized on each level. The invariance of the selection is an indicator of whether or not the same selection mechanism is applied across clusters. The informativeness of the selection is an indicator of how biased the selection is. We summarize our findings and recommend a multi-stage procedure based on the MPML method that can be used in practical applications.

Some Modifications for Choosing Ridge Parameters

Abstract: Standard least square regression can produce estimates having a large mean squares error (MSE) when predictor variables are highly correlated or multicollinear. In this article, we propose four modifications to choose the ridge parameter (K) when multicollinearity exists among the columns of the design matrix. The proposed new estimators are extended versions of that suggested by Khalaf and Shukur (2005). The properties of these estimators are compared with those of Hoerl and Kennard (1970a) and the OLS using the MSE criterion. All estimators under consideration are evaluated using simulation techniques under certain conditions where a number of factors that may affect their properties have been varied. In addition, it is shown that at least one of the proposed estimators either has a smaller MSE than the others or is the next best otherwise.

Distribution-Free Quality Control Charts Based on Signed-Rank-Like Statistics

Abstract: In this article we propose three distribution-free (or nonparametric) statistical quality control charts for monitoring a process center when an in-control target center is not specified. These charts are of the Shewhart-type, the exponentially moving average-type, and the cumulative sum-type. The constructions of the proposed charts require the availability of an initial reference sample taken when the process was operating in-control to calculate an estimator for the unknown in-control target process center. This estimated center is then used in the calculation of signed-rank-like statistics based on grouped observations taken periodically from the process output. As long as the in-control process underlying distribution is continuous and symmetric, the proposed charts have a constant in-control average run length and a constant false alarm rate irrespective of the process underlying distribution. Other advantages of the proposed distribution-free charts include their robustness against outliers and the...

Density of the Ratio of Two Normal Random Variables and Applications

Abstract: In reply to a question raised in the literature, and to settle an argument debated in the last decades, we give the exact closed form expression of the density of X/Y, where X and Y are normal random variables, in terms of Hermite and confluent hypergeometric functions. All cases will be considered: standardized and nonstandardized variables, independent or correlated variables. Examples in applied disciplines are presented, and generalizations to ratios of variables from scale mixtures of bivariate normal distributions show the potential of further new applications in applied statistics and operations research.

A New Fatigue Life Model Based on the Family of Skew-Elliptical Distributions

Abstract: Fatigue is structural damage produced by cyclic stress and tension. An important statistical model for fatigue life is the Birnbaum–Saunders distribution, which was developed to model ruptured lifetimes of metals that had been subjected to fatigue. This model has been previously generalized and in this article we extend it starting from a skew-elliptical distribution, the incorporation of the elliptical aspect makes the kurtosis flexible, and the skewness makes the asymmetry flexible. In this work we found the probability density, reliability, and hazard functions; as well as its moments and variation, skewness, and kurtosis coefficients. In addition, some properties of this new distribution were found.

Bivariate time series modelling of financial count data

Abstract: A bivariate integer-valued moving average (BINMA) model is proposed. The BINMA model allows for both positive and nagative correlation between the counts. This model can be seen as an inverse of the conditional duration model in the sense that short durations in a time interval correspond to a large count and vice versa. The conditional mean, variance, and covariance of the BINMA model are given. Model extensions to include explanatory variables are suggested. Using the BINMA model for AstraZeneca and Ericsson B, it is found that there is positive correlation between the stock transactions series. Empirically, we find support for the use of long-lag bivariate moving average models for the two series.

The distribution of sums of certain I.I.D. pareto variates

Abstract: Though the Pareto distribution is important to actuaries and economists, an exact expression for the distribution of the sum of n i.i.d. Pareto variates has been difficult to obtain in general. This article considers Pareto random variables with common probability density function (pdf) f(x) = (α/β) (1 + x/β)α+1 for x > 0, where α = 1,2,… and β > 0 is a scale parameter. To date, explicit expressions are known only for a few special cases: (i) α = 1 and n = 1,2,3; (ii) 0 0, and α and n are positive integers. Laplace transforms and generalized exponential integrals are used to derive these expressions, which involve integrals of real valued functions on the positive real line. An important attribute of these expressions is that the integrands involved are non oscillating.

A New Estimator of Population Mean in Stratified Sampling

Abstract: Kadilar and Cingi (2005) have suggested a new ratio estimator in stratified sampling. The efficiency of this estimator is compared with the traditional combined ratio estimator on the basis of mean square error (MSE). We propose another estimator by utilizing a simple transformation introduced by Bedi (1996). The proposed estimator is found to be more efficient than the traditional combined ratio estimator as well as the Kadilar and Cingi (2005) ratio estimator.

Point and interval estimation for a generalized logistic distribution under progressive type II censoring

Abstract: The likelihood equations based on a progressively Type II censored sample from a Type I generalized logistic distribution do not provide explicit solutions for the location and scale parameters. We present a simple method of deriving explicit estimators by approximating the likelihood equations appropriately. We examine numerically the bias and variance of these estimators and show that these estimators are as efficient as the maximum likelihood estimators (MLEs). The probability coverages of the pivotal quantities (for location and scale parameters) based on asymptotic normality are shown to be unsatisfactory, especially when the effective sample size is small. Therefore we suggest using unconditional simulated percentage points of these pivotal quantities for the construction of confidence intervals. A wide range of sample sizes and progressive censoring schemes have been considered in this study. Finally, we present a numerical example to illustrate the methods of inference developed here.

Preliminary Goodness-of-Fit Tests for Normality do not Validate the One-Sample Student t

Abstract: One of the most basic topics in many introductory statistical methods texts is inference for a population mean, μ. The primary tool for confidence intervals and tests is the Student t sampling distribution. Although the derivation requires independent identically distributed normal random variables with constant variance, σ2, most authors reassure the readers about some robustness to the normality and constant variance assumptions. Some point out that if one is concerned about assumptions, one may statistically test these prior to reliance on the Student t. Most software packages provide optional test results for both (a) the Gaussian assumption and (b) homogeneity of variance. Many textbooks advise only informal graphical assessments, such as certain scatterplots for independence, others for constant variance, and normal quantile–quantile plots for the adequacy of the Gaussian model. We concur with this recommendation. As convincing evidence against formal tests of (a), such as the Shapiro–Wilk, we offer...

Linear prediction sufficiency for new observations in the general Gauss-Markov model

Abstract: We consider the prediction of new observations in a general Gauss–Markov model. We state the fundamental equations of the best linear unbiased prediction, BLUP, and consider some properties of the BLUP. Particularly, we focus on such linear statistics, which preserve enough information for obtaining the BLUP of new observations as a linear function of them. We call such statistics linearly prediction sufficient for new observations, and introduce some equivalent characterizations for this new concept.

Some Improvements in Numerical Evaluation of Symmetric Stable Density and Its Derivatives

Abstract: We propose improvements in numerical evaluation of symmetric stable density and its partial derivatives with respect to the parameters. They are useful for more reliable evaluation of maximum likelihood estimator and its standard error. Numerical values of the Fisher information matrix of symmetric stable distributions are also given. Our improvements consist of modification of the method of Nolan (1997) for the boundary cases, i.e., in the tail and mode of the densities and in the neighborhood of the Cauchy and the normal distributions.

The Multivariate Split Normal Distribution and Asymmetric Principal Components Analysis

Abstract: The multivariate split normal distribution extends the usual multivariate normal distribution by a set of parameters which allows for skewness in the form of contraction/dilation along a subset of the principal axes. This article derives some properties for this distribution, including its moment generating function, multivariate skewness, and kurtosis, and discusses its role as a population model for asymmetric principal components analysis. Maximum likelihood estimators and a complete Bayesian analysis, including inference on the number of skewed dimensions and their directions, are presented.

Inference for Log-Gamma Distribution Based on Progressively Type-II Censored Data

Abstract: We discuss the maximum likelihood estimates (MLEs) of the parameters of the log-gamma distribution based on progressively Type-II censored samples. We use the profile likelihood approach to tackle the problem of the estimation of the shape parameter κ. We derive approximate maximum likelihood estimators of the parameters μ and σ and use them as initial values in the determination of the MLEs through the Newton–Raphson method. Next, we discuss the EM algorithm and propose a modified EM algorithm for the determination of the MLEs. A simulation study is conducted to evaluate the bias and mean square error of these estimators and examine their behavior as the progressive censoring scheme and the shape parameter vary. We also discuss the interval estimation of the parameters μ and σ and show that the intervals based on the asymptotic normality of MLEs have very poor probability coverages for small values of m. Finally, we present two examples to illustrate all the methods of inference discussed in this paper.

Goodness-of-Fit Tests for Multivariate Distributions

Abstract: We propose three new statistics, Z p , C p , and R p for testing a p-variate (p ≥ 2) normal distribution and compare them with the prominent test statistics. We show that C p is overall most powerful and is effective against skew, long-tailed as well as short-tailed symmetric alternatives. We show that Z p and R p are most powerful against skew and long-tailed alternatives, respectively. The Z p and R p statistics can also be used for testing an assumed p-variate nonnormal distribution.

Evidence-Based Assessment and Application of Prognostic Markers: The Long Way from Single Studies to Meta-Analysis

Abstract: The identification and assessment of prognostic markers constitutes one of the major tasks in clinical research. Despite huge research effort, the prognostic value of most traditional factors under discussion is uncertain and the usefulness of many specific markers, prognostic indices, and classification schemes is still unproven. Results from different studies are often contradictory, and a general assessment of the usefulness of a specific marker is very difficult. One reason is that systematic reviews of prognostic marker studies have received rather little attention in the literature. It is obvious that a clinically useful and sensible systematic review of a prognostic marker is only possible if the published studies reflect the true nature of the marker and if sufficient details are given in each report. An important goal of a systematic review is to produce a quantitative summary of an effect of interest by a meta-analysis, a statistical approach that combines the results of individual primary studi...

Confidence Intervals for the Scale Parameter of the Power-Law Process

Abstract: The power-law process (PLP) is a two-parameter model widely used for modeling repairable system reliability. Results on exact point estimation for both parameters as well as exact interval estimation for the shape parameter are well known. In this paper, we investigate the interval estimation for the scale parameter. Asymptotic confidence intervals are derived using Fisher information matrix and theoretical results by Cocozza-Thivent (1997). The accuracy of the interval estimation for finite samples is studied by simulation methods.

Estimation of the Parameters of the Birnbaum–Saunders Distribution

Abstract: Some alternative estimators to the maximum likelihood estimators of the two parameters of the Birnbaum–Saunders distribution are proposed Most have high efficiencies as measured by root mean square error and are robust to departure from the model as well as to outliers In addition, the proposed estimators are easy to compute Both complete and right-censored data are discussed Simulation studies are provided to compare the performance of the estimators

On the Construction of Families of Absolutely Continuous Copulas with Given Restrictions

Abstract: The problem of constructing copulas with a given diagonal section has been studied by Sungur and Yang (1996) and Fredricks and Nelsen (1997a); (b); (2002). The results of Sungur and Yang are especially relevant because, among other results, they have proven that an Archimedean copula is characterized by its diagonal section. The results obtained by Fredricks and Nelsen allow one to build a singular copula with a given a diagonal section. In all cases, the resulting copulas are symmetric. In this article, we provide a family of absolutely continuous copulas with a fixed diagonal, which can differ from another absolutely continuous copula almost everywhere with respect to Lebesgue measure. It is important to mention that the asymmetry in the proposed methodology is not an issue.

Formulas for the Design of CUSUM Quality Control Charts

Abstract: In the design of CUSUM control charts, it is common to use charts, tables, or software to find an appropriate critical threshold (h). This article provides an approximate formula to calculate the threshold directly from prespecified values of the reference value (k) and the in-control average run length (ARL0). Formulas are also provided for choosing k and h from prespecified values of the in-control and out-of-control average run lengths.

Estimators for the Finite Mixture of Rayleigh Model Based on Progressively Censored Data

Abstract: In this article, based on progressively Type-II censored samples from a heterogeneous population that can be represented by a finite mixture of two-component Rayleigh lifetime model, the problem of estimating the parameters and some lifetime parameters (reliability and hazard functions) are considered. Both Bayesian and maximum likelihood estimators are of interest. A class of natural conjugate prior densities is considered in the Bayesian setting. The Bayes estimators are obtained using both the symmetric (squared error) loss function, and the asymmetric (LINEX and General Entropy) loss functions. It has been seen that the estimators obtained can be easily evaluated for this type of censoring by using suitable numerical methods. Finally, the performance of the estimates have been compared on the basis of their simulated maximum square error via a Monte Carlo simulation study.

Parameter Estimation of Stable Distributions

Abstract: In finance, economics, statistical physics, signal processing, telecommunications, etc., we frequently meet data sets with outliers that transport important information. α-stable distributions are found more suitable in modeling these kind of data. But the lack of simple and effective methods of estimating their parameters limited their applications to wider variety of fields. In this article we develop an unbiased estimator for the stable index α. With the structure of U-statistic, it inherits all the good statistical properties from U-statistics. A consistent estimator of its asymptotic variance is provided. The asymptotic normality of the given estimator holds when using the estimated variance for standardization. Simulation studies are performed. The results support our theory.

Stochastic Comparisons and Dependence of Spacings from Two Samples of Exponential Random Variables

Abstract: Let X 1,X 2,…,X n be independent exponential random variables such that X i has hazard rate λ for i = 1,…,p and X j has hazard rate λ∗ for j = p + 1,…,n, where 1 ≤ p < n. Denote by D i:n (λ, λ∗) = X i:n − X i−1:n the ith spacing of the order statistics X 1:n ≤ X 2:n ≤ ··· ≤ X n:n , i = 1,…,n, where X 0:n ≡ 0. It is shown that the spacings (D 1,n ,D 2,n ,…,D n:n ) are MTP2, strengthening one result of Khaledi and Kochar (2000), and that (D 1:n (λ2, λ∗),…,D n:n (λ2, λ∗)) ≤ lr (D 1:n (λ1, λ∗),…,D n:n (λ1, λ∗)) for λ1 ≤ λ∗ ≤ λ2, where ≤ lr denotes the multivariate likelihood ratio order. A counterexample is also given to show that this comparison result is in general not true for λ∗ < λ1 < λ2.

CUSUM Chart with Transformed Exponential Data

Abstract: Time Between Events (TBE) charts were proposed to monitor the time between events occur based on exponential distribution, and have been shown to be more effective than monitoring the fraction non conforming directly. In this article, we consider monitoring the TBE data with CUSUM scheme by transformation. The idea behind it is to transform the TBE data to normal, and then apply the CUSUM scheme for the approximate normal data. Several simple transformation methods are examined. The calculation of Average Run Length (ARL) with Markov chain approach is described. Comparative studies on the ARL performance show that the transformed CUSUM is superior to the X-MR (Moving Range) chart with transformation, the Cumulative Quantity Control (CQC) chart, and have comparable performance with exponential CUSUM charts. The design procedures of optimal CUSUM chart are also presented. This study provides another possible alternative for monitoring TBE data with easy design procedures and relatively good performance.

Efficient Estimation of Parameters of the Negative Binomial Distribution

Abstract: In this article we investigate a class of moment-based estimators, called power method estimators, which can be almost as efficient as maximum likelihood estimators and achieve a lower asymptotic variance than the standard zero term method and method of moments estimators. We investigate different methods of implementing the power method in practice and examine the robustness and efficiency of the power method estimators.

Binary Regression with Stochastic Covariates

Abstract: In binary regression the risk factor X has been treated in the literature as a non-stochastic variable. In most situations, however, X is stochastic. We present solutions applicable to such situations. We show that our solutions are more precise than those obtained by treating X as non-stochastic when, in fact, it is stochastic.

Inference for the Mean Residual Life Function via Empirical Likelihood

Abstract: In addition to the distribution function, the mean residual life (MRL) function is the other important function which can be used to characterize a lifetime in survival analysis and reliability. For inference on the MRL function, some procedures have been proposed in the literature. However, the coverage accuracy of such procedures may be low when the sample size is small. In this article, an empirical likelihood (EL) inference procedure of MRL function is proposed and the limiting distribution of the EL ratio for MRL function is derived. Based on the result, we obtain confidence interval/band for the MRL function. The proposed method is compared with the normal approximation based method through simulation study in terms of coverage probability.

Some Properties of Weighted Distributions in the Context of Repairable Systems

Abstract: In this article we introduce some structural relationships between weighted and original variables in the context of maintainability function and reversed repair rate. Furthermore, we prove some characterization theorems for specific models such as power, exponential, Pareto II, beta, and Pearson system of distributions using the relationships between the original and weighted random variables.

Statistical Inference for the Generalized Pareto Distribution: Maximum Likelihood Revisited

Abstract: The maximum likelihood system of equations for the shape and scale parameters ξ and β of a generalized Pareto distribution is summarized through a single equation h x (ρ) = 0 where x designates the n-sample of realizations and . The study of the function h x (ρ) in the neighborhood of 0 puts forward a simple statistic S(x) only based on the first two empirical moments which is independent from β. This statistic allows to build an immediate and easy to use statistical test for ξ. The power of this test appears as quite similar to the one of already known test procedures which are much more difficult to use in practice. Moreover, the maximum likelihood solution of h x (ρ) = 0, say ρ∘, and S(x) both have the same sign. This property allows us to propose a controlled use of standard optimization algorithms for characterizing ρ∘, and as a consequence, the maximum likelihood estimations of ξ and β, without any risk of convergence failure or aberrant solution.