Accent on Teaching Materials

Estimation of covariance components in the multivariate random-effect model with nested covariance structure is discussed. There are two covariance matrices to be estimated, namely, the between-group and the within-group covariance matrices. These two covariance matrices are most often estimated by forming a multivariate analysis of variance and equating mean square matrices to their expectations. Such a procedure involves taking the difference between the between-group mean square and the within-group mean square matrices, and often produces an estimated between-group covariance matrix that is not nonnegative definite. We present estimators of the two covariance matrices that are always proper covariance matrices. The estimators are the restricted maximum likelihood estimators if the random effects are normally distributed. The estimation procedure is extended to more complicated models, including the twofold nested and the mixed-effect models. A numerical example is presented to illustrate the use of the estimation procedure.     

Dot Plots

Dot plots represent individual observations in a batch of data with symbols, usually circular dots. They have been used for more than 100 years to depict distributions in detail. Hand-drawn examples show their authors' efforts to arrange symbols so that they are as near as possible to their proper locations on a scale without overlapping enough to obscure each other. Recent computer programs that attempt to reproduce these historical plots have unfortunately resorted to simple histogram binning instead of using methods that follow the rules for the hand-drawn examples. This article introduces an algorithm that more accurately represents the dot plots cited in the literature.     

Estimation of a discriminant function from a mixture of two inverse Weibull distributions

The classification of a random variable based on a mixture can be meaningfully discussed only if the class of all finite mixtures is identifiable. In this paper, we find the maximum-likelihood estimates of the parameters of the mixture of two inverse Weibull distributions by using classified and unclassified observations. Next, we estimate the nonlinear discriminant function of the underlying model. Also, we calculate the total probabilities of misclassification as well as the percentage bias. In addition, we investigate the performance of all results through a series of simulation experiments by means of relative efficiencies. Finally, we analyse some simulated and real data sets through the findings of the paper.     

Simulating from irregular data: Kernel Carlo Simulation

In this paper, we address the problem of simulating from a data-generating process for which the observed data do not follow a regular probability distribution. One existing method for doing this is bootstrapping, but it is incapable of interpolating between observed data. For univariate or bivariate data, in which a mixture structure can easily be identified, we could instead simulate from a Gaussian mixture model. In general, though, we would have the problem of identifying and estimating the mixture model. Instead of these, we introduce a non-parametric method for simulating datasets like this: Kernel Carlo Simulation. Our algorithm begins by using kernel density estimation to build a target probability distribution. Then, an envelope function that is guaranteed to be higher than the target distribution is created. We then use simple accept–reject sampling. Our approach is more flexible than others, can simulate intelligently across gaps in the data, and requires no subjective modelling decisions. With several univariate and multivariate examples, we show that our method returns simulated datasets that, compared with the observed data, retain the covariance structures and have distributional characteristics that are remarkably similar.     

The umvue and bayes estimate of reliability of mixed failure time distribution

We study the reliability estimates of the non-standard mixture of degenerate (degenerated at zero) and exponential distributions. The Uniformly Minimum Variance Unbiased Estimator (UMVUE) and Bayes estimator of the reliability for some selective prior when the mixing proportion is known and unknown are derived. The Bayes risk is computed for each Bayes estimator of the reliability. A simulated study is carried out to assess the performance of the estimators alongwith the true and Maximum Likelihood Estimate (MLE) of the reliability. An example from Vannman (1991) is also discussed at the end of the paper.     

A Practical Implementation and Bayesian Estimation in Franklin's Randomized Response Procedure

A Bayesian estimator based on Franklin's randomized response procedure is proposed for proportion estimation in surveys dealing with a sensitive character. The method is simple to implement and avoids the usual drawbacks of Franklin's estimator, i.e., the occurrence of negative estimates when the population proportion is small. A simulation study is considered in order to assess the performance of the proposed estimator as well as the corresponding credible interval.     

Change-Point Estimation in Long Memory Nonparametric Models with Applications

We consider the estimation of a change point or discontinuity in a regression function for random design model with long memory errors. We provide several change-point estimators and investigate the consistency of the estimators. Using the fractional ARIMA process as an example of long memory process, we report a small Monte Carlo experiment to compare the performance of the estimators in finite samples. We finish by applying the method to a climatological data example.     

Exponentiated power Lindley power series class of distributions: Theory and applications

ABSTRACT

We introduce a new four-parameter generalization of the exponentiated power Lindley (EPL) distribution, called the exponentiated power Lindley power series (EPLPS) distribution. The new distribution arises on a latent complementary risks scenario, in which the lifetime associated with a particular risk is not observable; rather, we observe only the minimum lifetime value among all risks. The distribution exhibits a variety of bathtub-shaped hazard rate functions. It contains as particular cases several lifetime distributions. Various properties of the distribution are investigated including closed-form expressions for the density function, cumulative distribution function, survival function, hazard rate function, the rth raw moment, and also the moments of order statistics. Expressions for the Rényi and Shannon entropies are also given. Moreover, we discuss maximum likelihood estimation and provide formulas for the elements of the Fisher information matrix. Finally, two data applications are given showing flexibility and potentiality of the EPLPS distribution.     

Modified weighted squared error estimation procedures with special emphasis on the stable laws

Two families of parameter estimation procedures for the stable laws based on a variant of the characteristic function are provided. The methodology which produces viable computational procedures for the stable laws is generally applicable to other families of distributions across a variety of settings. Both families of procedures may be described as a modified weighted chi-squared minimization procedure, and both explicitly take account of constraints on the parameter space. Influence func-tions for and efficiencies of the estimators are given. If x₁, x₂, …x_n random sample from an unknown distribution F , a method for determining the stable law to which F is attracted is developed. Procedures for regression and autoregres-sion with stable error structure are provided. A number of examples are given.     

Computational algorithms for the factor model

Algorithms for computing the maximum likelihood estimators and the estimated covariance matrix of the estimators of the factor model are derived. The algorithms are particularly suitable for large matrices and for samples that give zero estimates of some error variances. A method of constructing estimators for reduced models is presented. The algorithms can also be used for the multivariate errors-in-variables model with known error covariance matrix.