Adjustment with three continuous variables

Correlation is not causation. Spurious association between X and Y may be due to a confounding variable W. Statisticians may adjust for W using a variety of techniques. This article presents the results of simulations conducted to assess the performance of these techniques under various, elementary, data-generating processes. The results indicate that no technique is best overall and that specific techniques should be selected based on the particulars of the data-generating process. Here, we show how causal graphs can guide the selection or design of techniques for statistical adjustment. R programs are provided for researchers interested in generalization.     

A comparison of linear regression techniques in method comparison studies

In this study, the performances of linear regression techniques, which are especially used in clinical chemistry in method comparison studies, are compared via the Monte-Carlo simulation. The regression techniques that take the measurement errors of both dependent and independent variables into account are called Type II regression techniques. In this study, we also compare the performances of Type II and Type I (classical regression techniques that do not take the measurement errors of the independent variable into account) regression techniques for different sample sizes and different shape parameters of the Weibull distribution. The mean square error is used as a performance criterion of each technique. MATLAB 7.02 software is used in the simulation study. As a result, in all conditions, the ordinary least-square (OLS)-bisector regression technique, which bisects the OLS(Y | X) and the OLS(X | Y), shows the best performance.     

On approximating point spread distributions

We discuss some properties of the point spread distribution, defined as the distribution of the difference of two independent binomial random variables with the same parameter n including exact and approximate probabilities and related optimization issues. We use various approximation techniques for different distributions, special functions, and analytic, combinatorial and symbolic methods, such as multi-summation techniques. We prove that in case of unequal success rates, if these rates change with their difference kept fixed and small, and n is appropriately bounded, then the point spread distribution only slightly changes for small point differences. We also prove that for equal success rates p, the probability of a tie is minimized if p=1/2. Numerical examples are included for the case with n=12.     

Simulations and computations of nonparametric density estimates for the deconvolution problem

The nonparametric density function estimation using sample observations which are contaminated with random noise is studied. The particular form of contamination under consideration is Y = X + Z, where Y is an observable random variableZ is a random noise variable with known distribution, and X is an absolutely continuous random variable which cannot be observed directly. The finite sample size performance of a strongly consistent estimator for the density function of the random variable X is illustrated for different distributions. The estimator uses Fourier and kernel function estimation techniques and allows the user to choose constants which relate to bandwidth windows and limits on integration and which greatly affect the appearance and properties of the estimates. Numerical techniques for computation of the estimated densities and for optimal selection of the constant are given.     

Inference for Reliability and Stress-Strength for a Scaled Burr Type X Distribution

Inferencefor R=P(Y is considered when Xand Y are independently distributed as scaled Burrtype X random variables. Under this model, exact inference proceduresfor R cannot be found. Hence, based on the expectedFisher information matrix which is derived here, asymptotic inferenceprocedures for R and other general functions ofthe parameters are developed. A bootstrap method to estimatevariance for the maximum likelihood estimators is also discussed.To illustrate these techniques, an example using carbon fiberstrength data is given. Simulations to assess the effectivenessof these techniques, as well as other concerns, are presented.     

On using non identical and independent Bernoulli trials in human surveys

Unlike the usual randomized response techniques, as a pioneering attempt, this article focuses on using non identical independent Bernoulli trials in sensitive surveys. For this purpose, a general class of randomized response techniques is considered. The usual randomized response techniques are based on a fixed probability of having a yes answer. Contrary to usual techniques, in the proposed technique every respondent has a different probability of reporting a yes answer. With this setting, in most of the situations, the proposed technique is observed performing better in terms of variability. To illustrate and support the superiority of the proposed technique it is compared with models such as Warner (1965), Greenberg et al. (1969), Mangat and Singh (1990), and Mangat (1994) using identical Bernoulli trials. Relative efficiency and privacy protection are studied in detail using Warner (1965) and Mangat (1994) models.     

Approximate distribution and test of fit for the clustering effect in the dirichlet multinomial model

The Dirichlet-multinomial model is considered as a model for cluster sampling. The model assumes that the design's covariance matrix is a constant times the covariance under multinomial sampling. The use of this model requires estimating a parameter C, that measures the clustering effect. In this paper, a regression estimate for C is obtained. An approximate distribution of this estimator is obtained through the use of asymptotic techniques. A goodness of fit statistic for testing the fit of the Dirichlet Multinomial model is also obtained, based on those asymptotic techniques. These statistics provide a means of knowing when the data satisfy the model assumption. These results are used to analyze data concerning the authorship of Greek prose.     

A procedure for testing the equality of k dependent correlation coefficients

A procedure is proposed for testing the equality of k dependent correlation coefficients. The procedure is simulated utilizing Monte Carlo techniques; and, a method for post hoc probing is also suggested.     

FUNCTIONAL ESTIMATION BY ASYMPTOTIC REGRESSION

Through an appeal to asymptotic Gaussian representations of certain empirical stochastic processes, the techniques of continuous regression are applied to derive estimates for underlying parametric probability laws. This asymptotic regression approach yields estimates for a wide range of statistical problems, including estimation based on the empirical quantile function, Poisson process intensity estimation, and parametric density estimation.     

Some asymptotic distribution results in time-sequential estimation of the mean exponential survival time

The asymptotic distribution of the stopping time N in a time-sequential procedure for the estimation of the mean exponential survival time given by Gardiner, Susarla, and van Ryzin (1986) is obtained. The same techniques used to obtain this asymptotic distribution of N are used to obtain the asymptotic distribution of the statistic representing the time-on-test expended per unit item in the study.     

Conditional covariance penalties for mixed models

The prediction error for mixed models can have a conditional or a marginal perspective depending on the research focus. We introduce a novel conditional version of the optimism theorem for mixed models linking the conditional prediction error to covariance penalties for mixed models. Different possibilities for estimating these conditional covariance penalties are introduced. These are bootstrap methods, cross-validation, and a direct approach called Steinian. The behavior of the different estimation techniques is assessed in a simulation study for the binomial-, the t-, and the gamma distribution and for different kinds of prediction error. Furthermore, the impact of the estimation techniques on the prediction error is discussed based on an application to undernutrition in Zambia.     

Asymptotic and Resampling-Based Confidence Intervals for P(X < Y)

ABSTRACT

We consider asymptotic and resampling-based interval estimation procedures for the stress-strength reliability P(X < Y). We developed and studied several types of intervals. Their performances are investigated using simulation techniques and compared in terms of attainment of the nominal confidence level, symmetry of lower and upper error rates, and expected length. Recommendations concerning their use are given.     

Exact moments of the symmetric cubic assignment statistic

Hubert (1987Assignment Methods in Combinatorial Data Analysis) presented a class of permutation, or random assignment, techniques for assessing correspondence between general k-dimensional proximity measures on a set of “objects.” A major problem in higher-order assignment models is the prohibitive level of computation that is required. We present the first three exact moments of a test statistic for the symmetric cubic assignment model. Efficient computational formulas for the first three moments have been derived, thereby permitting approximation of the permutation distribution using well-known methods.     

Some small-sample properties of some recently proposed multivariate outlier detection techniques

Recently, several new robust multivariate estimators of location and scatter have been proposed that provide new and improved methods for detecting multivariate outliers. But for small sample sizes, there are no results on how these new multivariate outlier detection techniques compare in terms of p _n , their outside rate per observation (the expected proportion of points declared outliers) under normality. And there are no results comparing their ability to detect truly unusual points based on the model that generated the data. Moreover, there are no results comparing these methods to two fairly new techniques that do not rely on some robust covariance matrix. It is found that for an approach based on the orthogonal Gnanadesikan–Kettenring estimator, p _n can be very unsatisfactory with small sample sizes, but a simple modification gives much more satisfactory results. Similar problems were found when using the median ball algorithm, but a modification proved to be unsatisfactory. The translated-biweights (TBS) estimator generally performs well with a sample size of n≥20 and when dealing with p-variate data where p≤5. But with p=8 it can be unsatisfactory, even with n=200. A projection method as well the minimum generalized variance method generally perform best, but with p≤5 conditions where the TBS method is preferable are described. In terms of detecting truly unusual points, the methods can differ substantially depending on where the outliers happen to be, the number of outliers present, and the correlations among the variables.     

Model-Based Classification via Mixtures of Multivariate t-Factor Analyzers

A model-based classification technique is developed, based on mixtures of multivariate t-factor analyzers. Specifically, two related mixture models are developed and their classification efficacy studied. An AECM algorithm is used for parameter estimation, and convergence of these algorithms is determined using Aitken's acceleration. Two different techniques are proposed for model selection: the BIC and the ICL. Our classification technique is applied to data on red wine samples from Italy and to fatty acid measurements on Italian olive oils. These results are discussed and compared to more established classification techniques; under this comparison, our mixture models give excellent classification performance.     

On the validation of fiducial techniques

A structured model is essentially a family of random vectors X_θ defined on a probability space with values in a sample space. If, for a given sample value x and for each ω in the probability space, there is at most one parameter value θ for which X_θ(ω) is equal to x, then the model is called additive at x. When a certain conditional distribution exists, a frequency interpretation specific to additive structured models holds, and is summarized in a unique structured distribution for the parameter. Many of the techniques used by Fisher in deriving and handling his fiducial probability distribution are shown to be valid when dealing with a structured distribution.     

Stabilizing bootstrap‐t confidence intervals for small samples

A major use of the bootstrap methodology is in the construction of nonparametric confidence intervals. Although no consensus has yet been reached on the best way to proceed, theoretical and empirical evidence indicate that bootstra.‐t intervals provide a reasonable solution to this problem. However, when applied to small data sets, these intervals can be unusually wide and unstable. The author presents techniques for stabilizing bootstra.‐t intervals for small samples. His methods are motivated theoretically and investigated though simulations.     

Generalized geometric distribution of order k: A flexible choice to randomize the response

This article focuses on the improvement of a well-celebrated randomized response technique of Kuk. A generalized randomized response technique is suggested. In particular, the generalized geometric distribution of order k is introduced as a randomization device for estimating the population proportion of a rare sensitive attribute. The proposed randomized response technique includes Singh and Grewal and Hussain et al. techniques as its special cases. Through numerical illustrations, it is established that the suggested technique is superior to the Kuk, Singh and Grewal, and Hussain et al. techniques. Flexibility of the proposed technique is also discussed.     

Resistant selection of the smoothing parameter for smoothing splines

Robust automatic selection techniques for the smoothing parameter of a smoothing spline are introduced. They are based on a robust predictive error criterion and can be viewed as robust versions of C _p and cross-validation. They lead to smoothing splines which are stable and reliable in terms of mean squared error over a large spectrum of model distributions.     

Tests for the multivariate two-sample problem based on empirical probability measures

Nonparametric tests are proposed for the equality of two unknown p-variate distributions. Empirical probability measures are defined from samples from the two distributions and used to construct test statistics as the supremum of the absolute differences between empirical probabilities, the supremum being taken over all possible events. The test statistics are truly multivariate in not requiring the artificial ranking of multivariate observations, and they are distribution-free in the general p-variate case. Asymptotic null distributions are obtained. Powers of the proposed tests and a competitor are examined by Monte Carlo techniques.