Particle swarm optimization based Liu-type estimator

In this study, a new method for the estimation of the shrinkage and biasing parameters of Liu-type estimator is proposed. Because k is kept constant and d is optimized in Liu’s method, a (k, d) pair is not guaranteed to be the optimal point in terms of the mean square error of the parameters. The optimum (k, d) pair that minimizes the mean square error, which is a function of the parameters k and d, should be estimated through a simultaneous optimization process rather than through a two-stage process. In this study, by utilizing a different objective function, the parameters k and d are optimized simultaneously with the particle swarm optimization technique.     

The Frisch–Waugh–Lovell theorem for the lasso and the ridge regression

The Frisch–Waugh–Lovell (FWL) (partitioned regression) theorem is essential in regression analysis. This is partly because it is quite useful to derive theoretical results. The lasso regression and the ridge regression, both of which are penalized least-squares regressions, have become popular statistical techniques. This article describes that the FWL theorem remains valid for these penalized least-squares regressions. More precisely, we demonstrate that the covariates corresponding to unpenalized regression parameters in these penalized least-squares regression can be projected out. Some other results related to the FWL theorem in such penalized least-squares regressions are also presented.     

Characterization of Unacceptable Solutions in LP Discriminant Analysis*

Many linear programming models have been proposed for performing discriminant analysis. Partial characterizations for unacceptable solutions have been presented and new models proposed to circumvent these problems. In this paper those conditions leading to unacceptable solutions for all two-group models are characterized.     

Inference in a Binary Response Model with Applications to Data Analysis*

Standard errors of the coefficients of a logistic regression (a binary response model) based on the asymptotic formula are compared to those obtained from the bootstrap through Monte Carlo simulations. The computer intensive bootstrap method, a nonparametric alternative to the asymptotic estimate, overestimates the true value of the standard errors while the asymptotic formula underestimates it. However, for small samples the bootstrap estimates are substantially closer to the true value than their counterpart derived from the asymptotic formula. The methodology is discussed using two illustrative data sets. The first example deals with a logistic model explaining the log-odds of passing the ERA amendment by the 1982 deadline as a function of percent of women legislators and the percent vote for Reagan. In the second example, the probability that an ingot is ready to roll is modelled using heating time and soaking time as explanatory variables. The results agree with those obtained from the simulations. The value of the study to better decision making through accurate statistical inference is discussed.     

A flexible semiparametric transformation model for survival data

I suggest an extension of the semiparametric transformation model that specifies a time-varying regression structure for the transformation, and thus allows time-varying structure in the data. Special cases include a stratified version of the usual semiparametric transformation model. The model can be thought of as specifying a first order Taylor expansion of a completely flexible baseline. Large sample properties are derived and estimators of the asymptotic variances of the regression coefficients are given. The method is illustrated by a worked example and a small simulation study. A goodness of fit procedure for testing if the regression effects lead to a satisfactory fit is also suggested.     

A NEW CLASS OF MODELS FOR THE DISCRIMINANT PROBLEM

Discriminant analysis is relevant to business decision making in a variety of contexts, such as when one decides to make or buy a specified component, fund a venture project, or hire a particular person. Potential applications in artificial intelligence, particularly in the area of pattern recognition, have further underscored the importance of the field. A recent innovation in discriminant analysis is provided by special linear programming (LP) models, which offer attractive alternatives to classical statistical approaches. The scope of application in which discriminant analysis can be advantageously employed is broadened by the flexibility to tailor parameters in the LP approaches to reflect diverse goals and by the power to explore the sensitivity of these parameters. In spite of the promise of the LP formulations, however, limitations to their effectiveness have been uncovered in certain settings. A recent advance involving a normalization construct removes some of the limitations but entails solving the LP model twice (to allow for different signs of a normalization constant) and does not yield equivalent solutions for different rotations of the problem data. This paper introduces a new model and a new class of normalizations that remedy both remaining limitations, making it possible to take advantage of the modeling capabilities of the LP formulations without the attendant shortcomings encountered by earlier investigations. Our development shows by empirical testing and illustrative analysis that the quality of solutions from LP discriminant approaches is more favorable (relative to the classical model) than previously supposed.     

A Variable Selection Criterion in the Linear Programming Approaches to Discriminant Analysis*

There are numerous variable selection rules in classical discriminant analysis. These rules enable a researcher to distinguish significant variables from nonsignificant ones and thus provide a parsimonious classification model based solely on significant variables. Prominent among such rules are the forward and backward stepwise variable selection criteria employed in statistical software packages such as Statistical Package for the Social Sciences and BMDP Statistical Software. No such criterion currently exists for linear programming (LP) approaches to discriminant analysis. In this paper, a criterion is developed to distinguish significant from nonsignificant variables for use in LP models. This criterion is based on the “jackknife” methodology. Examples are presented to illustrate implementation of the proposed criterion.     

Noninformative priors for linear combinations of normal means with unequal variances

For normal populations with unequal variances, we develop matching priors and reference priors for a linear combination of the means. Here, we find three second-order matching priors: a highest posterior density (HPD) matching prior, a cumulative distribution function (CDF) matching prior, and a likelihood ratio (LR) matching prior. Furthermore, we show that the reference priors are all first-order matching priors, but that they do not satisfy the second-order matching criterion that establishes the symmetry and the unimodality of the posterior under the developed priors. The results of a simulation indicate that the second-order matching prior outperforms the reference priors in terms of matching the target coverage probabilities, in a frequentist sense. Finally, we compare the Bayesian credible intervals based on the developed priors with the confidence intervals derived from real data.     

Misspecifications of the Normal Distribution

Incorrect statements about the normal distribution are discussed and illustrated with counterexamples.