Parameter Estimation Through Weighted Least-Squares Rank Regression with Specific Reference to the Weibull and Gumbel Distributions

Probability plots are often used to estimate the parameters of distributions. Using large sample properties of the empirical distribution function and order statistics, weights to stabilize the variance in order to perform weighted least squares regression are derived. Weighted least squares regression is then applied to the estimation of the parameters of the Weibull, and the Gumbel distribution. The weights are independent of the parameters of the distributions considered. Monte Carlo simulation shows that the weighted least-squares estimators outperform the usual least-squares estimators totally, especially in small samples.     

A simulation study of ridge and other regression estimators

We consider a number of estimators of regression coefficients, all of generalized ridge, or 'shrinkage' type. Results of a simulation study indicate that with respect to two commonly used mean square error criteria, two ordinary ridge estimators, one proposed by Hoerl, Kennard and Baldwin, and the other introduced here, perform substantially better than both least squares and the other estimators discussed here     

Minimum average risk estimators for coefficients in linear models

We consider the problem of estimating the coefficient vector β of a linear regression model with quadratic loss function. Some biased estimators which utilize the prior information about β are considered. Also studied is the problem of estimating the parameters of an over-identified structural equation from undersized samples.     

Testing parallelism of regression lines against ordered alternatives

For the problem of testing the equality of slopes of several regression lines against the alternative that the slopes are in increasing (decreasing) order of magnitude, two types of tests are proposed. These are the likelihood ratio test and a test that depends on suitable linear combination of one group statistics. Rank analogues of the two tests are also examined.     

On a characterization of the dispersion matrix based on the properties of regression

This paper provides a partial solution to a problem posed by J. Neyman (1965) regarding the characterization of multivariate negative binomial distribution based on the properties of regression. It is shown that some of the properties of regression characterize the form of the nonsingular dispersion matrix of the parent distribution, which, interestingly enough, corresponds to only two types viz. those of positive and negative multivariate binomial distributions.     

A Nonstochastic Interpretation of Reported Significance Levels

Tests of significance are often made in situations where the standard assumptions underlying the probability calculations do not hold. As a result, the reported significance levels become difficult to interpret. This article sketches an alternative interpretation of a reported significance level, valid in considerable generality. This level locates the given data set within the spectrum of other data sets derived from the given one by an appropriate class of transformations. If the null hypothesis being tested holds, the derived data sets should be equivalent to the original one. Thus, a small reported significance level indicates an unusual data set. This development parallels that of randomization tests, but there is a crucial technical difference: our approach involves permuting observed residuals; the classical randomization approach involves permuting unobservable, or perhaps nonexistent, stochastic disturbance terms.     

Efficacy of Statistical Outlier Analysis for Monitoring Quality of Care

Researchers have proposed that hospitals with excessive statistically unexplained mortality rates are more likely to have quality-of-care problems. The U.S. Health Care Financing Administration currently uses this statistical “outlier” approach to screen for poor quality in hospitals. Little is known, however, about the validity of this technique, since direct measures of quality are difficult to obtain. We use Monte Carlo methods to evaluate the performance of the outlier technique as parameters of the true mortality process are varied. Results indicate that the screening ability of the technique may be very sensitive to how widespread quality-related mortality is among hospitals but insensitive to other factors generally thought to be important.     

Using regression mixture models with non-normal data: examining an ordered polytomous approach

Mild to moderate skew in errors can substantially impact regression mixture model results; one approach for overcoming this includes transforming the outcome into an ordered categorical variable and using a polytomous regression mixture model. This is effective for retaining differential effects in the population; however, bias in parameter estimates and model fit warrant further examination of this approach at higher levels of skew. The current study used Monte Carlo simulations; 3000 observations were drawn from each of two subpopulations differing in the effect of X on Y. Five hundred simulations were performed in each of the 10 scenarios varying in levels of skew in one or both classes. Model comparison criteria supported the accurate two-class model, preserving the differential effects, while parameter estimates were notably biased. The appropriate number of effects can be captured with this approach but we suggest caution when interpreting the magnitude of the effects.     

New difference-based estimator of parameters in semiparametric regression models

The paper introduces a new difference-based Liu estimator β?_Ldiff=([Xtilde]′[Xtilde]+I)^?1([Xtilde]′[ytilde]+η β?_diff) of the regression parameters β in the semiparametric regression model, y=Xβ+f+?. Difference-based estimator, β?_diff=([Xtilde]′[Xtilde])^?1[Xtilde]′[ytilde] and difference-based Liu estimator are analysed and compared with respect to mean-squared error (mse) criterion. Finally, the performance of the new estimator is evaluated for a real data set. Monte Carlo simulation is given to show the improvement in the scalar mse of the estimator.     

Semiparametric ROC surface estimation for continuous diagnostic tests via polytomous logistic regression procedures

In this paper, we propose a semiparametric method of estimating receiver operating characteristic (ROC) surfaces for continuous diagnostic tests under density ratio models. Implementation of our method is easy since the usual polytomous logistic regression procedures in many statistical software packages can be employed. A simulated example is provided to facilitate the implementation of our method. Simulation results show that the proposed semiparametric ROC surface estimator is more efficient than the nonparametric counterpart and the parametric counterpart whether the normality assumption of data holds or not. Moreover, some simulation results on the underlying semiparametric distribution function estimators are also reported. In addition, some discussions on the proposed method as well as analysis of a real data set are provided.