Bonifacio Salvador

An algorithm for isotonic median regression

Abstract In this paper we give an algorithm for computing the solutions of the Isotonic median regression problem, equivalent to the calculation of the maximum likelihood estimate for the parameter θ=(θ 1 ,…,θ k ) , where θ1 is the location parameter of the bilateral exponential distribution, under the assumption θ 1 ⩽θ 2 ⩽…⩽θ k .

The Loss of Efficiency Estimating Linear Functions under Restrictions

This article is motivated by the problem of estimating contrast in a one-way ANOVA model with restrictions in the parameter vector. We prove that when the restrictions are given by a tree order or a simple order the MLE of some contrast has greater MSE than the unrestricted estimator. A similar behaviour of the MLE is exhibited in a general restricted setting given by a multivariate normal distribution with mean vector constrained to belong to a circular cone. The approach we use focuses on the central direction of the cones. These directions appear to have the greater MSE when the dimension of the restricted cone is big enough.

Incorporating Additional Information to Normal Linear Discriminant Rules

The most useful and broadly known rule in the classical two-group linear normal discriminant analysis is Andersons rule. In this article we propose some alternative procedures that prove useful when prior constraints on the mean vectors are known. These rules are based on new estimators of the difference of means. We prove under mild conditions that the new rules perform better when the common covariance matrix is known. Simulated experiments show that the misclassification errors are lower for the restricted rules defined here in the general case of an unknown covariance matrix. The prior constraints on the mean vector restrict the parameter space to a cone. A family of estimators indexed by a parameter γ, with 0 ≤ γ ≤ 1, is defined using an iterative procedure in such a way that the estimator with a higher value for γ takes values closer to the center of the cone with a greater probability. When γ = 0, the restricted maximum likelihood estimator is given, although the most interesting rule from a theoretical and practical standpoint is obtained when the estimator chosen is given by γ = 1. The usefulness of the proposed rules with real data is demonstrated by their application to two medical examples, the first dealing with heart attack patients and the second dealing with a diabetes dataset. In the former case, restrictions among surviving and nonsurviving patients are used; in the latter, the restrictions arise from differences between the healthy and diabetic populations.

Testing non-oblique hypotheses

This paper considers a likelihood ratio test for testing hypotheses defined by non-oblique closed convex cones, satisfying the so called iteration projection property, in a set of k normal means. We obtain the critical values of the test using the Chi-Bar-Squared distribution. The obtuse cones are introduced as a particular class of cones which are non-oblique with every one of their faces. Examples with the simple tree order cone and the total order cone are given to illustrate the results.

Classification of samples into two or more ordered populations with application to a cancer trial

In many applications, especially in cancer treatment and diagnosis, investigators are interested in classifying patients into various diagnosis groups on the basis of molecular data such as gene expression or proteomic data. Often, some of the diagnosis groups are known to be related to higher or lower values of some of the predictors. The standard methods of classifying patients into various groups do not take into account the underlying order. This could potentially result in high misclassification rates, especially when the number of groups is larger than two. In this article, we develop classification procedures that exploit the underlying order among the mean values of the predictor variables and the diagnostic groups by using ideas from order-restricted inference. We generalize the existing methodology on discrimination under restrictions and provide empirical evidence to demonstrate that the proposed methodology improves over the existing unrestricted methodology. The proposed methodology is applied to a bladder cancer data set where the researchers are interested in classifying patients into various groups.

On the maximum likelihood estimator under order restrictions in uniform probability models

We consider random samples from independent uniform populations U(0, θ i ) where the parameters θ i follow a known partial order. The aim of this paper is to compare the restricted and the unrestricted MLE using the universal domination and the squared error criterions when linear functions of the parameter are estimated. We determine functions for which the unrestricted estimator performs better than the unrestricted estimator and a class of functions for which the reverse is true.

Simultaneous estimation by isotonic regression

Abstract We consider a random vector X with unimodal density, diagonal covariance matrix, Ω −1 , and location parameter θ subject to order restrictions. For this model the maximum likelihood estimator (MLE) X∗ is the so-called isotonic regression. It is well known that, in certain situations, X∗ is not better than X when coordinate estimation is considered. We focus on the comparison between the simultaneous constant length confidence intervals for the coordinates centered at X or at X∗ . We characterize situations where the coverage probability of the intervals is greater if X∗ is used instead of X. Special attention is given to the elliptically symmetric distributions with a total ordering or a tree ordering on θ.

Performance and estimation of the true error rate of classification rules built with additional information. An application to a cancer trial

Abstract Classification rules that incorporate additional information usually present in discrimination problems are receiving certain attention during the last years as they perform better than the usual rules. Fernández, M. A., C. Rueda and B. Salvador (2006): “Incorporating additional information to normal linear discriminant rules,” J. Am. Stat. Assoc., 101, 569–577, proved that these rules have lower total misclassification probability than the usual Fisher’s rule. In this paper we consider two issues; on the one hand, we compare these rules with those based on shrinkage estimators of the mean proposed by Tong, T., L. Chen and H. Zhao (2012): “Improved mean estimation and its application to diagonal discriminant analysis,” Bioinformatics, 28(4): 531–537. with regard to four criteria: total misclassification probability, area under ROC curve, well-calibratedness and refinement; on the other hand, we consider the estimation of the true error rate, which is a very interesting parameter in applications. We prove results on the apparent error rate of the rules that expose the need of new estimators of their true error rate. We propose four such new estimators. Two of them are defined incorporating the additional information into the leave-one-out-bootstrap. The other two are the corresponding cross-validation after bootstrap versions. We compare these estimators with the usual ones in a simulation study and in a cancer trial application, showing the good behavior of the rules that incorporate additional information and of the new leave-one-out bootstrap estimators of their true error rate.

Bayes Discriminant Rules with Ordered Predictors

We propose and discuss improved Bayes rules to discriminate between two populations using ordered predictors. To address the problem we propose an alternative formulation using a latent space that allows to introduce the information about the order in the theoretical rules. The rules are first defined when the marginal densities are fully known and then under normality when the parameters are unknown and training samples are available. Several numerical examples and simulations in the paper illustrate the methodology and show that the new rules handle the information appropriately. We compare the new rules with the classical Bayes and Fisher rules in these examples and we show that the misclassification probability is smaller for the new rules. The method is also applied to data from a diabetes study where we again show that the new rules improve over the usual Fisher rule.

Bootstrap adjusted estimators in a restricted setting

In the context of a normal model, where the mean is constrained to a polyhedral convex cone, a new methodology has been developed for estimating a linear combination of the mean components. The method is based on an application of adapted parametric bootstrap procedures to reduce the bias of the maximum likelihood estimator. The proposed method is likely to lead to estimators with low mean squared error. Simulation results which support this argument are included.