Dean M. Young

A representation of the general common solution to the matrix equations A1XB1 = C1 and A2XB2 = C2 with applications

We give new necessary and sufficient conditions for the existence of a common solution to the pair of linear matrix equations A1XB1 = C1 and A2XB2 = C2 and derive a new representation of the general common solution to these two equations. We apply this result to determine new necessary and sufficient conditions for the existence of an Hermitian solution and a representation of the general Hermitian solution to the matrix equation AXB = C.

A comparison of asymptotic error rate expansions for the sample linear discriminant function

Abstract Several asymptotic expansions for approximating the expected or unconditional probability of misclassification for the sample linear discriminant function are compared for accuracy in terms of yielding the smallest mean absolute deviation from the exact value for 104 population configurations. The actual expected probabilities of misclassification are found via Monte Carlo simulation. A simple and relatively obscure asymptotic expansion derived by Raudys ( Tech. Cybern. 4 , 168–174, 1972) is found to yield better approximation than the well-known asymptotic expansions.

Results in statistical discriminant analysis: a review of the former Soviet union literature

Much work in discriminant analysis and statistical pattern recognition has been performed in the former Soviet Union. However, most results derived by former Soviet Union researchers are unknown to statisticians and statistical pattern recognition researchers in the West. We attempt to give a succinct overview of important contributions by Soviet Block researchers to several topics in the discriminant analysis literature concerning the small training-sample size problem. We also include a partial review of corresponding work done in the West.

Confidence intervals for a binomial parameter based on binary data subject to false-positive misclassification

In this paper we derive five first-order likelihood-based confidence intervals for a population proportion parameter based on binary data subject to false-positive misclassification and obtained using a double sampling plan. We derive confidence intervals based on certain combinations of likelihood, Fisher-information types, and likelihood-based statistics. Using Monte Carlo methods, we compare the coverage properties and average widths of three new confidence intervals for a binomial parameter. We determine that an interval estimator derived from inverting a score-type statistic is superior in terms of coverage probabilities to three competing interval estimators for the parameter configurations examined here. Utilizing the expressions derived, we also determine confidence intervals for a binary parameter using real data subject to false-positive misclassification.

Linear dimension reduction and Bayes classification with unknown population parameters

Abstract Odell and Decell, Odell and Coberly gave necessary and sufficient conditions for the smallest dimension compression matrix B such that the Bayes classification regions are preserved. That is, they developed an explicit expression of a compression matrix B such that the Bayes classification assignment are the same for both the original space x and the compressed space Bx . Odell indicated that whenever the population parameters are unknown, then the dimension of Bx is the same as x with probability one. Furthermore, Odell posed the problem of finding a lower dimension q p which in some sense best fits the range space generated by the matrix M . The purpose of this paper is to discuss this problem and provide a partial solution.

Quadratic discrimination: Some results on optimal low-dimensional representation

Abstract A random vector is assumed to belong to one several multivariate normal distributions possibility having unequal covariance matrices. The goal is to find a low-dimensional hyperplane which preserves or nearly preserves the separation of the individual population. We present a computationally simple method of deriving a linear transformation for low-dimensional representation and give conditions under which the Bayes classification rule is preserved in the low-dimensional space. Finally, we give several examples to demonstrate the method.

The Euclidean distance classifier: an alternative to the linear discriminant function

The sample linear discriminant function (LDF) is known to perform poorly when the number of features p is large relative to the size of the training samples, A simple and rarely applied alternative to the sample LDF is the sample Euclidean distance classifier (EDC). Raudys and Pikelis (1980) have compared the sample LDF with three other discriminant functions, including thesample EDC, when classifying individuals from two spherical normal populations. They have concluded that the sample EDC outperforms the sample LDF when p is large relative to the training sample size. This paper derives conditions for which the two classifiers are equivalent when all parameters are known and employs a Monte Carlo simulation to compare the sample EDC with the sample LDF no only for the spherical normal case but also for several nonspherical parameter configurations. Fo many practical situations, the sample EDC performs as well as or superior to the sample LDF, even for nonspherical covariance configurations.

Bayesian sample-size determination for one and two Poisson rate parameters with applications to quality control

Abstract We formulate Bayesian approaches to the problems of determining the required sample size for Bayesian interval estimators of a predetermined length for a single Poisson rate, for the difference between two Poisson rates, and for the ratio of two Poisson rates. We demonstrate the efficacy of our Bayesian-based sample-size determination method with two real-data quality-control examples and compare the results to frequentist sample-size determination methods.

Extended critical vawes of the multivariate extreme deviate test for detecting a single spurious observation

The Institute of Mathematical Statistics has published a table of critical values for the multivariate extreme deviate test. However, the critical values, derived by a Monte Carlo simulation, are given for only the dimensions 2 through 5. We present new critical values for the dimensions 6 through 10, 12, 15, and 20. The results are presented in both table and graphical form. All critical values for the test statistic have been generated by a Monte Carlo simulation using 10,000 observations per case. An example is presented using the new critical values.

Estimation of disease prevalence, true positive rate, and false positive rate of two screening tests when disease verification is applied on only screen-positives: A hierarchical model using multi-center data

OBJECTIVES A model is proposed to estimate and compare cervical cancer screening test properties for third world populations when only subjects with a positive screen receive the gold standard test. Two fallible screening tests are compared, VIA and VILI. METHODS We extend the model of Berry et al. [1] to the multi-site case in order to pool information across sites and form better estimates for prevalences of cervical cancer, the true positive rates (TPRs), and false positive rates (FPRs). For 10 centers in five African countries and India involving more than 52,000 women, Bayesian methods were applied when gold standard results for subjects who screened negative on both tests were treated as missing. The Bayesian methods employed suitably correct for the missing screen negative subjects. The study included gold standard verification for all cases, making it possible to validate model-based estimation of accuracy using only outcomes of women with positive VIA or VILI result (ignoring verification of double negative screening test results) with the observed full data outcomes. RESULTS Across the sites, estimates for the sensitivity of VIA ranged from 0.792 to 0.917 while for VILI sensitivities ranged from 0.929 to 0.977. False positive estimates ranged from 0.056 to 0.256 for VIA and 0.085 to 0.269 for VILI. The pooled estimates for the TPR of VIA and VILI are 0.871 and 0.968, respectively, compared to the full data values of 0.816 and 0.918. Similarly, the pooled estimates for the FPR of VIA and VILI are 0.134 and 0.146, respectively, compared to the full data values of 0.144 and 0.146. Globally, we found VILI had a statistically significant higher sensitivity but no statistical difference for the false positive rates could be determined. CONCLUSION Hierarchical Bayesian methods provide a straight forward approach to estimate screening test properties, prevalences, and to perform comparisons for screening studies where screen negative subjects do not receive the gold standard test. The hierarchical model with random effects used to analyze the sites simultaneously resulted in improved estimates compared to the single-site analyses with improved TPR estimates and nearly identical FPR estimates to the full data outcomes. Furthermore, higher TPRs but similar FPRs were observed for VILI compared to VIA.