Additive estimators for probabilities of correct classification

Abstract Several methods for estimating a sample-based discriminant's probability of correct classification are compared with respect to bias, variance, robustness, and computation cost. “Smooth” modification of the counting estimator, or sample success proportion, is recommended to reduce bias and variance while retaining robustness. Also the “bootstrap” method of Efron (8) can approximately correct an additive estimator's bias using an ancillary computer simulation. In contrast, bias reduction achieved by the popular “leave-one-out” modification of counting method is vitiated by corresponding increase in variance.

[1]  G. McLachlan.  AN ASYMPTOTIC EXPANSION OF THE EXPECTATION OF THE ESTIMATED ERROR RATE IN DISCRIMINANT ANALYSIS1 , 1973 .

[2]  G. McLachlan.  An Asymptotic Unbiased Technique for Estimating the Error Rates in Discriminant Analysis , 1974 .

[3]  David A. Landgrebe,et al.  Variance comparisons for unbiased estimators of probability of correct classification (Corresp.) , 1976, IEEE Trans. Inf. Theory.

[4]  J. S. Williams,et al.  Minimum variance estimators for misclassification probabilities in discriminant analysis , 1973 .

[5]  S. Schwartz.  Estimation of Probability Density by an Orthogonal Series , 1967 .

[6]  T. Cacoullos.  Estimation of a multivariate density , 1966 .

[7]  T. W. Anderson,et al.  An Introduction to Multivariate Statistical Analysis , 1959 .

[8]  B. Efron.  Bootstrap Methods: Another Look at the Jackknife , 1979 .

[9]  Godfried T. Toussaint,et al.  Bibliography on estimation of misclassification , 1974, IEEE Trans. Inf. Theory.

[10]  T. W. Anderson.  An Introduction to Multivariate Statistical Analysis , 1959 .

[11]  Ned Glick,et al.  Sample-based classification procedures related to empiric distributions , 1976, IEEE Trans. Inf. Theory.

[12]  Kishan Mehrotra.  Some Further Considerations on the Estimation of Error of Misclassification Based on the Design Set. , 1976 .

[13]  G. McLachlan.  The bias of the apparent error rate in discriminant analysis , 1976 .

[14]  O. J. Dunn.  Some Expected Values for Probabilities of Correct Classification in Discriminant Analysis , 1971 .

[15]  M. Sorum,et al.  Estimating the Expected and the Optimal Probabilities of Misclassification , 1972 .

[16]  D. Specht.  Series Estimation of a Probability Density Function , 1971 .

[17]  M. Rosenblatt.  Remarks on Some Nonparametric Estimates of a Density Function , 1956 .

[18]  P. Lachenbruch.  An almost unbiased method of obtaining confidence intervals for the probability of misclassification in discriminant analysis. , 1967, Biometrics.

[19]  King-Sun Fu,et al.  Error estimation in pattern recognition via LAlpha -distance between posterior density functions , 1976, IEEE Trans. Inf. Theory.

[20]  G. McLachlan.  Estimation of the Errors of Misclassification on the Criterion of Asymptotic Mean Square Error , 1974 .

[21]  C. Quesenberry,et al.  A nonparametric estimate of a multivariate density function , 1965 .

[22]  P. Révész.  On empirical density function , 1972 .

[23]  R. Kronmal,et al.  The Estimation of Probability Densities and Cumulatives by Fourier Series Methods , 1968 .

[24]  R. Kronmal,et al.  An Introduction to the Implementation and Theory of Nonparametric Density Estimation , 1976 .

[25]  M. R. Mickey,et al.  Estimation of Error Rates in Discriminant Analysis , 1968 .

[26]  G. McLachlan.  The asymptotic distributions of the conditional error rate and risk in discriminant analysis , 1974 .

[27]  M. Okamoto.  An Asymptotic Expansion for the Distribution of the Linear Discriminant Function , 1963 .

[28]  N. Glick.  Sample-Based Multinomial Classification , 1973 .

[29]  M. Sorum.  Estimating the Conditional Probability of Misclassification , 1971 .

[30]  N. Glick.  Sample-Based Classification Procedures Derived from Density Estimators , 1972 .

[31]  M. Sorum.  Three Probabilities of Misclassification , 1972 .

[32]  O. J. Dunn,et al.  Probabilities of Correct Classification in Discriminant Analysis , 1966 .

[33]  M. Okamoto,et al.  Estimation of the probabilities of misclassification for a linear discriminant function in the univariate normal case , 1971 .

[34]  R. Kronmal,et al.  On Multivariate Density Estimates Based on Orthogonal Expansions , 1970 .

[35]  E. Wegman.  Nonparametric probability density estimation , 1972 .

[36]  E. Wegman.  Nonparametric Probability Density Estimation: I. A Summary of Available Methods , 1972 .

[37]  Keinosuke Fukunaga,et al.  Nonparametric Bayes error estimation using unclassified samples , 1972, IEEE Trans. Inf. Theory.

[38]  Laveen N. Kanal,et al.  Patterns in pattern recognition: 1968-1974 , 1974, IEEE Trans. Inf. Theory.

[39]  E. Parzen.  On Estimation of a Probability Density Function and Mode , 1962 .

[40]  M. Hills.  Allocation Rules and Their Error Rates , 1966 .

[41]  G. McLachlan.  AN ASYMPTOTIC EXPANSION FOR THE VARIANCE OF THE ERRORS OF MISCLASSIFICATION OF THE LINEAR DISCRIMINANT FUNCTION1 , 1972 .

[42]  T. Wagner,et al.  Nonparametric estimates of probability densities , 1975, IEEE Trans. Inf. Theory.

[43]  Keinosuke Fukunaga,et al.  Application of optimum error-reject functions (Corresp.) , 1972, IEEE Trans. Inf. Theory.

[44]  G. McLachlan.  The relationship in terms of asymptotic mean square error between the separate problems of estimating each of the three types of error rate of the linear discriminant function , 1974 .