Showing papers on "Asymptotically optimal algorithm published in 1972"
TL;DR: In this paper, a sample-based rule is asymptotically optimal under very general conditions, and the apparent non-error rate is even optimistically biased as an estimator of the (unknown) optimal rule's nonerror rate; but the apparent rate converges to this optimum.
Abstract: Parametric or nonparametric density estimators from a mixture of K distributions can be used to estimate the non-error rate of an arbitrary classification rule—and to construct a rule which maximizes estimated probability of correct classification. (For two multivariate normal distributions with common covariance matrix, this general criterion yields the usual linear discriminant.) Such a sample-based rule is asymptotically optimal under very general conditions. Often its “apparent” non-error rate exceeds its true rate and is even optimistically biased as an estimator of the (unknown) optimal rule's non-error rate; but the apparent rate converges to this optimum.
94 citations
TL;DR: In this paper, it was shown that every one-sided rank test is asymptotically optimal for a certain nonparametric subclass of contiguous alternatives, provided the test and the associated subclass of alternatives are generated by certain square-integrable functions defined on the unit square.
Abstract: For the one-sample independence problem, the one-sample symmetry problem, and the two-sample problem it is shown that every one-sided rank test is asymptotically optimal for a certain nonparametric subclass of contiguous alternatives, provided the test and the associated subclass of alternatives are generated by certain square-integrable functions defined on the unit square. Then the asymptotic normality of the respective rank statistics under every alternative contiguous to the hypothesis is used in order to give necessary and sufficient conditions for local asymptotic unbiasedness of such tests. Finally, for locally asymptotically unbiased tests there are given necessary and sufficient conditions for having bounds for their asymptotic relative efficiency under contiguous alternatives.
30 citations
TL;DR: This paper uses truncated Taylor series expansions to evaluate the misadjustment, or extra probability of error, that results when these algorithms are used to design a linear classifier with a finite number of patterns.
Abstract: Adaptive algorithms for designing two-category linear pattern classifiers have been widely used on nonseparable pattern sets even though they do not directly minimize the number of classification errors and their optimality for pattern classification is not completely known. Many of these algorithms have been shown to be asymptotically optimal for patterns from Gaussian distributions with equal-covariance matrices. However, their relative efficiencies for design with a finite number of patterns have not been known. This paper uses truncated Taylor series expansions to evaluate the misadjustment, or extra probability of error, that results when these algorithms are used to design a linear classifier with a finite number of patterns. The expressions have been evaluated for three algorithms-- the fixed-increment error-correction algorithm, the relaxation error-correction algorithm, and the least-mean-square (LMS) algorithm--used with patterns from Gaussian distributions with equal-covariance matrices.
10 citations
5 citations
TL;DR: The numerical solution of integral equations of Hammerstein type results in solving systems of equationsx+Kf(x)=0 with constant matrixK and diagonal mappingf. as mentioned in this paper showed that two-step iterative methods are asymptotically optimal if K is positive semi-definite and f is isotone and continuously differentiable.
Abstract: The numerical solution of integral equations of Hammerstein type results in solving systems of equationsx+Kf(x)=0 with constant matrixK and diagonal mappingf. It is shown that two-step iterative methods are asymptotically optimal ifK is positive semi-definite andf is isotone and continuously differentiable.
3 citations
TL;DR: In this paper, the power functions of the classical X2 test and the locally asymptotically optimal test of homogeneity of a binomial series are derived for the case of known binomial parameter.
Abstract: Approximations to the power functions of the classical X2 test and the locally asymptotically optimal test of homogeneity of a binomial series are obtained for the case of known binomial parameter; asymptotic power functions are also derived. The results are compared with values of power obtained by simulation.
3 citations
TL;DR: In this article, the asymptotic efficiency of simple group rank test is examined in the case of testing that the location parameter of a symmetric distribution is zero against the alternative it is positive.
Abstract: The asymptotic efficiency of a simple group rank test is examined in the case of testing that the location parameter of a symmetric distribution is zero against the alternative it is positive. Asymptotic optimality is shown subject to a condition on the density function. Efficiencies in relation to the sample mean and the sign test are found in several cases.