In many problems, especially when data are collected over a long period of time, hypothesis testing is done repeatedly as new data arrive. It is shown both for one particular problem and for a more general class of problems that if testing is performed each time on the total amount of data accumulated, the true simple null hypothesis will be rejected at least once as the number of tests approaches infinity. Furthermore, it is conjectured that the conclusion holds for most problems of interest in which the null hypothesis is simple.

In pattern recognition problems it has been noted that beyond a certain point the inclusion of additional parameters (that have been estimated) leads to higher probabilities of error. A simple problem has been formulated where the probability of error approaches zero as the dimensionality increases and all the parameters are known; on the other hand, the probability of error approaches one-half as the dimensionality increases and parameters are estimated.

Let W be an N-dimensional vector space and let the signal locus V be a K-dimensional topological hypersurface in W. The intrinsic dimensionality problem can be stated as follows. Given M randomly selected points (signals) vi, viϵ V, estimate K, which is the dimensionality of V and is called the intrinsic dimensionality of the points vi. A statistical method, which is developed from geometric considerations, is used to estimate the dimensionality. This ad hoc statistical method avoids the approximations and assumptions required by the maximum likelihood solution. The problem of estimating dimensionality in the presence of additive white noise is also considered. A pseudo, signal-to-noise ratio, which has meaning with respect to estimating the dimensionality of a noisy signal collection, is defined. A filtering method, based on this ratio, is used to estimate the dimensionality of a noisy signal collection. The accuracy of the method is demonstrated by estimating the dimensionality of a collection of pulsed signals which have four free parameters.

LetWbe anN-dimensional vector space andVbe aK- dimensional topological hypersurface inW. The intrinsic dimensionality problem can be stated as follows. GivenMrandomly selected points\nu_i, \nu_i \in V, estimateK, which is the dimensionality ofVand is called the intrinsic dimensionality of the points\nu_i. This problem has been attacked by several authors. If signals are represented by vectors in an abstract signal spaceW, rather than as time or frequency functions, the locus of signals that are generated by a hypothetical signal generator possessingKfree parameters is aK-dimensional hypersurfaceVin the signal space. The purpose of this paper is to identify some of the free parameters of the signals. LetHbe a one-parameter group that acts onW, HW = W. To say that a parameter associated with a groupHis a free parameter ofVmeans thatVis closed underH, i.e.,HV = V. To decide whetherHV = V, the estimated dimensionalities ofHVandVare compared. The method is illustrated by presenting several examples from the field of signal analysis. Finally, a slight modification is suggested so that parameters can he identified even when the vectors of interest do not represent time or frequency functions.

The mean and median detectors are compared at both small- and large-(ARE)-sample sizes for use with envelope-detected data. For the Rayleigh distribution, the mean is the better detector for all sample sizes. However, for the noncoherent contaminated-normal distribution, the mean is the better small-sample detector, whereas the median is the better large-sample detector. Hence, the ARE cannot always be used to deduce the small-sample behavior of detectors.