Showing papers on "Symmetric probability distribution published in 1971"

Robust Estimates of Location: Symmetry and Asymmetric Contamination

Abstract: The problem of finding location estimators which are "robust" against deviations from normality has received increasing attention in the last several years. See, for example, Tukey (1960), Huber (1968), and papers cited therein. In the theoretical work done on the estimation of a location parameter, the underlying distribution is usually assumed to be symmetric, and the estimand is taken to be the center of symmetry, a natural quantity to estimate in this situation. Since the finite sample size properties of many proposed estimators are difficult to study analytically, most research has focussed on their more easily ascertainable asymptotic properties, which, it is hoped, will provide useful approximations to the finite sample size case. Most of the estimators commonly studied are, under suitable regularity conditions, asymptotically normal about the center of symmetry, with asymptotic variance depending on the underlying distribution. We thus have a simple criterion, the asymptotic variance, for comparing the performance of different estimators for a given underlying distribution, and of a given estimator for different underlying distributions. Huber (1964) has formulated and solved some minimax problems, in which the estimators are judged by their asymptotic variance. In Section 2 we define and state the asymptotic variances which have been found for the three most commonly studied types of location estimators. In Section 3 we demonstrate some relationships among the three types of estimators, and in Section 4 we show that Huber's minimax result applies to all three types. Then, in Section 5 we consider an aspect of the more general estimation problem in which the distributions are not assumed symmetric. A model of asymmetric contamination of a symmetric distribution is formulated, in which the amount of asymmetry tends to zero as the sample size increases. The estimators here are thought of as estimating the center of the symmetric component of the distribution. The maximum likelihood type estimators are shown to be asymptotically normal under this model, but with a bias that tends to zero as the sample size increases. The estimators may be judged by their asymptotic mean squared error, a concept which is made meaningful by the model. We conclude in Section 6 with a minimax result analogous to Huber's, for which we allow both symmetric and asymmetric contamination of a given distribution and judge the estimators by their asymptotic mean squared error.

Some Flexible Estimates of Location

Abstract: This paper considers two procedures for estimating the center of a symmetric distribution, which use the observations themselves to choose the form of the estimator. Both procedures begin with a family of possible estimators. We use the observations to estimate the asymptotic variance of each member of the family of estimators. We then choose the estimator in the family with smallest estimated asymptotic variance and use the value given by that estimator as the location estimate. These procedures are shown to be asymptotically as good as knowing beforehand which estimator in the family is best for the given distribution, and using that estimator.

Higher moment equations and the distribution function of the solar-wind plasma

Abstract: Study of the higher-moment equations for a collisionless fully ionized plasma. For a collisionless, heat-conducting plasma, the distribution function f is cylindrically symmetric about the direction of the magnetic field. It is shown that under a certain assumption the fourth moments of f can be expressed as simple functions of lower moments. Thus no higher-moment terms appear in the third-moment equations. The two third-moment equations, which are obtained in a simple form, join other lower-moment equations to form a closed set of moment equations. The new equations can be used to study the thermal anisotropy and the heat flux of the solar-wind proton. A special case of the cylindrically symmetric distribution function f is found to resemble the proton distribution function reconstructed from solar-wind data, and this resemblance justifies the assumption needed for decoupling the moment equations.

A stochastic model for the occurrence of main-sequence earthquakes

Abstract: If a long earthquake sequence is considered to be a stationary stochastic process, the stored elastic energy of deformation can be shown to be an independent variable in the usual ‘backward’ equation. Three unknown probability functions are introduced: the probability that the stored energy of deformation is at a certain level; the probability that, if this energy is at a given level, an earthquake will occur; and the transition probability that, if the earthquake occurs, the final energy state will be at a certain level. It is assumed that the frequency-energy distribution is known. The equations can be solved, if the transition probability is assumed to be known; and they have been solved for the model in which the transition probability is a function of the energy released in the shock but is not otherwise dependent on the final energy state. In this case, the results can be used to describe the earthquake history for some time after a great shock, and possibly for times just before a great shock. The results have some features of inconsistency with observations.

On the “zero-two” law

Abstract: Results of Ornstein-Sucheston, are extended to non-separable measure spaces and operators that are not induced by a transition probability.

Probability Distribution of Number of Networks in Topologically Random Network Patterns

Abstract: The probability distribution of the number of channel networks for Werner's model of topologically random network patterns is shown to obey a shifted negative or inverse hypergeometric probability law and, when the number of exterior links is large, to be approximated by a shifted negative binomial probability law.

Optimum sequential search with discrete locations and random acceptance errors

Abstract: : Much work has been done in search theory. However, very little effort has occurred where an object's presence at a location can be accepted when no object is present there. The case analyzed is of this type. The number of locations is finite, a single object is stationary at one location, and only one location is observed each step of the search. The object's location has a known prior probability distribution. Also known are the conditional probability of acceptance given the object's absence (small) and the conditional probability of rejection given the object's presence (not too large); these probabilities remain fixed for all searching and locations. The optimum sequential search policy specifies that the next location observed is one with the largest posterior probability of the object's presence (evaluated after each step from Bayes Rule) and that the object is at the first location where acceptance occurs. Placement at the first acceptance seems appropriate when the conditional probability of acceptance given the object's absence is sufficiently small. The policy is optimum in that, for any number of steps, it minimizes the probability of no acceptances and, simultaneously, maximizes the probability that an acceptance occurs and the object is accurately located. Search always terminates (with probability one). Optimum truncated sequential policies are also considered. Methods are given for evaluating some pertinent properties and for investigating the possibility that no object occurs at any location. (Author)

Some Upper Bounds on Error Probability for Multiclass Pattern Recognition

Abstract: An upper bound on the probability of error for the general pattern recognition problem is obtained as a functional of the pairwise Kolmogorov variational distances. Evaluation of the bound requires knowledge of a priori probabilities and of the class-conditional probability density functions. A tighter bound is obtained for the case of equal a priori probabilities, and a further bound is obtained that is independent of the a priori probabilities.

A Necessary Condition on the Infinite Divisibility of Probability Distributions

Abstract: For a probability distribution function on the real line, a necessary condition on its infinite divisibility is given which deals with its one-sided asymptotic behavior; the proof is based on properties of characteristic functions which are analytic in the upper (lower) half plane.

Asymptotic Distribution of the Sample Size for a Sequential Probability Ratio Test

Abstract: The derivation is presented of the asymptotic distribution of the number N of observations required to terminate the classical sequential probability ratio test (SPRT) with bounds a, b (a > 0 > b) of a simple parametric hypothesis H: θ = θ1 against a simple parametric alternative K:θ=θ1+Δ(Δ>0) when Δ tends to zero. It is shown that if the common probability density function (p.d.f.) of the basic random variables belongs to the Darmois-Koopman exponential class and if the true value of θ0 of θ is different from θ1 then as Δ→0, NΔ tends in probability to a constant A > 0 which is a specified multiple of a if θ0 > θ1 and of - b if θ0 0 which is finite and depends on a if θ0 < θ1 and on − b if θ0 < θ1 such that tends to N(0, 1) as Δ→0. Finally, a few numerical examples are worked to illustrate the magnitudes of A and B.

A Counterexample on Translation Invariant Estimators

Abstract: It seems to be generally known that the proof of the continuity part of Theorem 1 in Hodges' and Lehmann's paper (1963) is incorrect. The fact that the theorem is incorrect is--perhaps--not so well known. We show this by constructing independent real random variables $X_1,\cdots, X_n$, each having the same non-atomic symmetric distribution, and an odd translation invariant estimator $h(X_1,\cdots, X_n)$ such that $P(h(X_1,\cdots, X_n) = 0) > 0. h$ may be chosen symmetric provided $n \geqq 3$.