Thomas S. Ferguson

BAYESIAN DENSITY ESTIMATION BY MIXTURES OF NORMAL DISTRIBUTIONS

Publisher Summary This chapter discusses Bayesian density estimation by mixtures of normal distributions and discusses the estimation of an arbitrary density f(x) on the real line. This density is modeled as a mixture of a countable number of normal distributions. Using such mixtures, any distribution on the real line can be approximated to within any preassigned accuracy in the Levy metric and any density on the real line can be approximated similarly in the L1 norm. Thus, the problem can be considered nonparametric. Asymptotic theory for kernel estimators involves the problems of letting the window size tend to zero at some rate as the sample size tends to infinity.

An Inconsistent Maximum Likelihood Estimate

Abstract An example is given of a family of distributions on [— 1, 1] with a continuous one-dimensional parameterization that joins the triangular distribution (when Θ = 0) to the uniform (when Θ = 1), for which the maximum likelihood estimates exist and converge strongly to Θ = 1 as the sample size tends to infinity, whatever be the true value of the parameter. A modification that satisfies Cramers conditions is also given.

Game Theory and Decision Theory

This chapter discusses the Game Theory and the Decision Theory. The elements of decision theory are similar to those of the theory of games. Decision theory may be considered as the theory of a two-person game, in which nature takes the role of one of the players. The normal form of a zero-sum two-person game consists of three basic elements: (1) a nonempty set of possible states of nature, sometimes referred to as the parameter space, (2) a nonempty set of actions available to the statistician, and (3) a loss function. There are certain differences between game theory and decision theory that arise from the philosophical interpretation of the elements. The chapter discusses optimal decision rules and the fundamental problem of decision theory. It is a natural reaction to search for a best decision rule, a rule that has the smallest risk no matter what the true state of nature. However, the situations in which a best decision rule exists are rare and uninteresting.

Sequential classification on partially ordered sets

SEQUENTIAL CLASSIFICATION ON PARTIALLY ORDERED SETS Curtis Tatsuoka 1 and Thomas Ferguson 2 Department of Statistics, The George Washington University, Washington, DC 20052, USA Department of Statistics, UCLA, Los Angeles, CA 90095, USA Abstract A general theorem on asymptotically optimal sequential selection of experiments is pre- sented and applied to a Bayesian classification problem when the parameter space is a finite partially ordered set. The main results include establishing conditions under which the posterior probability of the true state converges to one almost surely, and determining optimal rates of convergence. Properties of various classes of experiment selection rules are explored. KEY WORDS: partially ordered set, cognitive diagnosis, group testing, sequential selection of experiment, optimal rates of convergence, Kullback-Leibler information. 1. Introduction: Background and Motivation. Partially ordered sets are natural models for many statistical applications. As an example of how partially ordered sets (posets) can be used in cognitively diagnostic analysis and computerized intelligent tutoring systems, consider the following example (e.g. see K. Tatsuoka (1995)). Suppose a student is to be tested on a certain subject domain for which there is a known finite set of knowledge states, denoted by S. It is of interest to determine the student’s knowledge state in S. Responses from sequentially selected test items (i.e. experiments) will be used to classify the person into one of the states. A natural model for S is to assume that certain states are at higher levels than others. Two states i and j in S may be related to each other in the following manner. If a student in state i has the knowledge to answer correctly all the test items that a student can who is in state j, we denote this by j ≤ i. It is thus natural to assume that S is a partially ordered set. Another example of the use of partially ordered sets in statistics with a rich body of literature, is group testing, originated by Dorfman (1943) (see also Ungar (1960), Sobel and Groll (1959) and (1966), Yao and Hwang (1990), and Gastwirth and Johnson (1994)). This is the problem of identifying all defectives in a set of finite objects by experiments that find for a given subset if there is at least one defective in the subset. Note that the classification states, consisting of all subsets of the objects, can also be viewed as partially ordered. In general, an experiment consists in observing a random variable or vector, X, whose distribution depends on the true unknown state, call it s ∈ S. We assume that for each experiment e and state s ∈ S, the corresponding class conditional response distribution of X has some density f(x|e, s). We assume that the prior distribution of the true state is known, and consider the Bayesian approach. The basic problem is to choose a sequence of experiments sequentially and a stopping rule to determine the true state as quickly as possible.

On sums of graph games with last player losing

The purpose of this paper is to find the general class of graph games with last player losing which may be solved by an analogue ofBoutons [1901] solution. Moreover, it can be shown that this class contains all subtraction games, as well asLaskers [1931] nim and several other games. Games such asKayles andDawsons [1935] game with last player losing are not treated by the method of this paper and are still unsolved.

Minimizing the expected rank with full information

The full-information secretary problem in which the objective is to minimize the expected rank is seen to have a value smaller than 7/3 for all n (the number of options). This can be achieved by a simple memoryless threshold rule. The asymptotically optimal value for the class of such rules is about 2.3266. For a large finite number of options, the optimal stopping rule depends on the whole sequence of observations and seems to be intractable. This raises the question whether the influence of the history of all observations may asymptotically fade. We have not solved this problem, but we show that the values for finite n are nondecreasing in n and exhibit a sequence of lower bounds that converges to the asymptotic value which is not smaller than 1.908. §

Maximum Likelihood Estimates of the Parameters of the Cauchy Distribution for Samples of Size 3 and 4

Abstract Expressions are given for the joint maximum likelihood estimates of the location and scale parameters of a Cauchy distribution based on samples of size 3 and 4.

Mate with bishop and knight in kriegspiel

Abstract It has long been an unsolved problem whether the king, bishop and knight can win against the king alone in the game of kriegspiel. In this paper, it is shown that in general (say, with king initially guarding both bishop and knight), the player with bishop and knight can win with probability one. The proof is constructive; a general procedure for winning is explicitly exhibited. In addition, it may be assumed that the player with king alone plays with full knowledge of the past moves of his opponent. The winning procedure involves randomization and so no upper bound can be placed on the number of moves required to mate. With king, bishop and knight initially on h8, g8 and h7, respectively, the expected number of moves required by the proposed winning strategy is less than 100. Whether or not there exists a strategy that guarantees mate within a fixed number of moves is still unknown.

Misère annihilation games

Abstract Graph games with an annihilation rule, as introduced by Conway, Fraenkel and Yesha, are studied under the misere play rule for progressively finite graphs that satisfy a condition on the reversibility of non-terminal Sprague-Grundy zeros to Sprague-Grundy ones. Two general theorems on the Sprague-Grundy zeros and ones are given, followed by two theorems characterizing the set of P-positions under certain additional conditions. Application is made to solving many subtraction games, and solutions to two games not covered by the general theory are presented indicating a direction for future research.

Selection by Committee

The many-player game of selling an asset, introduced by Sakaguchi and extended to monotone voting procedures by Yasuda, Nakagami and Kurano, is reviewed. Conditions for a unique equilibrium among stationary threshold strategies are given.