Ferguson, Thomas S. - Departments of Mathematics & Statistics, University of California at Los Angeles

• Inverting Dirichlet Tessellations Frederic Paik Schoenberg1
• Solutions to Exercises 7.3.2 through 7.3.5, and 7.3.7 through 7.3.11. E,(dN (TN )) = E,{E(dN (TN )|N, TN )}
• Solution to Exercises 7.7.12 through 7.7.15. 7.7.12. (a) Let Z1, Z2, . . . be i.i.d. with M(t) = EetZ
• Stat 200C, Spring 2010 Ferguson Solutions to Exercise Set 3.
• Large Sample Theory Exercises, Section 1, Modes of Convergence.
• GAME THEORY Thomas S. Ferguson
• On a Rao-Shanbhag Characterization of the Exponential/Geometric Distribution
• BEST-CHOICE PROBLEMS WITH DEPENDENT CRITERIA Thomas S. Ferguson
• Large Sample Theory Exercises, Section 21, Asymptotic Normality of Posterior Distributions.
• Solutions to the Exercises of Section 4.8. 4.8.1. We have g(F ) = F (-1
• MULTIPLE BUYING OR SELLING WITH VECTOR OFFERS
• Maximizing the duration of owning a relatively best object1 T. S. Ferguson, UCLA
• A Problem of Minimax Estimation with Directional Information
• Solution to Exercises 7.4.6 and 7.4.7. 7.4.6.(a) The problem is invariant under scale changes, gb(x1, . . ., xj) = (bx1, . . ., bxj) for b > 0,
• Stat 200C, Spring 2010 Ferguson Solutions to Exercise Set 4.
• Solutions to the Exercises of Section 1.7. 1.7.1(a) The minimax point is the intersection of the line joining (-2, 3) to (-3/4, -9/4) and the line
• Abdel-Hamid, A. R., 2.7 Additive Damage Model, 5.24
• Stat 200C, Spring 2010 Ferguson Solutions to Exercise Set 10.
• Solutions to Exercises 5.10.1 through 5.10.6. 5.10.1. Let x Er , Er c0 0, and > 0.
• Chapter 3. THE EXISTENCE OF OPTIMAL STOPPING RULES. Consider the general stopping rule problem of Chapter 1 with observations X1, X2, . . .
• Solutions of Exercises 5.5.1 to 5.5.3. 5.5.1. (a) If X C(0, 1), and U = 2X/(1+X2
• Solutions to the Exercises of Section 4.2. 4.2.1. Reflexivity: The identity transformation, e(), is in G and e() = .
• Solutions to Exercises 4.6.1 to 4.6.7, and 4.6.9 to 4.6.12. 4.6.1. Sufficiency reduces the problem to choosing rules based on T = min{X1, . . ., Xn}. This has
• Solutions to the Exercises of Section 5.7. 5.7.1. If X1, . . ., XN are i.i.d. from the density f(x|) = ex
• GAMES WITH FINITE RESOURCES Games with finite resources are two-person zero-sum multistage games defined by Gale
• Gambler's ruin in three dimensions Three gamblers with initial fortunes, a, b, and c, play a sequence of fair games until
• 7.2 The Continuous Case. The simplest extension of the minimax theorem to the continuous case is to assume that X and Y are compact subsets of Euclidean spaces, and
• On a Kreigspiel Problem of Lloyd Shapley. Thomas Ferguson, Mathematics Department, UCLA
• Appendix 1, Part 2. Subjective Probability. We have assumed that the "rational decision maker" knows
• Appendix 1: Utility Theory Much of the theory presented is based on utility theory at a fundamental level. This
• Solutions to Exercises 7.2.2 to 7.2.4, and 7.2.6 to 7.2.9. 7.2.2. At stage 0, the Bayes expected loss is
• GAME THEORY Thomas S. Ferguson
• Solutions to the Exercises of Section 3.5. 3.5.1. (a) f(x|) = e-
• Large Sample Theory Exercises, Section 18, Asymptotic Normality of the Maximum Likelihood
• The Kelly Betting System for Favorable Games. Thomas Ferguson, Statistics Department, UCLA
• SOME TIME-INVARIANT STOPPING RULE PROBLEMS Thomas S. Ferguson
• Chapter 5. MONOTONE STOPPING RULE PROBLEMS. The ease with which the various problems of Chapter 4 were solved may be mislead-
• Solutions to Exercises 5.2.2 through 5.2.11. 5.2.2. To show that U(, + 1) has monotone likelihood ratio, take 1 < 2 and consider two cases.
• Solutions to the Exercises of Section 2.11. 2.11.1. Proof. Let be an arbitrary positive number. Since r(n, n) C , we can find an integer n
• Solutions to the Exercises of Section 3.6. 3.6.1. The distribution function of T = max1in Xi is
• Solutions to the Exercises of Section 1.8. 1.8.1. E(Z -b)2
• MATE WITH BISHOP AND KNIGHT IN KRIEGSPIEL Thomas S. Ferguson
• Second Midterm Examination Mathematics 167, Game Theory
• On the Borel and von Neumann Poker Models Chris Ferguson, Bright Trading, Westwood, California
• MATE WITH THE TWO BISHOPS IN KRIEGSPIEL T. S. Ferguson, 03/08/95
• Asymptotic Joint Distribution of Sample Mean and a Sample Quantile Thomas S. Ferguson
• Half-Prophets and Robbins' Problem of Minimizing the Expected Rank
• GLEASON'S GAME T. S. Ferguson and L. S. Shapley
• Solutions to Exercises 5.9.3 through 5.9.9. = (Xij --i -j)2
• House-Hunting Without Second Moments Thomas S. Ferguson, University of California, Los Angeles
• The Sum-the-Odds Theorem with Application to a Stopping Game of Sakaguchi
• THE ENDGAME IN POKER Chris Ferguson, Full Tilt Poker
• Thomas S. Ferguson and Christian Genest University of California at Los Angeles and Universite Laval
• HIGH RISK AND COMPETITIVE INVESTMENT MODELS F. THOMAS BRUSS and THOMAS S. FERGUSON
• The Canadian Journal of Statistics Vol. 28, No. ?, 2000, Pages ???-???
• LINEAR PROGRAMMING A Concise Introduction
• A Class of Symmetric Bivariate Uniform Distributions Thomas S. Ferguson, 07/08/94
• Minimax Estimation of a Variance Thomas S. Ferguson
• Choice of Weapons for the Truel Thomas S. Ferguson, UCLA
• A GENERAL INVESTMENT MODEL Thomas S. Ferguson and C. Zachary Gilstein
• OPTIMAL INVESTMENT POLICIES FOR THE HORSE RACE MODEL Thomas S. Ferguson and C. Zachary Gilstein
• GAME THEORY Thomas S. Ferguson
• GAME THEORY Thomas S. Ferguson
• GAME THEORY Thomas S. Ferguson
• Stat 200C, Spring 2010 Ferguson Solutions to Exercise Set 2.
• Stat 200C, Spring 2010 Ferguson Solutions to Exercise Set 5.
• Stat 200C, Spring 2010 Ferguson Solutions to Exercise Set 6.
• Midterm Examination Statistics 200C
• 4.7 Approximating the Solution: Fictitious Play. The method of fictitious play may be used to approximate the value and optimal
• 7. Infinite Games. In this Chapter, we treat infinite two-person, zero-sum games. These are games
• 7.4 Solving Games. There are many interesting games that are more complex and that require a good deal of thought and ingenuity to find the solution. There is one tool
• Large Sample Theory Exercises, Section 2, Partial Converses to Theorem 1.
• Large Sample Theory Exercises, Section 3, Convergence in Law.
• Large Sample Theory Exercises, Section 6, Slutsky Theorems.
• Large Sample Theory Exercises, Section 8, The Sample Correlation Coefficient.
• Large Sample Theory Exercises, Section 10, Asymptotic Power of the Pearson's Chi-Square Test.
• Large Sample Theory Exercises, Section 11, Stationary m-Dependent Sequences.
• Large Sample Theory Exercises, Section 13, Asymptotic Distribution of Sample Quantiles.
• Large Sample Theory Exercises, Section 14, Asymptotic Theory of Extreme Order Statistics.
• Large Sample Theory Exercises, Section 17, Strong Consistency of Maximum Likelihood Esti-
• Large Sample Theory Exercises, Section 20, Asymptotic Efficiency.
• Large Sample Theory Exercises, Section 22, Asymptotic Distribution of the Likelihood Ratio
• Large Sample Theory Exercises, Section 24, General Chi-Square Tests.
• A COURSE IN LARGE SAMPLE THEORY by Thomas S. Ferguson
• Solutions to the Exercises of Section 1.4. 1.4.1. Proof. (i) From linearity of , either p p or p p. Thus, p p and p p.
• Solutions to the Exercises of Section 1.6. 1.6.1. If 0 is minimax, then by definition, sup R(, 0) = infD sup R(, ). This implies that
• Solutions to the Exercises of Section 2.1. 2.1.1. Proof. Let C be a complete class of decision rules, and let A denote the class of admissible
• Solutions to the Exercises of Section 2.5. 2.5.1. Let S be convex set in Ek , and let S denote its closure. Let x S and y S, and let 0 1.
• Solutions to the Exercises of Section 2.7. 2.7.1. Let S1 be a convex set in Ek , let A = S1 , let = {1, . . ., k} and consider the game (, A, L)
• Solutions to the Exercises of Section 3.2. 3.2.1. The (i, j)-element of YA is h l Yihalj . Its expectation is h l E(Yih)alj , which is the (i, j)
• Solutions to the Exercises of Section 3.3. 3.3.1. (a) Independent Xj B(nj, p) for j = 1, . . ., n.
• Solutions to the Exercises of Section 3.4. 3.4.1. We are to show E(median(Xi)|T) = X when X1, . . ., Xn is a sample from N(, 1) and T =
• Solutions to the Exercises of Section 4.7. 4.7.1. If the coin comes up heads, then d2 is minimax. It guarantees the statistician a loss of (at most)
• Solutions to the Exercises of Section 5.1. 5.1.1. Let us use the notation, 0() = R(0, ) = E0 ((X)) and 1() = R(1, ) = 1 -E1 ((X)).
• Solutions to the Exercises of Section 5.4. 5.4.1. Suppose is an unbiased test of size that is admissible within the class of unbiased tests.
• Solutions to Exercises 5.6.1 through 5.6.8, and 5.6.12. 5.6.1. This problem is invariant under a change of location, gc(x1, x2) = (x1 +c, x2 +c), and a maximal
• Solution to Exercises 6.3.3 through 6.3.5. 6.3.3. (a) The joint density of Y1, . . ., Yk-1 under Hi for i = 0 is given by (6.22). Let Z1 = |Y1|, Z2 =
• OPTIMAL STOPPING AND APPLICATIONS Chapter 1. STOPPING RULE PROBLEMS
• A. R. Abdel-Hamid, J. A. Bather, and G. B. Trustrum (1982) "The secretary problem with an unknown number of candidates", J. Appl. Prob. 19, 619-630.
• First Midterm Examination Mathematics 167, Game Theory
• Large Sample Theory Exercises, Section 12, Some Rank Statistics.
• Chapter 4. APPLICATIONS. MARKOV MODELS.
• Exercise 6. The Wallet Game. Two players each put a random amount with mean one into their wallets. The player whose wallet contains the smaller amount wins
• LAST ROUND BETTING THOMAS S. FERGUSON, University of California, Los Angeles
• Solutions to the Exercises of Section 4.1. 4.1.1. Consider a decision problem with parameter space , action space A, and observations X with
• Large Sample Theory Exercises, Section 9, Pearson's Chi-Square.
• Chapter 7. BANDIT PROBLEMS. Bandit problems are problems in the area of sequential selection of experiments, and
• A Poisson Fishing Model1 Thomas S. Ferguson, 08/30/942
• U-STATISTICS Notes for Statistics 200C, Spring 2005
• 7.3 Convex Games. If in Theorem 7.2, we add the assumption that the payoff function A(x, y) is convex in y for all x or concave in x for all y, then we can say a lot more
• Chapter 2. FINITE HORIZON PROBLEMS. A stopping rule problem has a finite horizon if there is a known upper bound on
• Stat 200C Ferguson Statistical Theory Spring 2008
• Solutions to the Exercises of Section 3.7. 3.7.1. The spelling of Dvoretzky should be corrected (twice). The density of X is
• NIM, TRIM and RIM James A. Flanigan
• Large Sample Theory Exercises, Section 4, Laws of Large Numbers.
• Solutions to the Exercises of Section 2.9. 2.9.1. Let be any randomized decision rule. Then is a distribution on A = {1, 2, . . .}. Let > 0
• Large Sample Theory Exercises, Section 5, Central Limit Theorems.
• Stat 200C, Spring 2010 Ferguson Solutions to Exercise Set 9.
• Games with Finite Resources Thomas S. Ferguson1
• Chapter 6. MAXIMIZING THE RATE OF RETURN. In stopping rule problems that are repeated in time, it is often appropriate to maximize
• I Stopping games 1 1 Selection by Committee
• Solutions to the Exercises of Chapter 1. 1. Let Xj = (X1j , X2j) denote the jth
• Stat 200C, Spring 2010 Ferguson Solutions to Exercise Set 8.
• Some Chip Transfer Games Thomas S. Ferguson
• Large Sample Theory Exercises, Section 7, Functions of the Sample Moments.
• Solutions to the Exercises of Section 2.8. 2.8.1. Let z = (z1, . . ., zk+1)T
• Large Sample Theory Exercises, Section 19, The Cramer-Rao Lower Bound.
• Solutions to the Exercises of Section 2.10. 2.10.1. (a) r(, (x, y)) = R(1/3, (x, y)) + (1 -)R(2/3, (x, y)),
• Solutions to the Exercises of Section 4.3. 4.3.1. The parameter space is = {(, j) : 0 1, j = 1, . . ., n}, the action space is A = [0, 1], and
• A Note on Dawson's Chess Thomas S. Ferguson, UCLA
• Solutions to Exercises 4.5.1 through 4.5.7, and 4.5.10. 4.5.1. The median, , of the Cauchy distribution is a location parameter, and the loss, L(, a) =
• Solutions to Exercises 5.8.2 through 5.8.4 and 5.8.7. 5.8.2. Let (x) denote the density of the standard normal distribution and let 1 < 2 . The likelihood
• ON THE INSPECTION GAME Thomas S. Ferguson and Costis Melolidakis
• Solutions to Exercises 6.1.1 through 6.1.3. 6.1.1. f(x|) = (5
• Solutions to the Exercises of Section 3.1. 3.1.1. (a) The joint distribution of X and Y is a mixed discrete and continuous density,
• UNIFORM(0,1) TWO-PERSON POKER MODELS Chris Ferguson, TiltWare, Los Angeles
• Int J Game Theory (2002) 31:223228 Notes on a stochastic game with information structure
• Stat 200C, Spring 2010 Ferguson Solutions to Exercise Set 7.
• Minimizing the Expected Rank with Full Information F. Thomas Bruss and Thomas S. Ferguson
• Solutions to the Exercises of Section 6.2. 6.2.1. First we show the hint: for z > 0, 1 -z + log z 0 (note the inequality is backward in the text).