Statistical thinking: From Tukey to Vardi and beyond

Data miners (minors?) and neural networkers tend to eschew modelling, misled perhaps by misinterpretation of strongly expressed views of John Tukey. I discuss Vardi's views of these issues as well as other aspects of Vardi's work in emision tomography and in sampling bias.Comment: Published at http://dx.doi.org/10.1214/074921707000000210 in the IMS Lecture Notes Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

On the Time for Brownian Motion to Visit Every Point on a Circle

Consider a Wiener process $W$ on a circle of circumference $L$. We prove the rather surprising result that the Laplace transform of the distribution of the first time, $\theta_L$, when the Wiener process has visited every point of the circle can be solved in closed form using a continuous recurrence approach.Comment: 8 pages, 1 figur

Revisiting a Theorem of L.A. Shepp on Optimal Stopping

Using a bondholder who seeks to determine when to sell his bond as our motivating example, we revisit one of Larry Shepp's classical theorems on optimal stopping. We offer a novel proof of Theorem 1 from from \cite{Shepp}. Our approach is that of guessing the optimal control function and proving its optimality with martingales. Without martingale theory one could hardly prove our guess to be correct.Comment: 5 page

Connectedness of Certain Random Graphs

L. Dubins conjectured in 1984 that the graph on vertices {1, 2, 3, ...} where an edge is drawn between verticesi andj with probability pij=λ / max(i, j) independently for each pairi andj is a.s. connected for λ=1. S. Kalikow and B. Weiss proved that the graph is a.s. connected for any λ\u3e1. We prove Dubin’s conjecture and show that the graph is a.s. connected for anyλ\u3e1/4. We give a proof based on a recent combinatorial result that forλ ≦ 1/4 the graph is a.s. disconnected. This was already proved for λ \u3c 1/4 by Kalikow and Weiss. Thus λ= 1/4 is the critical value for connectedness, which is surprising since it was believed that the critical value is at λ=1

Radon-Nikodym Derivatives of Gaussian Measures

We give simple necessary and sufficient conditions on the mean and covariance for a Gaussian measure to be equivalent to Wiener measure. This was formerly an unsolved problem [26]. An unsolved problem is to obtain the Radom-Nikodym derivative dμ/dν where μ and ν are equivalent Gaussian measure [28]. We solve this problem for many cases of μ and ν, by writing dμ/dν in terms of Fredholm determinants and resolvents. The problem is thereby reduced to the calculation of these classical quantities, and explicit formulas can often be given. Our method uses Wiener measure μw as a catalyst; that is, we compute derivatives with respect to μw and then use the chain rule: dμ/dν = (dμ/dμw) / (dν/dμw). Wiener measure is singled out because it has a simple distinctive property--the Wiener process has a random Fourier-type expansion in the integrals of any complete orthonormal system. We show that any process equivalent to the Wiener process W can be realized by a linear transformation of W. This transformation necessarily involves stochastic integration and generalizes earlier nodulation transformations studied by Legal [21] and others [4], [27]. New variants of the Wiener process are introduced, both conditioned Wiener processes and free n-fold integrated Wiener processes. We given necessary and sufficient conditions for a Gaussian process to be equivalent to any one of the variants and also give the corresponding Radon-Niels (R-N) derivative. Last, some novel uses of R-N derivatives are given. We calculate explicitly: (i) the probability that W cross a slanted line in a finite time, (ii) the first passage probability for the process W (T + 1) − W(t), and (iii) a class of function space integrals. Using (iii) we prove a zero-one law for convergence of certain integrals on Wiener paths

First Passage Time for a Particular Gaussian Process

We find an explicit formula for the first passage probability, Qa(T|x) = Pr(S(t) \u3c a, 0 ≦ t ≦ T | S(0) = x), for all T \u3e 0, where S is the Gaussian process with mean zero and covariance ES(τ)S(t) = max (1-| t - τ|, 0). Previously, Qa(T | x) was known only for T ≦ 1. In particular for T = n an integer and - ∞ \u3c x \u3c a \u3c ∞, Qa(T | x) = 1⁄φ(x) ∫D . . . ∫ det φ(yi - yj+1 + a) dy2 . . . dyn+1, where the integral is a n-fold integral of y2, . . . , yn+1 over the region D given by D = {a - x \u3c y2 \u3c y1 \u3c . . . n+1} and the determinant is of size (n + 1)x(n + 1), 0 \u3c i, j ≦ n, with y0 ≡ 0, y1 ≡ a - x

A Model for Stock Price Fluctuations Based on Information

The author presents a new model for stock price fluctuations based on a concept of information. In contrast, the usual Black-Scholes-Merton-Samuelson (1965, 1973) model is based on the explicit assumption that information is uniformly held by everyone and plays no role in stock prices. The new model is based on the evident nonuniformity of information in the market and the evident time delay until new information becomes generally known. A second contribution of the paper is to present some problems with explicit solutions which are of value in obtaining insights. Several problems of mathematical interest are compared in order to better understand which optimal stopping problems have explicit solution