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

Permutation graphs, fast forward permutations, and sampling the cycle structure of a permutation

A permutation P on {1,..,N} is a_fast_forward_permutation_ if for each m the computational complexity of evaluating P^m(x)$ is small independently of m and x. Naor and Reingold constructed fast forward pseudorandom cycluses and involutions. By studying the evolution of permutation graphs, we prove that the number of queries needed to distinguish a random cyclus from a random permutation on {1,..,N} is Theta(N) if one does not use queries of the form P^m(x), but is only Theta(1) if one is allowed to make such queries. We construct fast forward permutations which are indistinguishable from random permutations even when queries of the form P^m(x) are allowed. This is done by introducing an efficient method to sample the cycle structure of a random permutation, which in turn solves an open problem of Naor and Reingold.Comment: Corrected a small erro

ROC and the bounds on tail probabilities via theorems of Dubins and F. Riesz

For independent $X$ and $Y$ in the inequality $P(X\leq Y+\mu)$, we give sharp lower bounds for unimodal distributions having finite variance, and sharp upper bounds assuming symmetric densities bounded by a finite constant. The lower bounds depend on a result of Dubins about extreme points and the upper bounds depend on a symmetric rearrangement theorem of F. Riesz. The inequality was motivated by medical imaging: find bounds on the area under the Receiver Operating Characteristic curve (ROC).Comment: Published in at http://dx.doi.org/10.1214/08-AAP536 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Chain Plot: A Tool for Exploiting Bivariate Temporal Structures

In this paper we present a graphical tool useful for visualizing the cyclic behaviour of bivariate time series. We investigate its properties and link it to the asymmetry of the two variables concerned. We also suggest adding approximate confidence bounds to the points on the plot and investigate the effect of lagging to the chain plot. We conclude our paper by some standard Fourier analysis, relating and comparing this to the chain plot

Drift rate control of a Brownian processing system

A system manager dynamically controls a diffusion process Z that lives in a finite interval [0,b]. Control takes the form of a negative drift rate \theta that is chosen from a fixed set A of available values. The controlled process evolves according to the differential relationship dZ=dX-\theta(Z) dt+dL-dU, where X is a (0,\sigma) Brownian motion, and L and U are increasing processes that enforce a lower reflecting barrier at Z=0 and an upper reflecting barrier at Z=b, respectively. The cumulative cost process increases according to the differential relationship d\xi =c(\theta(Z)) dt+p dU, where c(\cdot) is a nondecreasing cost of control and p>0 is a penalty rate associated with displacement at the upper boundary. The objective is to minimize long-run average cost. This problem is solved explicitly, which allows one to also solve the following, essentially equivalent formulation: minimize the long-run average cost of control subject to an upper bound constraint on the average rate at which U increases. The two special problem features that allow an explicit solution are the use of a long-run average cost criterion, as opposed to a discounted cost criterion, and the lack of state-related costs other than boundary displacement penalties. The application of this theory to power control in wireless communication is discussed.Comment: Published at http://dx.doi.org/10.1214/105051604000000855 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org