Given a parametric statistical model, an improper prior distribution can often be used to induce a proper posterior distribution (an inference). This inference can then be used to solve decision problems once an action space and loss have been specified. One way to evaluate the inference is to ask for which estimation problems does the above formal Bayes method produce admissible estimators. The relationship of this problem to the recurrence of an associated symmetric Markov chain is reviewed.

In this review of estimation problems in restricted parameter spaces, we focus through a series of illustrations on a number of methods that have proven to be successful. These methods relate to the decision-theoretic aspects of admissibility and minimaxity, as well as to the determination of dominating estimators of inadmissible procedures obtained for instance from the criteria of unbiasedness, maximum likelihood, or minimum risk equivariance. Finally, we accompany the presentation of these methods with various related historical developments.

We generalize a set of axioms introduced by Rubin, A weak system of axioms for “rational” behavior and the nonseparability of utility from prior, Statistics & Decisions, to the case of partial preference. That is, we consider cases in which not all uncertain acts are comparable to each other. We demonstrate some relations between these axioms and a decision theory based on sets of probability/utility pairs. We illustrate by example how comparisons solely between pairs of acts is not sufficient to distinguish between decision makers who base their choices on distinct sets of probability/utility pairs.

For two integers chosen independently at random from {1, 2, . . . , x}, we give expansions for the distribution and the moments of their greatest common divisor and the least common multiple, with explicit error rates. The expansion involves Riemann’s zeta function. Application to a statistical question is briefly discussed.

Alzaid et al., An application of the Perron-Frobenius theorem to a damage model problem, and Rao et al., Damage models: A Martin boundary connection. Basu Memorial Volume, have shown that several of the results on damage models have links with certain results on nonnegative matrices. In the present article, we deal with integral equations met in damage model studies via specialized versions of de Finetti’s theorem and extend further the theorems of Rao and Rubin, On a characterization of the Poisson distribution, and Shanbhag , An extension of the Rao-Rubin characterization of the Poisson distribution, on damage models.

We present a history of the development of the theory of Stochastic Integration, starting from its roots with Brownian motion, up to the introduction of semimartingales and the independence of the theory from an underlying Markov process framework. We show how the development has influenced and in turn been influenced by the development of Mathematical Finance Theory. The calendar period is from 1880 to 1970.

We consider non-linear filtering problem with Gaussian martingales as a noise process, and obtain iterative equations for the optimal filter. We apply that result in the case of fractional Browian motion noise process and derive Kalman type equations in the linear case.

Self-similar stochastic processes are used for stochastic modeling whenever it is expected that long range dependence may be present in the phenomenon under consideration. After discussing some basic concepts of self-similar processes and fractional Brownian motion, we review some recent work on parametric and nonparametric inference for estimation of parameters for linear systems of stochastic differential equations driven by a fractional Brownian motion.

This paper shows how the invariance of the arc-sine distribution on (0, 1) under a family of rational maps is related on the one hand to various integral identities with probabilistic interpretations involving random variables derived from Brownian motion with arc-sine, Gaussian, Cauchy and other distributions, and on the other hand to results in the analytic theory of iterated rational maps.

A short variation of the original proof of Dubins and Schwarz of their result, that all continuous martingales can be time changed to Brownian motion, is given.

A sequence of independent random variables $\{X_1,X_2,\ldots\}$ is called a $B-$harmonic Bernoulli sequence if $P(X_i=1)=1-P(X_i=0) = 1/(i+B)\ i=1,2,\ldots$, with $B\ge 0$. For $k\ge 1$, the count variable $Z_k$ is the number of occurrences of the $k$-string $(1,\protect\underbrace{0,\ldots,0}_{k-1},1)$\vadjust{\vspace*{-2pt}} in the Bernoulli sequence\ldots\$. This paper gives the joint distribution $P_B$ of the count vector ${\bf Z} = (Z_1,Z_2,\ldots)$ of strings of all lengths in a $B-$harmonic Bernoulli sequence. This distribution can be described as follows. There is random variable $V$ with a Beta$(B,1)$ distribution, and given $V=v$, the conditional distribution of ${\bf Z}$ is that of independent Poissons with intensities $(1 -v),\ (1 - v^2)/2,\ (1-v^3)/3, \ldots$.

Around 1996, Persi Diaconis stated and proved that when $B=0$, the distribution of $Z_1$ is Poisson with intensity $1$. Emery gave an alternative proof a few months later. For the case $B=0$, it was also recognized that $Z_1,Z_2,\ldots,Z_n$ are independent Poissons with intensities $1, \frac{1}{2},\ldots, \frac{1}{n}$. Proofs up until this time made use of hard combinational techniques. A few years later, Joffe et al, obtained the marginal distribution of $Z_1$ as a Beta-Poisson mixture when $B\geq 0$. Their proof recognizes an underlying inhomogeneous Markov chain and uses moment generating functions.

In this note, we give a compact expression for the joint factorial moment of\break $(Z_1,\ldots,Z_N)$ which leads to the joint distribution given above. One might feel that if $Z_1$ is large, it will exhaust the number of $1$'s in the Bernoulli sequence $(X_1,X_2,\ldots)$ and this in turn would favor smaller values for $Z_2$ and introduce some negative dependence. We show that, on the contrary, the joint distribution of ${\bf Z}$ is positively associated or possesses the FKG property.

We address a more general version of a classic question in probability theory. Suppose $\bf {X \sim N_p({\mu},\Sigma )}$. What functions of $\bf {X}$ also have the $N_p({\mu},\Sigma )$ distribution? For $p = 1$, we give a general result on functions that cannot have this special property. On the other hand, for the $p = 2,3$ cases, we give a family of new nonlinear and non-analytic functions with this property by using the Chebyshev polynomials of the first, second and the third kind. As a consequence, a family of rational functions of a Cauchy-distributed variable are seen to be also Cauchy distributed. Also, with three i.i.d. $N(0,1)$ variables, we provide a family of functions of them each of which is distributed as the symmetric stable law with exponent $\frac{1}{2}$. The article starts with a result with astronomical origin on the reciprocal of the square root of an infinite sum of nonlinear functions of normal variables being also normally distributed; this result, aside from its astronomical interest, illustrates the complexity of functions of normal variables that can also be normally distributed.

We characterize the sets where an n-dimensional, infinitely differentiable characteristic function can have its real part zero, positive, and negative, and where it can have its imaginary part zero, positive, and negative.

Using an identity of Stein, Approximate computation of expectations, this article gives an exact expression for the characteristic function of Pearson type IV distributions in terms of confluent hypergeometric functions.

We present a general proposal for testing for goodness of fit, based on resampling and subsampling methods, and illustrate it with graphical and analytical tests for the problems of testing for univariate or multivariate normality. The proposal shows promising, and in some cases dramatic, success in detecting nonnormality. Compared to common competitors, such as a Q-Q plot or a likelihood ratio test against a specified alternative, our proposal seems to be the most useful when the sample size is small, such as 10 or 12, or even very small, such as 6! We also show how our proposal provides tangible information about the nature of the true cdf from which one is sampling. Thus, our proposal also has data analytic value. Although only the normality problem is addressed here, the scope of application of the general proposal should be much broader.

The main inequality (Theorem 1) here involves the Hellinger distance of a statistical model of an observation $X$, which imposes bounds on the mean of any estimator in terms of its variance. We use this inequality to explain some of the bias-variance trade-off phenomena studied in Doss and Sethuraman (1989) and Liu and Brown (1993). We provide some quantified results about how the reduction of bias would increase the variance of an estimator.

There are many applications in which one seeks to combine multiple estimators of the same parameter. If the constituent estimators are unbiased, then the fixed linear combination which is minimum variance unbiased is well-known, and may be written in terms of the covariance matrix of the constituent estimators. In general, the covariance matrix is unknown, and one computes a composite estimate of the unknown parameter with the covariance matrix replaced by its maximum likelihood estimator. The efficiency of this composite estimator relative to the constituent estimators has been investigated in the special case for which the constituent estimators are uncorrelated. For the general case in which the estimators are normally distributed and correlated, we give an explicit expression relating the variance of the composite estimator computed using the covariance matrix, and the variance of the composite estimator computed using the maximum likelihood estimate of the covariance matrix. This result suggests that the latter composite estimator may be useful in applications in which only a moderate sample size is available. Details of one such application are presented: combining estimates of agricultural yield obtained from multiple surveys into a single yield prediction.

When estimating the mean of a normal distribution with squared error loss and a cost for each observation, the optimal (fixed) sample size depends on the variance $\sigma^2$. A two-stage sequential procedure is to first conduct a pilot study from which $\sigma^2$ can be estimated, and then estimate the desired sample size. Here an asymptotic formula for the initial sample size in a two-stage sequential estimation procedure is derived-asymptotic as the cost of a single observation becomes small compared to the loss from estimation error. The experimenter must specify only the sample size, $n_0$ say, that would be used in a fixed sample size experiment; the initial sample size of the two-stage procedure is then the least integer greater than or equal to $\sqrt{n_0/2}$. The resulting procedure is shown to minimize the maximum Bayes regret, where the maximum is taken over prior distributions that are consistent with the initial guess $n_0$; and the minimax solution is shown to provide an asymptotic lower bound for optimal Bayesian choices for a wide class of prior distributions.

With applications to cluster analysis in mind, we suggest new approaches to constructing tree diagrams that describe associations among points in a scatterplot. Our most basic tree diagram results in two data points being associated with one another if and only if their respective curves of steepest ascent up the density or intensity surface lead toward the same mode. The representation, in the sample space, of the set of steepest ascent curves corresponding to the data, is called the gradient tree. It has a regular, octopuslike structure, and is consistently estimated by its analogue computed from a nonparametric estimator which gives consistent estimation of both the density surface and its derivatives. We also suggest ‘forests’, in which data are linked by line segments which represent good approximations to portions of the population gradient tree. A forest is closely related to a minimum spanning tree, or MST, defined as the graph of minimum total length connecting all sample points. However, forests use a larger bandwidth for constructing the density-surface estimate than is implicit in the MST, with the result that they are substantially more orderly and are more readily interpreted. The effective bandwidth for the MST is so small that even the corresponding density-surface estimate, let alone its derivatives, is inconsistent. As a result, relationships that are suggested by the MST can change considerably if relatively small quantities of data are added or removed. Our trees and forests do not suffer from this problem. They are related to the concept of gradient traces, introduced by Wegman, Carr and Luo (1993) and Wegman and Carr (1993) for purposes quite different from our own.

With very large sample sizes the conventional calculations for tests of the equality of two probabilities can lead to very small P-values. In those cases, the large deviation effects make inappropriate the asymptotic normality approximations on which those calculations are based. While reasonable interpretations of the data would tend to reject the hypothesis in those cases, it is desireable to have conservative estimates which don’t underestimate the P-value. The calculation of such estimates is presented here.

Consider an imaging situation with extremely high noise levels, hidden in the noise there may or may not be a signal; the signal--when present--is so faint that it cannot be reliably detected from a single frame of imagery. Suppose now multiple frames of imagery are available. Within each frame, there is only one pixel possibly containing a signal while all other pixels contain purely Gaussian noise; in addition, the position of the signal moves around randomly from frame to frame. Our goal is to study how to reliably detect the existence of the signal by combining all different frames together, or by “multiple looks”. In other words, we are considering the following testing problem: test whether all normal means are zeros versus the alternative that one normal mean per frame is non-zero. We identified an interesting range of cases in which either the number of frames or the contrast size of the signal is not large enough, so that the alternative hypothesis exhibits little noticeable effect on the bulk of the tests or for the few most highly significant tests. With careful calibration, we carried out detailed study of the log-likelihood ratio for a precisely-specified alternative. We found that there is a threshold effect for the above detection problem: for a given amplitude of the signal, there is a critical value for the number of frames--the detection boundary--above which it is possible to detect the presence of the signals, and below which it is impossible to reliably detect it. The detection boundary is explicitly specified and graphed.

In addition, we show that above the detection boundary, the likelihood ratio test would succeed by simply accepting the alternative when the loglikelihood ratio exceeds 0. We also show that the newly proposed Higher Criticism statistic in Higher Criticism for detecting sparse heterogeneous mixtures (Donoho, D. and Jin, J. (2004), to appear), is successful throughout the same region of number (of frames) vs. amplitude where the likelihood ratio test would succeed. Since Higher Criticism does not require a specification of the alternative, this implies that Higher Criticism is in a sense optimally adaptive for the above detection problem. The phenomenon found for the Gaussian setting above also exists for several non-Gaussian settings.

Studies of inhomogeneities in long DNA sequences can be insightful to the organization of the human genome (or any genome). Questions about the spacings of a marker array and general issues of sequence heterogeneity in our studies of DNA and protein sequences led us to statistical considerations of $r$-scan lengths, the distances between marker $i$ and marker $i+r$, $i=1,2,3,\ldots\,$. It is interesting to characterize the $r$-scan lengths harboring clusters or indicating regions of over-dispersion of the markers along the sequence. Applications are reviewed for certain words in the Haemophilus genome and the Cyanobacter genome.

This paper considers a problem of estimating an unknown symmetric region in $R^k$ based on $n$ points randomly drawn from it. The domain of interest is characterized by two parameters: size parameter $r$ and shape parameter $p$. Three methods are investigated which are the maximum likelihood, Bayesian procedures, and a composition of these two. A modification of Wald's theorem as well as a Bayesian version of it are given in this paper to demonstrate the strong consistency of these estimates. We use the measures of symmetric differences and the Hausdorff distance to assess the performance of the estimates. The results reveal that the composite method does the best. Discussion on the convergence in distribution is also given.

The contact process--and more generally interacting particle systems--are useful and interesting models for a variety of statistical problems. This paper is a report on past, present and future of research by the authors concerning the problem of estimating the parameters of the contact process. A brief review of published work on an ad-hoc estimator for the case where the process is observed at a single (large) time $t$ is given in Section 1. In Section 2 we discuss maximum likelihood estimation for the case where the process is observed during a long time interval $[0,t]$. We construct the estimator and state its asymptotic properties as $t \rightarrow \infty$, but spare the reader the long and tedious proof. In Section 3 we return to the case where the process is observed at a single time $t$ and obtain the likelihood equation for the estimator. Much work remains to be done to find a workable approximation to the estimator and study its properties. Our prime interest is to find out whether it is significantly better than the ad-hoc estimator in Section1.

It was a joy to write this paper for Herman Rubin's festschrift. To this is added the bonus that Herman will doubtless solve our remaining problems immediately.

Berk and Jones (1979) introduced a goodness of fit test statistic $R_n$ which is the supremum of pointwise likelihood ratio tests for testing $H_0 : F(x) = F_0 (x)$ versus $H_1 : F (x) \not= F_0 (x)$. They showed that their statistic does not always converge almost surely to a constant under alternatives $F$, and, in fact that there exists an alternative distribution function $F$ such $R_n \rightarrow_d \sup_{t>0} \NN(t)/t$ where $\NN$ is a standard Poisson process on $[0,\infty)$. We call the particular distribution function $F$ which leads to this limiting Poisson behavior the {\sl Poisson boundary distribution function for} $R_n$. We investigate Poisson boundaries for weighted Kolmogorov statistics $D_n (\psi)$ for various weight functions $\psi$ and comment briefly on the history of results concerning Bahadur efficiency of these statistics. One result of note is that the logarithmically weighted Kolmogorov statistic of Groeneboom and Shorack (1981) has the same Poisson boundary as the statistic of Berk and Jones (1979).

A general theorem on compatibility of two systems of subsets of a separable metric space is proved. This theorem is used to deduce results about points of continuity of functions, filtrations and operator semigroups among other things.

We introduce the Aleksandrov-Fenchel inequality, apply it to a tail bound for Gaussian processes, and speculate on a further connection.

This paper extends the Accept-Reject algorithm to allow the proposal distribution to change at each iteration. We first establish a necessary and sufficient condition for this generalized Accept-Reject algorithm to be valid, and then show how the resulting estimator can be improved by Rao-Blackwellization. An application of these results is to the perfect sampling technique of Fill (1998), An interruptible algorithm for perfect sampling via Markov chains, which is a generalized Accept-Reject algorithm.

Data mining is a process of discovering useful patterns (knowledge) hidden in extremely large datasets. Classification is a fundamental data mining function, and some other functions can be reduced to it. In this paper we propose a novel classification algorithm (classifier) called MIND (MINing in Databases). MIND can be phrased in such a way that its implementation is very easy using the extended relational calculus SQL, and this in turn allows the classifier to be built into a relational database system directly. MIND is truly scalable with respect to I/O efficiency, which is important since scalability is a key requirement for any data mining algorithm.

We have built a prototype of MIND in the relational database management system DB2 and have benchmarked its performance. We describe the working prototype and report the measured performance with respect to the previous method of choice. MIND scales not only with the size of datasets but also with the number of processors on an IBM SP2 computer system. Even on uniprocessors, MIND scales well beyond dataset sizes previously published for classifiers.We also give some insights that may have an impact on the evolution of the extended relational calculus SQL.

A simple proof is given for a theorem concerning the first difference and some linear functions of a cointegrated autoregressive process being stationary.

Predicting the outcome of a future game between two sports teams poses a challenging problem of interest to statistical scientists as well as the general public. To be effective such prediction must exploit special contextual features of the game. In this paper, we confront three such features and address the need to: (i) use all relevant sample information; (ii) reflect the home court advantage. To do so we use the relevance weighted likelihood of Hu and Zidek (2002). Finally we demonstrate the value of the method by showing how it could have been used to predict the 1996–1997 NBA Final series results. Our relevance likelihood-based method proves to be quite accurate.

Failure time distributions are derived for non-homogeneous compound Poisson cumulative damage processes. We focus attention on Weibull type processes with exponential damage size. The hazard functions are illustrated and their asymptotic behavior investigated. Moment equations and maximum likelihood estimates are studied for the homogeneous case.

Herman Rubin was born October 27, 1926 in Chicago, Illinois. He obtained his Ph.D. in Mathematics from the University of Chicago in 1948 at the age of 21. He has been on the faculty of Stanford University, the University of Oregon, Michigan State University and Purdue University, where he is currently Professor of Statistics and Professor of Mathematics. He is a Fellow of the Institute of Mathematical Statistics and of the American Association for the Advancement of Science as well as a member of Sigma Xi.

He is well known for his broad ranging mathematical research interests and for fundamental contributions in Bayesian decision theory, in set theory, in estimations for simultaneous equations, in probability and in asymptotic statistics.

These conversations took place during the 2003–2004 academic year at Purdue University.