This paper argues for the status of formal probability theory as a mathematical, rather than a scientific, theory. David Freedman and Philip Stark’s concept of model based probabilities is examined and is used as a bridge between the formal theory and applications.

In this expository paper we describe a relatively elementary method of establishing the existence of a Dutch book in a simple multivariate normal prediction setting. The method involves deriving a nonstandard predictive distribution that is motivated by invariance. This predictive distribution satisfies an interesting identity which in turn yields an elementary demonstration of the existence of a Dutch book for a variety of possible predictive distributions.

We give an example of a transient reversible Markov chain that almost surely has only a finite number of cutpoints. We explain how this is relevant to a conjecture of Diaconis and Freedman and a question of Kaimanovich. We also answer Kaimanovich’s question when the Markov chain is a nearest-neighbor random walk on a tree.

We solve the moment problem for convex distribution functions on [0, 1] in terms of completely alternating sequences. This complements a recent solution of this problem by Diaconis and Freedman, and relates this work to the Lévy-Khintchine formula for the Laplace transform of a subordinator, and to regenerative composition structures.

Motivated by Lévy’s characterization of Brownian motion on the line, we propose an analogue of Brownian motion that has as its state space an arbitrary closed subset of the line that is unbounded above and below: such a process will be a martingale, will have the identity function as its quadratic variation process, and will be “continuous” in the sense that its sample paths don’t skip over points. We show that there is a unique such process, which turns out to be automatically a reversible Feller-Dynkin Markov process. We find its generator, which is a natural generalization of the operator f↦½f''.

We then consider the special case where the state space is the self-similar set {±q^k: k∈ℤ}∪{0} for some q>1. Using the scaling properties of the process, we represent the Laplace transforms of various hitting times as certain continued fractions that appear in Ramanujan’s “lost” notebook and evaluate these continued fractions in terms of basic hypergeometric functions (that is, q-analogues of classical hypergeometric functions). The process has 0 as a regular instantaneous point, and hence its sample paths can be decomposed into a Poisson process of excursions from 0 using the associated continuous local time. Using the reversibility of the process with respect to the natural measure on the state space, we find the entrance laws of the corresponding Itô excursion measure and the Laplace exponent of the inverse local time – both again in terms of basic hypergeometric functions. By combining these ingredients, we obtain explicit formulae for the resolvent of the process. We also compute the moments of the process in closed form. Some of our results involve q-analogues of classical distributions such as the Poisson distribution that have appeared elsewhere in the literature.

Assessment of learning in higher education is a critical concern to policy makers, educators, parents, and students. And, doing so appropriately is likely to require including constructed response tests in the assessment system. We examined whether scoring costs and other concerns with using open-end measures on a large scale (e.g., turnaround time and inter-reader consistency) could be addressed by machine grading the answers. Analyses with 1359 students from 14 colleges found that two human readers agreed highly with each other in the scores they assigned to the answers to three types of open-ended questions. These reader assigned scores also agreed highly with those assigned by a computer. The correlations of the machine-assigned scores with SAT scores, college grades, and other measures were comparable to the correlations of these variables with the hand-assigned scores. Machine scoring did not widen differences in mean scores between racial/ethnic or gender groups. Our findings demonstrated that machine scoring can facilitate the use of open-ended questions in large-scale testing programs by providing a fast, accurate, and economical way to grade responses.

The U.S. Census Bureau provides an estimate of the true population as a supplement to the basic census numbers. This estimate is constructed from data in a post-censal survey. The overall procedure is referred to as dual system estimation. Dual system estimation is designed to produce revised estimates at all levels of geography, via a synthetic estimation procedure.

We design three alternative formulas for dual system estimation and investigate the differences in area estimates produced as a result of using those formulas. The primary target of this exercise is to better understand the nature of the homogeneity assumptions involved in dual system estimation and their consequences when used for the enumeration data that occurs in an actual large scale application like the Census. (Assumptions of this nature are sometimes collectively referred to as the “synthetic assumption” for dual system estimation.)

The specific focus of our study is the treatment of the category of census counts referred to as imputations in dual system estimation. Our results show the degree to which varying treatment of these imputation counts can result in differences in population estimates for local areas such as states or counties.

This paper tries to tell the story of the general linear model, which saw the light of day 200 years ago, and the assumptions underlying it. We distinguish three principal stages (ignoring earlier more isolated instances). The model was first proposed in the context of astronomical and geodesic observations, where the main source of variation was observational error. This was the main use of the model during the 19th century.

In the 1920’s it was developed in a new direction by R.A. Fisher whose principal applications were in agriculture and biology. Finally, beginning in the 1930’s and 40’s it became an important tool for the social sciences. As new areas of applications were added, the assumptions underlying the model tended to become more questionable, and the resulting statistical techniques more prone to misuse.

Over the past two decades, a variety of methods have been used to count the homeless in large metropolitan areas. In this paper, we report on an effort to count the homeless in Los Angeles County, one that employed the sampling of census tracts. A number of complications are discussed, including the need to impute homeless counts to areas of the County not sampled. We conclude that, despite their imperfections, estimated counts provided useful and credible information to the stakeholders involved.

The Women’s Health Initiative randomized clinical trial of hormone therapy found no benefit of hormones in preventive cardiovascular disease, a finding in striking contrast with a large body of observational research. Understanding whether better methodology and/or statistical adjustment might have prevented the erroneous conclusions of observational research is important. This is a re-analysis of data from a case-control study examining the relationship of postmenopausal hormone therapy and the risks of myocardial infarction (MI) and ischemic stroke in which we reported no overall increase or decrease in the risk of either event. Variables measuring health behavior/lifestyle that are not likely to be causally with the risks of MI and stroke (e.g., sunscreen use) were included in multivariate analysis along with traditional confounders (age, hypertension, diabetes, smoking, body mass index, ethnicity, education, prior coronary heart disease for MI and prior stroke/TIA for stroke) to determine whether adjustment for the health behavior/lifestyle variables could reproduce or bring the results closer to the findings in a large and definitive randomized clinical trial of hormone therapy, the Women’s Health Initiative.

For both MI and stroke, measures of health behavior/lifestyle were associated with odds ratios (ORs) less than 1.0. Adjustment for traditional cardiovascular disease confounders did not alter the magnitude of the ORs for MI or stroke. Addition of a subset of these variables selected using stepwise regression to the final MI or stroke models along with the traditional cardiovascular disease confounders moved the ORs for estrogen and estrogen/progestin use closer to values observed in the Women Health Initiative clinical trial, but did not reliably reproduce the clinical trial results for these two endpoints.

We propose a general and formal statistical framework for multiple tests of association between known fixed features of a genome and unknown parameters of the distribution of variable features of this genome in a population of interest. The known gene-annotation profiles, corresponding to the fixed features of the genome, may concern Gene Ontology (GO) annotation, pathway membership, regulation by particular transcription factors, nucleotide sequences, or protein sequences. The unknown gene-parameter profiles, corresponding to the variable features of the genome, may be, for example, regression coefficients relating possibly censored biological and clinical outcomes to genome-wide transcript levels, DNA copy numbers, and other covariates. A generic question of great interest in current genomic research regards the detection of associations between biological annotation metadata and genome-wide expression measures. This biological question may be translated as the test of multiple hypotheses concerning association measures between gene-annotation profiles and gene-parameter profiles. A general and rigorous formulation of the statistical inference question allows us to apply the multiple hypothesis testing methodology developed in [Multiple Testing Procedures with Applications to Genomics (2008) Springer, New York] and related articles, to control a broad class of Type I error rates, defined as generalized tail probabilities and expected values for arbitrary functions of the numbers of Type I errors and rejected hypotheses. The resampling-based single-step and stepwise multiple testing procedures of [Multiple Testing Procedures with Applications to Genomics (2008) Springer, New York] take into account the joint distribution of the test statistics and provide Type I error control in testing problems involving general data generating distributions (with arbitrary dependence structures among variables), null hypotheses, and test statistics.

The proposed statistical and computational methods are illustrated using the acute lymphoblastic leukemia (ALL) microarray dataset of [Blood 103 (2004) 2771–2778], with the aim of relating GO annotation to differential gene expression between B-cell ALL with the BCR/ABL fusion and cytogenetically normal NEG B-cell ALL. The sensitivity of the identified lists of GO terms to the choice of association parameter between GO annotation and differential gene expression demonstrates the importance of translating the biological question in terms of suitable gene-annotation profiles, gene-parameter profiles, and association measures. In particular, the results reveal the limitations of binary gene-parameter profiles of differential expression indicators, which are still the norm for combined GO annotation and microarray data analyses. Procedures based on such binary gene-parameter profiles tend to be conservative and lack robustness with respect to the estimator for the set of differentially expressed genes. Our proposed statistical framework, with general definitions for the gene-annotation and gene-parameter profiles, allows consideration of a much broader class of inference problems, that extend beyond GO annotation and microarray data analysis.

This paper presents exploratory techniques for multivariate data, many of them well known to French statisticians and ecologists, but few well understood in North American culture. We present the general framework of duality diagrams which encompasses discriminant analysis, correspondence analysis and principal components, and we show how this framework can be generalized to the regression of graphs on covariates.

A quarter-century of statistical research has shown that census coverage surveys, valuable as they are in offering a report card on each decennial census, do not provide usable estimates of geographical differences in coverage. The determining reason is the large number of “doubly missing” people missing both from the census enumeration and from coverage survey estimates. Future coverage surveys should be designed to meet achievable goals, foregoing efforts at spatial specificity. One implication is a sample size no more than about 30,000, setting free resources for controlling processing errors and investing in coverage improvement. Possible integration of coverage measurement with the American Community Survey would have many benefits and should be given careful consideration.

Hawaiian monk seals (Monachus schauinslandi) are endemic to the Hawaiian Islands and are the most endangered species of marine mammal that lives entirely within the jurisdiction of the United States. The species numbers around 1300 and has been declining owing, among other things, to poor juvenile survival which is evidently related to poor foraging success. Consequently, data have been collected recently on the foraging habitats, movements, and behaviors of monk seals throughout the Northwestern and main Hawaiian Islands.

Our work here is directed to exploring a data set located in a relatively shallow offshore submerged bank (Penguin Bank) in our search of a model for a seal’s journey. The work ends by fitting a stochastic differential equation (SDE) that mimics some aspects of the behavior of seals by working with location data collected for one seal. The SDE is found by developing a time varying potential function with two points of attraction. The times of location are irregularly spaced and not close together geographically, leading to some difficulties of interpretation. Synthetic plots generated using the model are employed to assess its reasonableness spatially and temporally. One aspect is that the animal stays mainly southwest of Molokai. The work led to the estimation of the lengths and locations of the seal’s foraging trips.

This paper develops projection pursuit for discrete data using the discrete Radon transform. Discrete projection pursuit is presented as an exploratory method for finding informative low dimensional views of data such as binary vectors, rankings, phylogenetic trees or graphs. We show that for most data sets, most projections are close to uniform. Thus, informative summaries are ones deviating from uniformity. Syllabic data from several of Plato’s great works is used to illustrate the methods. Along with some basic distribution theory, an automated procedure for computing informative projections is introduced.

When a defendant’s DNA matches a sample found at a crime scene, how compelling is the match? To answer this question, DNA analysts typically use relative frequencies, random-match probabilities or likelihood ratios. They compute these quantities for the major racial or ethnic groups in the United States, supplying prosecutors with such mind-boggling figures as “one in nine hundred and fifty sextillion African Americans, one in one hundred and thirty septillion Caucasians, and one in nine hundred and thirty sextillion Hispanics.” In People v. Prince, a California Court of Appeals rejected this practice on the theory that only the perpetrator’s race is relevant to the crime; hence, it is impermissible to introduce statistics about other races. This paper critiques this reasoning. Relying on the concept of likelihood, it presents a logical justification for referring to a range of races and identifies some problems with the one-race-only rule. The paper also notes some ways to express the probative value of a DNA match quantitatively without referring to variations in DNA profile frequencies among races or ethnic groups.

Statistical tests of earthquake predictions require a null hypothesis to model occasional chance successes. To define and quantify ‘chance success’ is knotty. Some null hypotheses ascribe chance to the Earth: Seismicity is modeled as random. The null distribution of the number of successful predictions – or any other test statistic – is taken to be its distribution when the fixed set of predictions is applied to random seismicity. Such tests tacitly assume that the predictions do not depend on the observed seismicity. Conditioning on the predictions in this way sets a low hurdle for statistical significance. Consider this scheme: When an earthquake of magnitude 5.5 or greater occurs anywhere in the world, predict that an earthquake at least as large will occur within 21 days and within an epicentral distance of 50 km. We apply this rule to the Harvard centroid-moment-tensor (CMT) catalog for 2000–2004 to generate a set of predictions. The null hypothesis is that earthquake times are exchangeable conditional on their magnitudes and locations and on the predictions–a common “nonparametric” assumption in the literature. We generate random seismicity by permuting the times of events in the CMT catalog. We consider an event successfully predicted only if (i) it is predicted and (ii) there is no larger event within 50 km in the previous 21 days. The P-value for the observed success rate is <0.001: The method successfully predicts about 5% of earthquakes, far better than ‘chance,’ because the predictor exploits the clustering of earthquakes – occasional foreshocks – which the null hypothesis lacks. Rather than condition on the predictions and use a stochastic model for seismicity, it is preferable to treat the observed seismicity as fixed, and to compare the success rate of the predictions to the success rate of simple-minded predictions like those just described. If the proffered predictions do no better than a simple scheme, they have little value.

It has been widely realized that Monte Carlo methods (approximation via a sample ensemble) may fail in large scale systems. This work offers some theoretical insight into this phenomenon in the context of the particle filter. We demonstrate that the maximum of the weights associated with the sample ensemble converges to one as both the sample size and the system dimension tends to infinity. Specifically, under fairly weak assumptions, if the ensemble size grows sub-exponentially in the cube root of the system dimension, the convergence holds for a single update step in state-space models with independent and identically distributed kernels. Further, in an important special case, more refined arguments show (and our simulations suggest) that the convergence to unity occurs unless the ensemble grows super-exponentially in the system dimension. The weight singularity is also established in models with more general multivariate likelihoods, e.g. Gaussian and Cauchy. Although presented in the context of atmospheric data assimilation for numerical weather prediction, our results are generally valid for high-dimensional particle filters.

We present a theory of point and interval estimation for nonlinear functionals in parametric, semi-, and non-parametric models based on higher order influence functions (Robins (2004), Section 9; Li et al. (2004), Tchetgen et al. (2006), Robins et al. (2007)). Higher order influence functions are higher order U-statistics. Our theory extends the first order semiparametric theory of Bickel et al. (1993) and van der Vaart (1991) by incorporating the theory of higher order scores considered by Pfanzagl (1990), Small and McLeish (1994) and Lindsay and Waterman (1996). The theory reproduces many previous results, produces new non-$\sqrt{n}$ results, and opens up the ability to perform optimal non-$\sqrt{n}$ inference in complex high dimensional models. We present novel rate-optimal point and interval estimators for various functionals of central importance to biostatistics in settings in which estimation at the expected $\sqrt{n}$ rate is not possible, owing to the curse of dimensionality. We also show that our higher order influence functions have a multi-robustness property that extends the double robustness property of first order influence functions described by Robins and Rotnitzky (2001) and van der Laan and Robins (2003).