Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

Genetic studies are under way for many complex traits, spurred by the recent feasibility of whole genome scans. Clear guidelines for the interpretation of linkage results are needed to avoid a flood of false positive claims. At the same time, an overly cautious approach runs the risk of causing true hints of linkage to be missed. We address this problem by proposing specific standards designed to maintain rigor while also promoting communication.

Genetic dissection of complex traits: guidelines for interpreting and reporting linkage results

This paper considers tests for parameter instability and structural change with unknown change point. The results apply to a wide class of parametric models that are suitable for estimation by generalized method of moments procedures. The asymptotic distributions of the test statistics considered here are nonstandard because the change point parameter only appears under the alternative hypothesis and not under the null. The tests considered here are shown to have nontrivial asymptotic local power against all alternatives for which the parameters are nonconstant. The tests are found to perform quite well in a Monte Carlo experiment reported elsewhere. Copyright 1993 by The Econometric Society.

/pdf/tests-for-parameter-instability-and-structural-change-with-26amgrdoht.pdf

Tests for Parameter Instability and Structural Change with Unknown Change Point.

The identification of changes in the recent trend is an important issue in the analysis of cancer mortality and incidence data. We apply a joinpoint regression model to describe such continuous changes and use the grid-search method to fit the regression function with unknown joinpoints assuming constant variance and uncorrelated errors. We find the number of significant joinpoints by performing several permutation tests, each of which has a correct significance level asymptotically. Each p-value is found using Monte Carlo methods, and the overall asymptotic significance level is maintained through a Bonferroni correction. These tests are extended to the situation with non-constant variance to handle rates with Poisson variation and possibly autocorrelated errors. The performance of these tests are studied via simulations and the tests are applied to U.S. prostate cancer incidence and mortality rates.

Permutation tests for joinpoint regression with applications to cancer rates

This book is downloadable from http://www.irisa.fr/sisthem/kniga/. Many monitoring problems can be stated as the problem of detecting a change in the parameters of a static or dynamic stochastic system. The main goal of this book is to describe a unified framework for the design and the performance analysis of the algorithms for solving these change detection problems. Also the book contains the key mathematical background necessary for this purpose. Finally links with the analytical redundancy approach to fault detection in linear systems are established. We call abrupt change any change in the parameters of the system that occurs either instantaneously or at least very fast with respect to the sampling period of the measurements. Abrupt changes by no means refer to changes with large magnitude; on the contrary, in most applications the main problem is to detect small changes. Moreover, in some applications, the early warning of small - and not necessarily fast - changes is of crucial interest in order to avoid the economic or even catastrophic consequences that can result from an accumulation of such small changes. For example, small faults arising in the sensors of a navigation system can result, through the underlying integration, in serious errors in the estimated position of the plane. Another example is the early warning of small deviations from the normal operating conditions of an industrial process. The early detection of slight changes in the state of the process allows to plan in a more adequate manner the periods during which the process should be inspected and possibly repaired, and thus to reduce the exploitation costs.

Detection of abrupt changes: theory and application

Summary. The false discovery rate (FDR) is a multiple hypothesis testing quantity that describes the expected proportion of false positive results among all rejected null hypotheses. Benjamini and Hochberg introduced this quantity and proved that a particular step-up p-value method controls the FDR. Storey introduced a point estimate of the FDR for fixed significance regions. The former approach conservatively controls the FDR at a fixed predetermined level, and the latter provides a conservatively biased estimate of the FDR for a fixed predetermined significance region. In this work, we show in both finite sample and asymptotic settings that the goals of the two approaches are essentially equivalent. In particular, the FDR point estimates can be used to define valid FDR controlling procedures. In the asymptotic setting, we also show that the point estimates can be used to estimate the FDR conservatively over all significance regions simultaneously, which is equivalent to controlling the FDR at all levels simultaneously. The main tool that we use is to translate existing FDR methods into procedures involving empirical processes. This simplifies finite sample proofs, provides a framework for asymptotic results and proves that these procedures are valid even under certain forms of dependence.

Strong control, conservative point estimation and simultaneous conservative consistency of false discovery rates: a unified approach

I Introduction and Examples.- II The Sequential Probability Ratio Test.- III Brownian Approximations and Truncated Tests.- IV Tests with Curved Stopping Boundaries.- V Examples of Repeated Significance Tests.- VI Allocation of Treatments.- VII Interval Estimation of Prescribed Accuracy.- VIII Random Walk and Renewal Theory.- IX Nonlinear Renewal Theory.- X Corrected Brownian Approximations.- XI Miscellaneous Boundary Crossing Problems.- Appendix 1 Brownian Motion.- Appendix 2 Queueing and Insurance Risk Theory.- Appendix 3 Martingales and Stochastic Integrals.- Appendix 4 Renewal Theory.- Bibliographical Notes.- References.

Sequential Analysis: Tests and Confidence Intervals

Great expectations: The theory of optimal stopping

Two samples can be compared by selecting a cut point and then forming a 2 x 2 table of the numbers of observations above and below the cut point in each sample. When the cut point is selected so as to maximize the standard chi square statistic, the chi square percentile points are inappropriate. Actual significance levels are computed for large samples, and correct percentile pOilltS are tabulated.

https://www.jstor.org/stable/pdfplus/2529881.pdf

Maximally Selected Chi Square Statistics

In the analysis of data generated by change-point processes, one critical challenge is to determine the number of change-points. The classic Bayes information criterion (BIC) statistic does not work well here because of irregularities in the likelihood function. By asymptotic approximation of the Bayes factor, we derive a modified BIC for the model of Brownian motion with changing drift. The modified BIC is similar to the classic BIC in the sense that the first term consists of the log likelihood, but it differs in the terms that penalize for model dimension. As an example of application, this new statistic is used to analyze array-based comparative genomic hybridization (array-CGH) data. Array-CGH measures the number of chromosome copies at each genome location of a cell sample, and is useful for finding the regions of genome deletion and amplification in tumor cells. The modified BIC performs well compared to existing methods in accurately choosing the number of regions of changed copy number. Unlike existing methods, it does not rely on tuning parameters or intensive computing. Thus it is impartial and easier to understand and to use.

David Siegmund

Papers

Strong control, conservative point estimation and simultaneous conservative consistency of false discovery rates: a unified approach

Sequential Analysis: Tests and Confidence Intervals

Great expectations: The theory of optimal stopping

Maximally Selected Chi Square Statistics

A modified Bayes information criterion with applications to the analysis of comparative genomic hybridization data.