Physical Sciences and Mathematics Commons^™

Invariant Sets And Inverse Limits, William Thomas Ingram

Mathematics and Statistics Faculty Research & Creative Works

In this paper we investigate the nature of inverse limits from the point of view of invariant sets. We then introduce a special class of examples of inverse limits on [0,1] using Markov bonding maps determined by members of the group of permutations on n elements. © 2002 Elsevier Science B.V. All rights reserved.

Recurrent Events Analysis In The Presence Of Time Dependent Covariates And Dependent Censoring, Maja Miloslavsky, Sunduz Keles, Mark J. Van Der Laan, Steve Butler

U.C. Berkeley Division of Biostatistics Working Paper Series

Recurrent events models have lately received a lot of attention in the literature. The majority of approaches discussed show the consistency of parameter estimates under the assumption that censoring is independent of the recurrent events process of interest conditional on the covariates included into the model. We provide an overview of available recurrent events analysis methods, and present an inverse probability of censoring weighted estimator for the regression parameters in the Andersen-Gill model that is commonly used for recurrent event analysis. This estimator remains consistent under informative censoring if the censoring mechanism is estimated consistently, and generally improves on the …

Construction Of Counterfactuals And The G-Computation Formula, Zhuo Yu, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

Robins' causal inference theory assumes existence of treatment specific counterfactual variables so that the observed data augmented by the counterfactual data will satisfy a consistency and a randomization assumption. Gill and Robins [2001] show that the consistency and randomization assumptions do not add any restrictions to the observed data distribution. In particular, they provide a construction of counterfactuals as a function of the observed data distribution. In this paper we provide a construction of counterfactuals as a function of the observed data itself. Our construction provides a new statistical tool for estimation of counterfactual distributions. Robins [1987b] shows that the …

On Choosing And Bounding Probability Metrics, Alison L. Gibbs, Francis E. Su

All HMC Faculty Publications and Research

When studying convergence of measures, an important issue is the choice of probability metric. We provide a summary and some new results concerning bounds among some important probability metrics/distances that are used by statisticians and probabilists. Knowledge of other metrics can provide a means of deriving bounds for another one in an applied problem. Considering other metrics can also provide alternate insights. We also give examples that show that rates of convergence can strongly depend on the metric chosen. Careful consideration is necessary when choosing a metric.

Multi-Level Decomposition Of Probalistic Relations, Stanislaw Grygiel, Martin Zwick, Marek Perkowski

Systems Science Faculty Publications and Presentations

Two methods of decomposition of probabilistic relations are presented in this paper. They consist of splitting relations (blocks) into pairs of smaller blocks related to each other by new variables generated in such a way so as to minimize a cost function which depends on the size and structure of the result. The decomposition is repeated iteratively until a stopping criterion is met. Topology and contents of the resulting structure develop dynamically in the decomposition process and reflect relationships hidden in the data.

Analysis Of Longitudinal Marginal Structural Models , Jennifer F. Bryan, Zhuo Yu, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

In this article we construct and study estimators of the causal effect of a time-dependent treatment on survival in longitudinal studies. We employ a particular marginal structural model (MSM), and follow a general methodology for constructing estimating functions in censored data models. The inverse probability of treatment weighted (IPTW) estimator is used as an initial estimator and the corresponding treatment-orthogonalized, one-step estimator is consistent and asymptotically linear when the treatment mechanism is consistently estimated. We extend these methods to handle informative censoring. A simulation study demonstrates that the the treatment-orthogonalized, one-step estimator is superior to the IPTW estimator in terms …

Association Between Mean Residual Life (Mrl) And Failure Rate Functions For Continuous And Discrete Lifetime Distributions, Leonid Bekker

FIU Electronic Theses and Dissertations

The purpose of this study was to correct some mistakes in the literature and derive a necessary and sufficient condition for the MRL to follow the roller-coaster pattern of the corresponding failure rate function. It was also desired to find the conditions under which the discrete failure rate function has an upside-down bathtub shape if corresponding MRL function has a bathtub shape. The study showed that if discrete MRL has a bathtub shape, then under some conditions the corresponding failure rate function has an upside-down bathtub shape. Also the study corrected some mistakes in proofs of Tang, Lu and Chew …

Locally Efficient Estimation With Bivariate Right Censored Data , Christopher M. Quale, Mark J. Van Der Laan, James M. Robins

U.C. Berkeley Division of Biostatistics Working Paper Series

Estimation for bivariate right censored data is a problem that has had much study over the past 15 years. In this paper we propose a new class of estimators for the bivariate survivor function based on locally efficient estimation. The locally efficient estimator takes bivariate estimators Fn and Gn of the distributions of the time variables T1,T2 and the censoring variables C1,C2, respectively, and maps them to the resulting estimator. If Fn and Gn are consistent estimators of F and G, respectively, then the resulting estimator will be nonparametrically efficient (thus the term ``locally efficient''). However, if either Fn or …

An Empirical Study Of Marginal Structural Models For Time-Independent Treatment, Tanya A. Henneman, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

In non-randomized treatment studies a significant problem for statisticians is determining how best to adjust for confounders. Marginal structural models (MSMs) and inverse probability of treatment weighted (IPTW) estimators are useful in analyzing the causal effect of treatment in observational studies. Given an IPTW estimator a doubly robust augmented IPTW (AIPTW) estimator orthogonalizes it resulting in a more e±cient estimator than the IPTW estimator. One purpose of this paper is to make a practical comparison between the IPTW estimator and the doubly robust AIPTW estimator via a series of Monte- Carlo simulations. We also consider the selection of the optimal …

Numeric Issues In Test Software Correctness, Robert G. Hayes, Gary B. Hughes, Phillip M. Dorin, Raymond J. Toal

Statistics

Test system designers are comfortable with the concepts of precision and accuracy with regard to measurements achieved with modern instrumentation. In a well-designed test system, great care is taken to ensure accurate measurements, with rigorous attention to instrument specifications and calibration. However, measurement values are subjected to representation and manipulation as limited precision floating-point numbers by test software. This paper investigates some of the issues related to floating point representation of measurement values, as well as the consequences of algorithm selection. To illustrate, we consider the test case of standard deviation calculations as used in the testing of infrared focal …

Student Fact Book, Fall 2002, Twenty-Sixth Annual Edition, Wright State University, Office Of Student Information Systems, Wright State University

Wright State University Student Fact Books

The student fact book has general demographic information on all students enrolled at Wright State University for Fall Quarter, 2002.

Student Fact Book, Fall 2002, Wright State University Lake Campus, Office Of Student Information Systems, Wright State University

Wright State University Student Fact Books

The student fact book has general demographic information on all students enrolled at Wright State University Lake Campus for Fall Quarter, 2002.

Stochastic 2-D Navier-Stokes Equation, J. L. Menaldi, S. S. Sritharan

Mathematics Faculty Research Publications

In this paper we prove the existence and uniqueness of strong solutions for the stochastic Navier-Stokes equation in bounded and unbounded domains. These solutions are stochastic analogs of the classical Lions-Prodi solutions to the deterministic Navier-Stokes equation. Local monotonicity of the nonlinearity is exploited to obtain the solutions in a given probability space and this signi cantly improves the earlier techniques for obtaining strong solutions, which depended on pathwise solutions to the Navier-Stokes martingale problem where the probability space is also obtained as a part of the solution.

The Kpss Test With Seasonal Dummies, Sainan Jin, Sainan Jin

Research Collection School Of Economics

It is shown that the KPSS test for stationarity may be applied without change to regressions with seasonal dummies. In particular, the limit distribution of the KPSS statistic is the same under both the null and alternative hypotheses whether or not seasonal dummies are used.

The Analysis Of Placement Values For Evaluating Discriminatory Measures, Margaret S. Pepe, Tianxi Cai

UW Biostatistics Working Paper Series

The idea of using measurements such as biomarkers, clinical data, or molecular biology assays for classification and prediction is popular in modern medicine. The scientific evaluation of such measures includes assessing the accuracy with which they predict the outcome of interest. Receiver operating characteristic curves are commonly used for evaluating the accuracy of diagnostic tests. They can be applied more broadly, indeed to any problem involving classification to two states or populations (D = 0 or D = 1). We show that the ROC curve can be interpreted as a cumulative distribution function for the discriminatory measure Y in the …

Accelerated Hazards Model: Method, Theory And Applications, Ying Qing Chen, Nicholas P. Jewell, Jingrong Yang

U.C. Berkeley Division of Biostatistics Working Paper Series

In an accelerated hazards model, the hazard functions of a failure time are related through the time scale-change, which is often a function of covariates and associated parameters. When the hazard functions have special properties, such as monotonicity in time, the parameters may be clinically meaningful in measuring a treatment effect. This paper reviews methodological and theoretical development of this model. Applications of the accelerated hazards model including sample size calculation in clinical trials, are also explored.

Locally Efficient Estimation Of Regression Parameters Using Current Status Data, Chris Andrews, Mark J. Van Der Laan, James M. Robins

U.C. Berkeley Division of Biostatistics Working Paper Series

In biostatistics applications interest often focuses on the estimation of the distribution of a time-variable T. If one only observes whether or not T exceeds an observed monitoring time C, then the data structure is called current status data, also known as interval censored data, case I. We consider this data structure extended to allow the presence of both time-independent covariates and time-dependent covariate processes that are observed until the monitoring time. We assume that the monitoring process satisfies coarsening at random.

Our goal is to estimate the regression parameter beta of the regression model T = Z*beta+epsilon where the …

Case-Control Current Status Data, Nicholas P. Jewell, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

Current status observation on survival times has recently been widely studied. An extreme form of interval censoring, this data structure refers to situations where the only available information on a survival random variable, T, is whether or not T exceeds a random independent monitoring time C, a binary random variable, Y. To date, nonparametric analyses of current status data have assumed the availability of i.i.d. random samples of the random variable (Y, C), or a similar random sample at each of a set of fixed monitoring times. In many situations, it is useful to consider a case-control sampling scheme. Here, …

Why Prefer Double Robust Estimates? Illustration With Causal Point Treatment Studies, Romain Neugebauer, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

In point treatment marginal structural models with treatment A, outcome Y and covariates W, causal parameters can be estimated under the assumption of no unobserved confounders. Three estimates can be used: the G-computation, Inverse Probability of Treatment Weighted (IPTW) or Double Robust (DR) estimates. The properties of the IPTW and DR estimates are known under an assumption on the treatment mechanism that we name "Experimental Treatment Assignment" (ETA) assumption. We show that the DR estimating function is unbiased when the ETA assumption is violated if the model used to regress Y on A and W is correctly specified. The practical …

Bivariate Current Status Data, Mark J. Van Der Laan, Nicholas P. Jewell

U.C. Berkeley Division of Biostatistics Working Paper Series

In many applications, it is often of interest to estimate a bivariate distribution of two survival random variables. Complete observation of such random variables is often incomplete. If one only observes whether or not each of the individual survival times exceeds a common observed monitoring time C, then the data structure is referred to as bivariate current status data (Wang and Ding, 2000). For such data, we show that the identifiable part of the joint distribution is represented by three univariate cumulative distribution functions, namely the two marginal cumulative distribution functions, and the bivariate cumulative distribution function evaluated on the …

Current Status Data: Review, Recent Developments And Open Problems, Nicholas P. Jewell, Mark J. Van Der Laan

U.C. Berkeley Division of Biostatistics Working Paper Series

Researchers working with survival data are by now adept at handling issues associated with incomplete data, particular those associated with various forms of censoring. An extreme form of interval censoring, known as current status observation, refers to situations where the only available information on a survival random variable T is whether or not T exceeds a random independent monitoring time C. This article contains a brief review of the extensive literature on the analysis of current status data, discussing the implications of response-based sampling on these methods. The majority of the paper introduces some recent extensions of these ideas to …

Matroid Duality From Topological Duality In Surfaces Of Nonnegative Euler Characteristic, Dan Slilaty

Mathematics and Statistics Faculty Publications

Let G be a connected graph that is 2-cell embedded in a surface S, and let G* be its topological dual graph. We will define and discuss several matroids whose element set is E(G), for S homeomorphic to the plane, projective plane, or torus. We will also state and prove old and new results of the type that the dual matroid of G is the matroid of the topological dual G*.

Semiparametric Regression Analysis On Longitudinal Pattern Of Recurrent Gap Times, Ying Qing Chen, Mei-Cheng Wang, Yijian Huang

U.C. Berkeley Division of Biostatistics Working Paper Series

In longitudinal studies, individual subjects may experience recurrent events of the same type over a relatively long period of time. The longitudinal pattern of the gaps between the successive recurrent events is often of great research interest. In this article, the probability structure of the recurrent gap times is first explored in the presence of censoring. According to the discovered structure, we introduce the proportional reverse-time hazards models with unspecified baseline functions to accommodate heterogeneous individual underlying distributions, when the ongitudinal pattern parameter is of main interest. Inference procedures are proposed and studied by way of proper riskset construction. The …

Multiple Hypothesis Testing In Microarray Experiments, Sandrine Dudoit, Juliet Popper Shaffer, Jennifer C. Boldrick

U.C. Berkeley Division of Biostatistics Working Paper Series

DNA microarrays are a new and promising biotechnology which allows the monitoring of expression levels in cells for thousands of genes simultaneously. An important and common question in microarray experiments is the identification of differentially expressed genes, i.e., genes whose expression levels are associated with a response or covariate of interest. The biological question of differential expression can be restated as a problem in multiple hypothesis testing: the simultaneous test for each gene of the null hypothesis of no association between the expression levels and the responses or covariates. As a typical microarray experiment measures expression levels for thousands of …

Estimation Of The Bivariate Survival Function With Generalized Bivariate Right Censored Data Structures, Sunduz Keles, Mark J. Van Der Laan, James M. Robins

U.C. Berkeley Division of Biostatistics Working Paper Series

We propose a bivariate survival function estimator for a general right censored data structure that includes a time dependent covariate process. Firstly, an initial estimator that generalizes Dabrowska's (1988) estimator is introduced. We obtain this estimator by a general methodology of constructing estimating functions in censored data models. The initial estimator is guaranteed to improve on Dabrowska's estimator and remains consistent and asymptotically linear under informative censoring schemes if the censoring mechanism is estimated consistently. We then construct an orthogonalized estimating function which results in a more robust and efficient estimator than our initial estimator. A simulation study demonstrates the …

Reconciling Experimental Incoherence With Real-World Coherence In Punitive Damages, Theodore Eisenberg, Jeffrey J. Rachlinski, Martin T. Wells

Cornell Law Faculty Publications

Experimental evidence generated in controlled laboratory studies suggests that the legal system in general, and punitive damages awards in particular, should display an incoherent pattern. According to the prediction, inexperienced decisionmakers, such as juries, should fail to convert their qualitative judgments of defendants' conduct into consistent, meaningful dollar amounts. This Article tests this prediction and finds modest support for the thesis that experience across different types of cases will lead to greater consistency in awards. Despite this support, numerous studies of damage awards in real cases detect a generally sensible pattern of damage awards. This Article tries to reconcile the …

Trial Outcomes And Demographics: Is There A Bronx Effect?, Theodore Eisenberg, Martin T. Wells

Cornell Law Faculty Publications

Minorities favor injured plaintiffs and give them inflated awards. This folk wisdom in the legal community influences choice of trial locale and the screening of jurors. A Los Angeles court is said to be known by local lawyers as "the bank" because of the frequency and size of its anti-corporate awards. A newspaper article summarizing court results suggests, somewhat jokingly, that the "Bronx County Courthouse should post a warning: People who get sued here run an increased risk of suffering staggering losses." Beliefs about the influence of factors other than race, such as income and urbanization, also are common.

This …

Secured Debt And The Likelihood Of Reorganization, Clas Bergström, Theodore Eisenberg, Stefan Sundgren

Cornell Law Faculty Publications

Theory suggests that secured creditors may increasingly oppose a debtor’s reorganization as the value of their collateral approaches the amount of their claims. If reorganization occurs and the value of the firm appreciates, the secured creditor receives only part of the gain. But if the firm’s value depreciates, the secured creditor bears all of the cost. Secured claimants, thus, often have more to lose than to gain in reorganizations. This study of Finnish reorganizations filed in districts that account for most of the country’s reorganizations finds that creditor groups most likely to be well-secured are most likely to oppose reorganization. …

Inference For Proportional Mean Residual Life Model In The Presence Of Censoring, Ying Q. Chen, Nicholas P. Jewell

U.C. Berkeley Division of Biostatistics Working Paper Series

As a function of time t, mean residual life is defined as remaining life expectancy of a subject given its survival to t. It plays an important role in many research areas to characterise stochastic behavior of survival over time. Similar to the Cox proportional hazard model, the proportional mean residual life model were proposed in statistical literature to study association between the mean residual life and individual subject's explanatory covariates. In this article, we will study this model and develop appropriate inference procedures in presence of censoring. Numerical studies including simulation and real data analysis are presented as well.

On The Existence Of Multiple Positive Solutions For A Semilinear Problem In Exterior Domains, Yinbin Deng, Yi Li

Mathematics and Statistics Faculty Publications

In this paper, we study the existence and nonexistence of multiple positive solutions for problem where Ω=^N\ω is an exterior domain in ^N, ω⊂^N is a bounded domain with smooth boundary, and N>2. μ⩾0, p>1 are some given constants. K(x) satisfies: K(x)∈C^α_loc(Ω) and ∃C, ϵ, M>0 such that |K(x)|⩽C |x|^l for any |x|⩾M, with l⩽ −2−ϵ. Some existence and …