Erol A. Peköz

Degree asymptotics with rates for preferential attachment random graphs.

We provide optimal rates of convergence to the asymptotic distribution of the (properly scaled) degree of a fixed vertex in two preferential attachment random graph models. Our approach is to show that these distributions are unique fixed points of certain distributional transformations which allows us to obtain rates of convergence using a new variation of Steins method. Despite the large literature on these models, there is surprisingly little known about the limiting distributions so we also provide some properties and new representations, including an explicit expression for the densities in terms of the confluent hypergeometric function of the second kind.

New rates for exponential approximation and the theorems of Rényi and Yaglom

We introduce two abstract theorems that reduce a variety of complex exponential distributional approximation problems to the construction of couplings. These are applied to obtain new rates of convergence with respect to the Wasserstein and Kolmogorov metrics for the theorem of Renyi on random sums and generalizations of it, hitting times for Markov chains, and to obtain a new rate for the classical theorem of Yaglom on the exponential asymptotic behavior of a critical Galton―Watson process conditioned on nonextinction. The primary tools are an adaptation of Steins method, Stein couplings, as well as the equilibrium distributional transformation from renewal theory.We introduce two abstract theorems that reduce a variety of complex exponential distributional approximation problems to the construction of couplings. These are applied to obtain new rates of convergence with respect to the Wasserstein and Kolmogorov metrics for the theorem of Renyi on random sums and generalizations of it, hitting times for Markov chains, and to obtain a new rate for the classical theorem of Yaglom on the exponential asymptotic behavior of a critical Galton―Watson process conditioned on nonextinction. The primary tools are an adaptation of Steins method, Stein couplings, as well as the equilibrium distributional transformation from renewal theory.

Estimating a Composite Measure of Hospital Quality From the Hospital Compare Database Differences When Using a Bayesian Hierarchical Latent Variable Model Versus Denominator-Based Weights

Background:A single composite measure calculated from individual quality indicators (QIs) is a useful measure of hospital performance and can be justified conceptually even when the indicators are not highly correlated with one another. Objective:To compare 2 basic approaches for calculating a composite measure: an extension of the most widely-used approach, which weights individual indicators based on the number of people eligible for the indicator (referred to as denominator-based weights, DBWs), and a Bayesian hierarchical latent variable model (BLVM). Methods:Using data for 15 QIs from 3275 hospitals in the Hospital Compare database, we calculated hospital ranks using several versions of DBWs and 2 BLVMs. Estimates in 1 BLVM were driven by differences in variances of the QIs (BLVM1) and estimates in the other by differences in the signal-to-noise ratios of the QIs (BLVM2). Results:There was a high correlation in ranks among all of the DBW approaches and between those approaches and BLVM1. However, a high correlation does not necessarily mean that the same hospitals were ranked in the top or bottom quality deciles. In general, large hospitals were ranked in higher quality deciles by all of the approaches, though the effect was most apparent using BLVM2. Conclusions:Both conceptually and practically, hospital-specific DBWs are a reasonable approach for calculating a composite measure. However, this approach fails to take into account differences in the reliability of estimates from hospitals of different sizes, a big advantage of the Bayesian models.

STEIN'S METHOD FOR GEOMETRIC APPROXIMATION

The Stein-Chen method for Poisson approximation is adapted to the setting of the geometric distribution. This yields a convenient method for assessing the accuracy of the geometric approximation to the distribution of the number of failures preceding the first success in dependent trials. The results are applied to approximating waiting time distributions for patterns in coin tossing, and to approximating the distribution of the time when a stationary Markov chain first visits a rare set of states. The error bounds obtained are sharper than those obtainable using related Poisson approximations.

Person-centered Care Practices and Quality in Department of Veterans Affairs Nursing Homes Is There a Relationship?

Objective:To examine variation in culture change to a person-centered care (PCC) model, and the association between culture change and a composite measure of quality in 107 Department of Veterans Affairs nursing homes. Methods:We examined the relationship between a composite quality measure calculated from 24 quality indicators (QIs) from the Minimum Data Set (that measure unfavorable events), and PCC summary scores calculated from the 6 domains of the Artifact of Culture Change Tool, using 3 different methods of calculating the summary scores. We also use a Bayesian hierarchical model to analyze the relationship between a latent construct measuring extent of culture change and the composite quality measure. Results:Using the original Artifacts scores, the highest performing facility has a 2.9 times higher score than the lowest. There is a statistically significant relationship between the composite quality measure and each of the 3 summary Artifacts scores. Depending on whether original scores, standardized scores, or optimal scores are used, a facility at the 10th percentile in terms of culture change compared with one at the 90th percentile has 8.0%, 8.9%, or 10.3% more QI events. When PCC implementation is considered as a latent construct, 18 low performance PCC facilities have, on an average, 16.3% more QI events than 13 high performance facilities. Conclusions:Our results indicate that culture change to a PCC model is associated with higher Minimum Data Set-based quality. Longitudinal data are needed to better assess whether there is a causal relationship between the extent of culture change and quality.

A simple derivation of exact reliability formulas for linear and circular consecutive-k-of-n: F systems

Exact reliability formulas for linear and circular consecutive-k-of-n : F systems are derived in the case of equal component reliabilities. SYSTEM OF COMPONENTS; EQUAL COMPONENT RELIABILITIES

Generalized gamma approximation with rates for urns, walks and trees

We study a new class of time inhomogeneous Polya-type urn schemes and give optimal rates of convergence for the distribution of the properly scaled number of balls of a given color to nearly the full class of generalized gamma distributions with integer parameters, a class which includes the Rayleigh, half-normal and gamma distributions. Our main tool is Stein’s method combined with characterizing the generalized gamma limiting distributions as fixed points of distributional transformations related to the equilibrium distributional transformation from renewal theory. We identify special cases of these urn models in recursive constructions of random walk paths and trees, yielding rates of convergence for local time and height statistics of simple random walk paths, as well as for the size of random subtrees of uniformly random binary and plane trees.

Some results for skip-free random walk

A random walk that is skip-free to the left can only move down one level at a time but can skip up several levels. Such random walk features prominently in many places in applied probability including queuing theory and the theory of branching processes. This article exploits the special structure in this class of random walk to obtain a number of simplified derivations for results that are much more difficult in general cases. Although some of the results in this article have appeared elsewhere, our proof approach is different.

Optimal Policies for Multi-server Non-preemptive Priority Queues

We consider a multi-server non-preemptive queue with high and low priority customers, and a decision maker who decides when waiting customers may enter service. The goal is to minimize the mean waiting time for high-priority customers while keeping the queue stable. We use a linear programming approach to find and evaluate the performance of an asymptotically optimal policy in the setting of exponential service and inter-arrival times.

Joint degree distributions of preferential attachment random graphs

Abstract We study the joint degree counts in linear preferential attachment random graphs and find a simple representation for the limit distribution in infinite sequence space. We show weak convergence with respect to the p-norm topology for appropriate p and also provide optimal rates of convergence of the finite-dimensional distributions. The results hold for models with any general initial seed graph and any fixed number of initial outgoing edges per vertex; we generate nontree graphs using both a lumping and a sequential rule. Convergence of the order statistics and optimal rates of convergence to the maximum of the degrees is also established.