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Donald Richards

Publications and source records attributed to Donald Richards.

At least 37 records · Page 2Linked to original sources
arXiv2022

A Continuous-Time Markov Chain Model for the Spread of COVID-19

Since late 2019 the novel coronavirus, also known as COVID-19, has caused a pandemic that persists. This paper shows how a continuous-time Markov chain model for the spread of COVID-19 can be used to explain, and justify to undergraduate students, strategies now being used in attempts to control the virus. The material in the paper is written at the level of students who are taking an introductory course on the theory and applications of stochastic processes.

stat.AP↗
arXiv2022

A Basic Treatment of the Distance Covariance

The distance covariance of Székely, et al. [23] and Székely and Rizzo [21], a powerful measure of dependence between sets of multivariate random variables, has the crucial feature that it equals zero if and only if the sets are mutually independent. Hence the distance covariance can be applied to multivariate data to detect arbitrary types of non-linear associations between sets of variables. We provide in this article a basic, albeit rigorous, introductory treatment of the distance covariance. Our investigations yield an approach that can be used as the foundation for presentation of this important and timely topic even in advanced undergraduate- or junior graduate-level courses on mathematical statistics.

math.ST↗
arXiv2022

Product Inequalities for Multivariate Gaussian, Gamma, and Positively Upper Orthant Dependent Distributions

The Gaussian product inequality is an important conjecture concerning the moments of Gaussian random vectors. While all attempts to prove the Gaussian product inequality in full generality have been unsuccessful to date, numerous partial results have been derived in recent decades and we provide here further results on the problem. Most importantly, we establish a strong version of the Gaussian product inequality for multivariate gamma distributions in the case of nonnegative correlations, thereby extending a result recently derived by Genest and Ouimet [5]. Further, we show that the Gaussian product inequality holds with nonnegative exponents for all random vectors with positive components whenever the underlying vector is positively upper orthant dependent. Finally, we show that the Gaussian product inequality with negative exponents follows directly from the Gaussian correlation inequality.

math.PR↗
arXiv2019

The Distance Standard Deviation

The distance standard deviation, which arises in distance correlation analysis of multivariate data, is studied as a measure of spread. The asymptotic distribution of the empirical distance standard deviation is derived under the assumption of finite second moments. Applications are provided to hypothesis testing on a data set from materials science and to multivariate statistical quality control. The distance standard deviation is compared to classical scale measures for inference on the spread of heavy-tailed distributions. Inequalities for the distance variance are derived, proving that the distance standard deviation is bounded above by the classical standard deviation and by Gini's mean difference. New expressions for the distance standard deviation are obtained in terms of Gini's mean difference and the moments of spacings of order statistics. It is also shown that the distance standard deviation satisfies the axiomatic properties of a measure of spread.

math.ST↗
arXiv2019

Independence Properties of the Truncated Multivariate Elliptical Distributions

Truncated multivariate distributions arise extensively in econometric modelling when non-negative random variables are intrinsic to the data-generation process. More broadly, truncated multivariate distributions have appeared in censored and truncated regression models, simultaneous equations modelling, multivariate regression, and applications going back to the now-classic papers of Amemiya (1974) and Heckman (1976). In some applications of truncated multivariate distributions, there arises the problem of characterizing the distribution through correlation and independence properties of sub-vectors. In this paper, we characterize the truncated multivariate normal random vectors for which two complementary sub-vectors are mutually independent. Further, we characterize the multivariate truncated elliptical distributions, proving that if two complementary sub-vectors are mutually independent then the distribution of the joint vector is truncated multivariate normal, as is the distribution of each sub-vector. As an application, we apply the independence criterion to test the hypothesis of independence of the entrance examination scores and subsequent course averages achieved by a sample of university students; to do so, we verify the regularity conditions underpinning a classical theorem of Wilks on the asymptotic null distribution of the likelihood ratio test statistic.

math.ST↗
arXiv2019

Long-Term Implications of the Revenue Transfer Methodology in the Affordable Care Act

The Affordable Care Act introduced a revenue transfer formula that requires insurance plans with generally healthier enrollees to pay funds into a revenue transfer pool for to reimburse plans with generally less healthy enrollees. For a given plan, the issue arises of whether the plan will be a payer into or a receiver from the pool in a chosen future year. To examine that issue, we analyze data from The Actuary Magazine on transfer payments for 2014-2015, and we infer strong evidence of a statistical relationship between year-to-year transfer payments. We also apply to the data a Markov transition model to study annual changes in the payer-receiver statuses of insurance plans. We estimate that the limiting conditional probability that an insurance plan will pay into the pool, given that the plan had paid into the pool in 2014, is 55.6 percent. Further, that limiting probability is attained quickly because the conditional probability that an insurance plan will pay into the pool in 2024, given that the plan had paid into the pool in 2014, is estimated to be 55.7 percent. We also find the revenue transfer system to have the disturbing feature that once a plan enters the "state" of paying into the pool then it will stay in that state for an average period of 4.87 years; moreover, once a plan has received funds from the pool then it will stay in that state for an average period of 3.89 years.

stat.AP↗
arXiv2019

Integral Transform Methods in Goodness-of-Fit Testing, II: The Wishart Distributions

We initiate the study of goodness-of-fit testing when the data consist of positive definite matrices. Motivated by the recent appearance of the cone of positive definite matrices in numerous areas of applied research, including diffusion tensor imaging, models of the volatility of financial time series, wireless communication systems, and the analysis of polarimetric radar images, we apply the method of Hankel transforms of matrix argument to develop goodness-of-fit tests for Wishart distributions with given shape parameter and unknown scale matrix. We obtain the limiting null distribution of the test statistic and the corresponding covariance operator. We show that the eigenvalues of the operator satisfy an interlacing property, and we apply our test to some financial data. Moreover, we establish the consistency of the test against a large class of alternative distributions and we derive the asymptotic distribution of the test statistic under a sequence of contiguous alternatives. We establish the Bahadur and Pitman efficiency properties of the test statistic and we show the validity of a modified Wieand condition.

math.ST↗
arXiv2019

Loading Monotonicity of Weighted Premiums, and Total Positivity Properties of Weight Functions

We consider the construction of insurance premiums that are monotonically increasing with respect to a loading parameter. By introducing weight functions that are totally positive of higher order, we derive higher monotonicity properties of weighted transformed premiums. We deduce that the greater the degree of randomness of insured risks, the higher the order of total positivity that should be required for the chosen weight functions. We examine seven classes of weight functions that have appeared in the literature, and we ascertain the higher order total positivity properties of those functions.

math.CA↗
arXiv2019

Dependence Properties of B-Spline Copulas

We construct by using B-spline functions a class of copulas that includes the Bernstein copulas arising in Baker's distributions. The range of correlation of the B-spline copulas is examined, and the Frechet--Hoeffding upper bound is proved to be attained when the number of B-spline functions goes to infinity. As the B-spline functions are well-known to be an order-complete weak Tchebycheff system from which the property of total positivity of any order follows for the maximum correlation case, the results given here extend classical results for the Bernstein copulas. In addition, we derive in terms of the Stirling numbers of the second kind an explicit formula for the moments of the related B-spline functions for nonnegative real numbers.

math.ST↗
arXiv2018

Integral Transform Methods in Goodness-of-Fit Testing, I: The Gamma Distributions

We apply the method of Hankel transforms to develop goodness-of-fit tests for gamma distributions with given shape parameter and unknown rate parameter, thereby extending results of Baringhaus and Taherizadeh (2010) on the exponential distributions. We derive the limiting null distribution of the test statistic as an integrated squared Gaussian process, obtain the corresponding covariance operator, and oscillation properties of its eigenfunctions. We show that the eigenvalues of the operator satisfy an interlacing property, and we apply that property in approximating critical values of the test statistic in one of the two applications to data considered. Further, we establish the consistency of the test. In studying properties of the test statistic under a variety of contiguous alternatives, we obtain the asymptotic distribution of the test statistic for gamma alternatives with varying rate or shape parameters and for a class of contaminated gamma models. We investigate the approximate Bahadur slope of the test statistic under local alternatives and we establish the validity of the Wieand condition, under which the approaches through the approximate Bahadur efficiency and the Pitman efficiency are in accord.

math.ST↗
arXiv2018

Statistical Implications of the Revenue Transfer Methodology in the Affordable Care Act

The Affordable Care Act (ACA) includes a permanent revenue transfer methodology which provides financial incentives to health insurance plans that have higher than average actuarial risk. In this paper, we derive some statistical implications of the revenue transfer methodology in the ACA. We treat as random variables the revenue transfers between individual insurance plans in a given marketplace, where each plan's revenue transfer amount is measured as a percentage of the plan's total premium. We analyze the means and variances of those random variables, and deduce from the zero sum nature of the revenue transfers that there is no limit to the magnitude of revenue transfer payments relative to plans' total premiums. Using data provided by the American Academy of Actuaries and by the Centers for Medicare and Medicaid Services, we obtain an explanation for empirical phenomena that revenue transfers were more variable and can be substantially greater for insurance plans with smaller market shares. We show that it is often the case that an insurer which has decreasing market share will also have increased volatility in its revenue transfers.

stat.AP↗
arXiv2018

Constant Proportion Debt Obligations, Zeno's Paradox, and the Spectacular Financial Crisis of 2008

We study a coin-tossing model used by a ratings agency to justify the sale of constant proportion debt obligations (CPDOs), and prove that it was impossible for CPDOs to achieve in a finite lifetime the Cash-In event of doubling its capital. In the best-case scenario of a two-headed coin, we show that the goal of attaining the Cash-In event in a finite lifetime is precisely the goal, described more than two thousand years ago in Zeno's Paradox of the Dichotomy, of obtaining the sum of an infinite geometric series with only a finite number of terms. In the worst-case scenario of a two-tailed coin, we prove that the Cash-Out event occurs in exactly ten tosses. If the coin is fair, we show that if a CPDO were allowed to toss the coin without regard for the Cash-Out rule then the CPDO eventually has a high probability of attaining large net capital levels; however, hundreds of thousands of tosses may be needed to do so. Moreover, if after many tosses the CPDO shows a loss then the probability is high that it will Cash-Out on the very next toss. If a CPDO experiences a tail on the first toss or on an early toss, we show that, with high probability, the CPDO will have capital losses thereafter for hundreds of tosses; moreover, its sequence of net capital levels is a martingale. When the Cash-Out rule holds, we modify the Cash-In rule to mean that the CPDO attains a profit of 90 percent on its capital; then we prove that the CPDO game, almost surely, will end in finitely many tosses and the probability of Cash-Out is at least 89 percent. In light of our results, our fears about the durability of worldwide financial crises are heightened by the existence of other financial derivatives more arcane than CPDOs. In particular, we view askance all later-generation CPDOs that depend mean-reversion assumptions or use a betting strategy similar to their first-generation counterparts.

q-fin.MF↗
arXiv2017

Generalized Fréchet Bounds for Cell Entries in Multidimensional Contingency Tables

We consider the lattice, $\mathcal{L}$, of all subsets of a multidimensional contingency table and establish the properties of monotonicity and supermodularity for the marginalization function, $n(\cdot)$, on $\mathcal{L}$. We derive from the supermodularity of $n(\cdot)$ some generalized Fréchet inequalities complementing and extending inequalities of Dobra and Fienberg. Further, we construct new monotonic and supermodular functions from $n(\cdot)$, and we remark on the connection between supermodularity and some correlation inequalities for probability distributions on lattices. We also apply an inequality of Ky Fan to derive a new approach to Fréchet inequalities for multidimensional contingency tables.

math.ST↗
arXiv2017

The Multiple Roots Phenomenon in Maximum Likelihood Estimation for Factor Analysis

Multiple root estimation problems in statistical inference arise in many contexts in the literature. In the context of maximum likelihood estimation, the existence of multiple roots causes uncertainty in the computation of maximum likelihood estimators using hill-climbing algorithms, and consequent difficulties in the resulting statistical inference. In this paper, we study the multiple roots phenomenon in maximum likelihood estimation for factor analysis. We prove that the corresponding likelihood equations have uncountably many feasible solutions even in the simplest cases. For the case in which the observed data are two-dimensional and the unobserved factor scores are one-dimensional, we prove that the solutions to the likelihood equations form a one-dimensional real curve.

math.ST↗
arXiv2016

Distance Correlation Coefficients for Lancaster Distributions

We consider the problem of calculating distance correlation coefficients between random vectors whose joint distributions belong to the class of Lancaster distributions. We derive under mild convergence conditions a general series representation for the distance covariance for these distributions. To illustrate the general theory, we apply the series representation to derive explicit expressions for the distance covariance and distance correlation coefficients for the bivariate normal distribution and its generalizations of Lancaster type, the multivariate normal distributions, and the bivariate gamma, Poisson, and negative binomial distributions which are of Lancaster type.

math.ST↗
arXiv2016

Exact formulas for the normalizing constants of Wishart distributions for graphical models

Gaussian graphical models have received considerable attention during the past four decades from the statistical and machine learning communities. In Bayesian treatments of this model, the G-Wishart distribution serves as the conjugate prior for inverse covariance matrices satisfying graphical constraints. While it is straightforward to posit the unnormalized densities, the normalizing constants of these distributions have been known only for graphs that are chordal, or decomposable. Up until now, it was unknown whether the normalizing constant for a general graph could be represented explicitly, and a considerable body of computational literature emerged that attempted to avoid this apparent intractability. We close this question by providing an explicit representation of the G-Wishart normalizing constant for general graphs.

math.ST↗
arXiv2016

Maximum Likelihood Estimation for Linear Gaussian Covariance Models

We study parameter estimation in linear Gaussian covariance models, which are $p$-dimensional Gaussian models with linear constraints on the covariance matrix. Maximum likelihood estimation for this class of models leads to a non-convex optimization problem which typically has many local maxima. Using recent results on the asymptotic distribution of extreme eigenvalues of the Wishart distribution, we provide sufficient conditions for any hill-climbing method to converge to the global maximum. Although we are primarily interested in the case in which $n>\!\!>p$, the proofs of our results utilize large-sample asymptotic theory under the scheme $n/p \to γ> 1$. Remarkably, our numerical simulations indicate that our results remain valid for $p$ as small as $2$. An important consequence of this analysis is that for sample sizes $n \simeq 14 p$, maximum likelihood estimation for linear Gaussian covariance models behaves as if it were a convex optimization problem.

math.ST↗