Parameter Estimation for a Bidimensional Partially Observed Ornstein–Uhlenbeck Process with Biological Application

Abstract. We consider a bidimensional Ornstein–Uhlenbeck process to describe the tissue microvascularization in anti‐cancer therapy. Data are discrete, partial and noisy observations of this stochastic differential equation (SDE). Our aim is to estimate the SDE parameters. We use the main advantage of a one‐dimensional observation to obtain an easy way to compute the exact likelihood using the Kalman filter recursion, which allows to implement an easy numerical maximization of the likelihood. Furthermore, we establish the link between the observations and an ARMA process and we deduce the asymptotic properties of the maximum likelihood estimator. We show that this ARMA property can be generalized to a higher dimensional underlying Ornstein–Uhlenbeck diffusion. We compare this estimator with the one obtained by the well‐known expectation maximization algorithm on simulated data. Our estimation methods can be directly applied to other biological contexts such as drug pharmacokinetics or hormone secretions.     

A generalized Fleming and Harrington's class of tests for interval‐censored data

The class $G^{\rho,\lambda }$ of weighted log‐rank tests proposed by Fleming & Harrington [Fleming & Harrington (1991) Counting Processes and Survival Analysis, Wiley, New York] has been widely used in survival analysis and is nowadays, unquestionably, the established method to compare, nonparametrically, k different survival functions based on right‐censored survival data. This paper extends the $G^{\rho,\lambda }$ class to interval‐censored data. First we introduce a new general class of rank based tests, then we show the analogy to the above proposal of Fleming & Harrington. The asymptotic behaviour of the proposed tests is derived using an observed Fisher information approach and a permutation approach. Aiming to make this family of tests interpretable and useful for practitioners, we explain how to interpret different choices of weights and we apply it to data from a cohort of intravenous drug users at risk for HIV infection. The Canadian Journal of Statistics 40: 501–516; 2012 © 2012 Statistical Society of Canada     

The asymptotic distribution of numbers of observations near order statistics

In this paper, we discuss the limiting behavior of numbers of observations near an order statistic. We then derive an expression for the joint distribution of the numbers of observations that fall into the open right a-vicinity and left b-vicinity of k  th and (n-r)$(n-r)$th order statistics, respectively, from a sample of size n and establish the result that they are asymptotically independent under suitable conditions.     

THE EFFECTS OF CORRELATION AMONG OBSERVATIONS ON THE ACCURACY OF APPROXIMATING THE DISTRIBUTION OF SAMPLE MEAN BY ITS ASYMPTOTIC DISTRIBUTION1

In this paper, we investigate the effects of correlation among observations on the accuracy of approximating the distribution of sample mean by its asymptotic distribution. The accuracy is investigated by the Berry-Esseen bound (BEB), which gives an upper bound on the error of approximation of the distribution function of the sample mean from its asymptotic distribution for independent observations. For a given sample size (n₀) the BEB is obtained when the observations are independent. Let this be BEB. We then find the sample size (n_*) required to have BEB below BEB₀, when the observations are dependent. Comparison of n_* with n₀ reveals the effects of correlation among observations on the accuracy of the asymptotic distribution as an approximation. It is shown that the effects of correlation among observations are not appreciable if the correlation is moderate to small but it can be severe for extreme correlations.     

A Score Test for the Multivariate Burr and Other Weibull Mixture Distributions

The p -variate Burr distribution has been derived, developed, discussed and deployed by various authors. In this paper a score statistic for testing independence of the components, equivalent to testing for p independent Weibull against a p -variate Burr alternative, is obtained. Its null and non-null properties are investigated with and without nuisance parameters and including the possibility of censoring. Two applications to real data are described. The test is also discussed in the context of other Weibull mixture models.     

Further theoretical results and a comparison between two methods for approximating eigenvalues of perturbed covariance matrices

Covariance matrices, or in general matrices of sums of squares and cross-products, are used as input in many multivariate analyses techniques. The eigenvalues of these matrices play an important role in the statistical analysis of data including estimation and hypotheses testing. It has been recognized that one or few observations can exert an undue influence on the eigenvalues of a covariance matrix. The relationship between the eigenvalues of the covariance matrix computed from all data and the eigenvalues of the perturbed covariance matrix (a covariance matrix computed after a small subset of the observations has been deleted) cannot in general be written in closed-form. Two methods for approximating the eigenvalues of a perturbed covariance matrix have been suggested by Hadi (1988) and Wang and Nyquist (1991) for the case of a perturbation by a single observation. In this paper we improve on these two methods and give some additional theoretical results that may give further insight into the problem. We also compare the two improved approximations in terms of their accuracies.     

STATISTICAL CHALLENGES OF AIDS: (The 1990 Knibbs Lecture, Canberra, 27 November 1990)

Issues that are central to the understanding and management of the HIV epidemic have generated numerous statistical challenges. This paper considers questions concerning the incubation period, the effects of treatments, pre diction of AIDS cases, the choice of surrogate end points for the assessment of treatments and design of strategies for screening blood samples. These issues give rise to a broad range of intriguing problems for statisticians. We describe some of these problems, how they have been tackled so far and what remains to be done. The discussion touches on topical statistical methods such as smoothing, bootstrapping, interval censoring and the ill-posed inverse problem, as well as asking fundamental questions for frequentist statistics.     

Empirical Likelihood Based Synthetic Data Method for Censored Regression Analysis

In this article, we propose a new empirical likelihood method for linear regression analysis with a right censored response variable. The method is based on the synthetic data approach for censored linear regression analysis. A log-empirical likelihood ratio test statistic for the entire regression coefficients vector is developed and we show that it converges to a standard chi-squared distribution. The proposed method can also be used to make inferences about linear combinations of the regression coefficients. Moreover, the proposed empirical likelihood ratio provides a way to combine different normal equations derived from various synthetic response variables. Maximizing this empirical likelihood ratio yields a maximum empirical likelihood estimator which is asymptotically equivalent to the solution of the estimating equation that are optimal linear combination of the original normal equations. It improves the estimation efficiency. The method is illustrated by some Monte Carlo simulation studies as well as a real example.     

Microbiological and Modeling Approach to Derive Performance Objectives for Bacillus cereus Group in Ready‐to‐Eat Salads

In this article, the performance objectives (POs) for Bacillus cereus group (BC) in celery, cheese, and spelt added as ingredients in a ready‐to‐eat mixed spelt salad, packaged under modified atmosphere, were calculated using a Bayesian approach. In order to derive the POs, BC detection and enumeration were performed in nine lots of naturally contaminated ingredients and final product. Moreover, the impact of specific production steps on the BC contamination was quantified. Finally, a sampling plan to verify the ingredient lots' compliance with each PO value at a 95% confidence level (CL) was defined. To calculate the POs, detection results as well as results above the limit of detection but below the limit of quantification (i.e., censored data) were analyzed. The most probable distribution of the censored data was determined and two‐dimensional (2D) Monte Carlo simulations were performed. The PO values were calculated to meet a food safety objective of 4 log₁₀ cfu of BC for g of spelt salad at the time of consumption. When BC grows during storage between 0.90 and 1.90 log₁₀ cfu/g, the POs for BC in celery, cheese, and spelt ranged between 1.21 log₁₀ cfu/g for celery and 2.45 log₁₀ cfu/g for spelt. This article represents the first attempt to manage the concept of PO and 2D Monte Carlo simulation in the flow chart of a complex food matrix, including raw and cooked ingredients.