Statistical applications of the multivariate skew-normal distribution

Azzalini & Dalla Valle (1996) have recently discussed the multivariate skew-normal distribution which extends the class of normal distributions by the addition of a shape parameter. The first part of the present paper examines further probabilistic properties of the distribution, with special emphasis on aspects of statistical relevance. Inferential and other statistical issues are discussed in the following part, with applications to some multivariate statistics problems, illustrated by numerical examples. Finally, a further extension is described which introduces a skewing factor of an elliptical density.Comment: full-length version of the published paper, 32 pages, with 7 figures, uses psfra

The Falling Factorial Basis and Its Statistical Applications

We study a novel spline-like basis, which we name the "falling factorial basis", bearing many similarities to the classic truncated power basis. The advantage of the falling factorial basis is that it enables rapid, linear-time computations in basis matrix multiplication and basis matrix inversion. The falling factorial functions are not actually splines, but are close enough to splines that they provably retain some of the favorable properties of the latter functions. We examine their application in two problems: trend filtering over arbitrary input points, and a higher-order variant of the two-sample Kolmogorov-Smirnov test.Comment: Full version for the ICML paper with the same titl

Brillinger mixing of determinantal point processes and statistical applications

Stationary determinantal point processes are proved to be Brillinger mixing. This property is an important step towards asymptotic statistics for these processes. As an important example, a central limit theorem for a wide class of functionals of determinantal point processes is established. This result yields in particular the asymptotic normality of the estimator of the intensity of a stationary determinantal point process and of the kernel estimator of its pair correlation

Laws of large numbers in stochastic geometry with statistical applications

Given $n$ independent random marked $d$-vectors (points) $X_i$ distributed with a common density, define the measure $\nu_n=\sum_i\xi_i$, where $\xi_i$ is a measure (not necessarily a point measure) which stabilizes; this means that $\xi_i$ is determined by the (suitably rescaled) set of points near $X_i$. For bounded test functions $f$ on $R^d$, we give weak and strong laws of large numbers for $\nu_n(f)$. The general results are applied to demonstrate that an unknown set $A$ in $d$-space can be consistently estimated, given data on which of the points $X_i$ lie in $A$, by the corresponding union of Voronoi cells, answering a question raised by Khmaladze and Toronjadze. Further applications are given concerning the Gamma statistic for estimating the variance in nonparametric regression.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ5167 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Statistical Applications in Healthcare Systems

This thesis consists of three contributing manuscripts related to waiting times with possible applications in health care. The first manuscript is inspired by a practical problem related to decision making in an emergency department (ED). As short-run predictions of ED censuses are particularly important for efficient allocation and management of ED resources we model ED changes and present estimations for short term (hourly) ED censuses at each time point. We present a Markov-chain based algorithm to make census predictions in near future. Considering the variation in arrival pattern and service requirements, we apply and compare three models which best describe our data. We provide hourly predictions up to 24 hours in a day which will provide suggestions to ED managers on how to prevent over-crowding in their system. We illustrate our approach using 22 months data obtained from the ED of a hospital in south western Ontario. The next two manuscripts extend the theory underlying the Accumulating Priority queues (APQs). We focus on the queues with two classes of customers and Poisson arrivals. The first work in this topic derives the stationary waiting time distributions for the class of lowest priority customers in an Affine Accumulating Priority queues (Affine APQs). APQs were first studied by Kleinrock (1964) and later revisited by Stanford et al (2014) where they obtained explicit solution for the Laplace Stieltjes Transform (LST) of the stationary waiting times for all classes of customers. All subsequent publications on APQs, have assumed that all arriving customers accumulate priority credits over time starting from the same initial value (assumed, without loss of generality, to be 0). Whereas, our model studies Affine APQs which assume different initial priorities (without loss of generality in a two-class setting we assume the lowest class starts with 0 credit and the higher class customers with positive credit a. In this work we determine the waiting time distributions for the lower class of customers with Poisson arrivals and general service and present some numerical results for special cases of M/M/1, M/M/c and M/D/1. Inspired by health care applications, we have also considered a particular optimization problem related to the Affine APQ model, in order to select the optimum accumulation rate which allows for the lowest class customers to meet their associated KPIs. We next focus on the Analysis of the Maximum priority processes in the context of Affine APQ. Maximum Priority Processes were first introduced in the context of APQs in Stanford et al (2014). We derive the LST of the stationary steady state distributions of the Maximum Priority Processes as recursive functions and derive the explicit solutions for the LSTs in classical APQ (i.e. a = 0). We employ this argument to present a new approach to determine the LST of waiting time distribution for an APQ with two-classes of customers under the M=M=1 discipline. Since the Analysis of the Maximum Priority Processes in this work is done for the general class of Affine APQs, it has provided the grounds for future researches to obtain the LST of the waiting time distributions in the Affine APQs