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Articles 1 - 13 of 13
Full-Text Articles in Physical Sciences and Mathematics
The Probability Mass Function Of The Kaplan-Meier Product-Limit Estimator, Yuxin Qin, Heather Sasinowska, Lawrence Leemis
The Probability Mass Function Of The Kaplan-Meier Product-Limit Estimator, Yuxin Qin, Heather Sasinowska, Lawrence Leemis
Arts & Sciences Articles
Kaplan andMeier’s 1958 article developed a nonparametric estimator of the survivor function from a right censored dataset. Determining the size of the support of the estimator as a function of the sample size provides a challenging exercise for students in an advanced course in mathematical statistics. We devise two algorithms for calculating the support size and calculate the associated probability mass function for small sample sizes and particular probability distributions for the failure and censoring times.
Symbolic Arma Model Analysis, Keith H. Webb, Lawrence Leemis
Symbolic Arma Model Analysis, Keith H. Webb, Lawrence Leemis
Arts & Sciences Articles
ARMA models provide a parsimonious and flexible mechanism for modeling the evolution of a time series. Some useful measures of these models (e.g., the autocorrelation function or the spectral density function) are tedious to compute by hand. This paper uses a computer algebra system, not simulation, to calculate measures of interest associated with ARMA models.
Eigenvalue Pairing In The Response Matrix For A Class Of Network Models With Circular Symmetry, Miriam Farber, Charles R. Johnson, Wei Zhen
Eigenvalue Pairing In The Response Matrix For A Class Of Network Models With Circular Symmetry, Miriam Farber, Charles R. Johnson, Wei Zhen
Arts & Sciences Articles
We consider the response matrices in certain weighted networks that display a circular symmetry. It had been observed empirically that these exhibit several paired (multiplicity two) eigenvalues. Here, this pairing is explained analytically for a version of the model more general than the original. The exact number of necessarily paired eigenvalues is given in terms of the structure of the model, and the special structure of the eigenvectors is also described. Examples are provided.
Univariate Probability Distributions, Lawrence Leemis, Daniel J. Luckett, Austin Powell, Peter J. Vermeer
Univariate Probability Distributions, Lawrence Leemis, Daniel J. Luckett, Austin Powell, Peter J. Vermeer
Arts & Sciences Articles
We describe a web-based interactive graphic that can be used as a resource in introductory classes in mathematical statistics. This interactive graphic presents 76 common univariate distributions and gives details on (a) various features of the distribution such as the functional form of the probability density function and cumulative distribution function, graphs of the probability density function for various parameter settings, and values of population moments; (b) properties that the distribution possesses, for example, linear combinations of independent random variables from a particular distribution family also belong to the same distribution family; and (c) relationships between the various distributions, including …
Managing Magnetic Resonance Imaging Machines: Support Tools For Scheduling And Planning, Adam P. Carpenter, Lawrence Leemis, Alan .S. Papir, David J. Phillips, Grace S. Phillips
Managing Magnetic Resonance Imaging Machines: Support Tools For Scheduling And Planning, Adam P. Carpenter, Lawrence Leemis, Alan .S. Papir, David J. Phillips, Grace S. Phillips
Arts & Sciences Articles
We devise models and algorithms to estimate the impact of current and future patient demand for examinations on Magnetic Resonance Imaging (MRI) machines at a hospital radiology department. Our work helps improve scheduling decisions and supports MRI machine personnel and equipment planning decisions. Of particular novelty is our use of scheduling algorithms to compute the competing objectives of maximizing examination throughput and patient-magnet utilization. Using our algorithms retrospectively can help (1) assess prior scheduling decisions, (2) identify potential areas of efficiency improvement and (3) identify difficult examination types. Using a year of patient data and several years of MRI utilization …
Parametric Model Discrimination For Heavily Censored Survival Data, Lawrence Leemis, A. D. Block
Parametric Model Discrimination For Heavily Censored Survival Data, Lawrence Leemis, A. D. Block
Arts & Sciences Articles
Simultaneous discrimination among various parametric lifetime models is an important step in the parametric analysis of survival data. We consider a plot of the skewness versus the coefficient of variation for the purpose of discriminating among parametric survival models. We extend the method of Cox & Oakes from complete to censored data by developing an algorithm based on a competing risks model and kernel function estimation. A by-product of this algorithm is a nonparametric survival function estimate.
Algorithms For Computing The Distributions Of Sums Of Discrete Random Variables, D. L. Evans, Lawrence Leemis
Algorithms For Computing The Distributions Of Sums Of Discrete Random Variables, D. L. Evans, Lawrence Leemis
Arts & Sciences Articles
We present algorithms for computing the probability density function of the sum of two independent discrete random variables, along with an implementation of the algorithm in a computer algebra system. Some examples illustrate the utility of this algorithm.
Nonparametric Estimation Of The Cumulative Intensity Function For A Nonhomogeneous Poisson Process From Overlapping Realizations, Bradford L. Arkin, Lawrence Leemis
Nonparametric Estimation Of The Cumulative Intensity Function For A Nonhomogeneous Poisson Process From Overlapping Realizations, Bradford L. Arkin, Lawrence Leemis
Arts & Sciences Articles
A nonparametric technique for estimating the cumulative intensity function of a nonhomogeneous Poisson process from one or more realizations on an interval is extended here to include realizations that overlap. This technique does not require any arbitrary parameters from the modeler, and the estimated cumulative intensity function can be used to generate a point process for simulation by inversion.
Computing The Cumulative Distribution Function Of The Kolmogorov–Smirnov Statistic, John H. Drew, Andrew G. Glen, Lawrence Leemis
Computing The Cumulative Distribution Function Of The Kolmogorov–Smirnov Statistic, John H. Drew, Andrew G. Glen, Lawrence Leemis
Arts & Sciences Articles
We present an algorithm for computing the cumulative distribution function of the Kolmogorov–Smirnov test statistic Dn in the all-parameters-known case. Birnbaum (1952, J. Amer. Statist. Assoc. 47, 425–441), gives an n-fold integral for the CDF of the test statistic which yields a function defined in a piecewise fashion, where each piece is a polynomial of degree n. Unfortunately, it is difficult to determine the appropriate limits of integration for computing these polynomials. Our algorithm performs the required integrations in a manner that avoids calculating the same integrals repeatedly, resulting in shorter computation time. It can be used …
A Generalized Univariate Change-Of-Variable Transformation Technique, Andrew G. Glen, Lawrence Leemis, John H. Drew
A Generalized Univariate Change-Of-Variable Transformation Technique, Andrew G. Glen, Lawrence Leemis, John H. Drew
Arts & Sciences Articles
We present a generalized version of the univariate change-of-variable technique for transforming continuous random variables. Extending a theorem from Casella and Berger [1990. Statistical Inference, Wadsworth and Brooks/Cole, Inc., Pacific Grove, CA] for many-to-1 transformations, we consider more general univariate transformations. Specifically, the transformation can range from 1-to-1 to many-to-1 on various subsets of the support of the random variable of interest. We also present an implementation of the theorem in a computer algebra system that automates the technique. Some examples demonstrate the theorem's application.
Computational Algebra Applications In Reliability, G Hartless, Lawrence Leemis
Computational Algebra Applications In Reliability, G Hartless, Lawrence Leemis
Arts & Sciences Articles
Reliability analysts are typically forced to choose between using an 'algorithmic programming language' or a 'reliability package' for analyzing their models and lifetime data. This paper shows that computational languages can be used to bridge the gap to combine the flexibility of a programming language with the ease of use of a package. Computational languages facilitate the development of new statistical techniques and are excellent teaching tools. This paper considers three diverse reliability problems that are handled easily with a computational algebra language: system reliability bounds; lifetime data analysis; and model selection.
On The Minimum Of Independent Geometrically Distributed Random Variables, Gianfranco Ciardo, Lawrence Leemis, David Nicol
On The Minimum Of Independent Geometrically Distributed Random Variables, Gianfranco Ciardo, Lawrence Leemis, David Nicol
Arts & Sciences Articles
The expectations E[X(1)], E[Z(1)], and E[Y(1)] of the minimum of n independent geometric, modified geometric, or exponential random variables with matching expectations differ. We show how this is accounted for by stochastic variability and how E[X(1)]/E[Y(1)] equals the expected number of ties at the minimum for the geometric random variables. We then introduce the “shifted geometric distribution”, and show that there is a unique value of the shift for which the individual shifted geometric and exponential random variables match expectations both individually and in …
Variate Generation For Nonhomogeneous Poisson Processes With Time Dependent Covariates, Li-Hsing Shih, Lawrence Leemis
Variate Generation For Nonhomogeneous Poisson Processes With Time Dependent Covariates, Li-Hsing Shih, Lawrence Leemis
Arts & Sciences Articles
Algorithms are developed for generating a sequence of event times from a nonhomogeneous Poisson process that is influenced by the values of covariates that vary with time. Closed form expressions for random variate generation are shown for several baseline intensity and link functions. Two specific models linking the baseline process to the general model are considered: the accelerated time model and the proportional intensity model. In the accelerated time model, the cumulative intensity function of a nonhomogeneous Poisson process under covariate effects is [formula], where z is a covariate vector, ⋀0(t) is the baseline cumulative intensity function and …