Independent Poisson Random Variables Research Articles

Mean and covariance properties of dynamic PET reconstructions from list-mode data

We derive computationally efficient methods for the estimation of the mean and variance properties of penalized likelihood dynamic positron emission tomography (PET) images. This allows us to predict the accuracy of reconstructed activity estimates and to compare reconstruction algorithms theoretically. We combine a bin-mode approach in which data is modeled as a collection of independent Poisson random variables at each spatiotemporal bin with the space-time separabilities in the imaging equation and penalties to derive rapidly computable analytic mean and variance approximations. We use these approximations to compare bias/variance properties of our dynamic PET image reconstruction algorithm with those of multiframe static PET reconstructions.

Structure of large random hypergraphs

The theme of this paper is the derivation of analytic formulae for certain large combinatorial structures. The formulae are obtained via fluid limits of pure jump-type Markov processes, established under simple conditions on the Laplace transforms of their Lévy kernels. Furthermore, a related Gaussian approximation allows us to describe the randomness which may persist in the limit when certain parameters take critical values. Our method is quite general, but is applied here to vertex identifiability in random hypergraphs. A vertex $v$ is identifiable in $n$ steps if there is a hyperedge containing $v$ all of whose other vertices are identifiable in fewer steps. We say that a hyperedge is identifiable if every one of its vertices is identifiable. Our analytic formulae describe the asymptotics of the number of identifiable vertices and the number of identifiable hyperedges for a Poisson $(β)$ random hypergraph $Λ$ on a set $V$ of $N$ vertices, in the limit as $N→∞$. Here $β$ is a formal power series with nonnegative coefficients $β_0,β_1,…,$ and $(Λ(A))_{A⊆V}$ are independent Poisson random variables such that $Λ(A)$, the number of hyperedges on $A$, has mean $Nβ_j/\pmatrix{{N}\cr{j}}$ whenever $|A|=j$.

Hierarchical modeling of count data with application to nuclear fall-out

We propose a method for a Bayesian hierarchical analysis of count data that are observed at irregular locations in a bounded domain of R2. We model the data as having been observed on a fine regular lattice, where we do not have observations at all the sites. The counts are assumed to be independent Poisson random variables whose means are given by a log Gaussian process. In this article, the Gaussian process is assumed to be either a Markov random field (MRF) or a geostatistical model, and we compare the two models on an environmental data set. To make the comparison, we calibrate priors for the parameters in the geostatistical model to priors for the parameters in the MRF. The calibration is obtained empirically. The main goal is to predict the hidden Poisson-mean process at all sites on the lattice, given the spatially irregular count data; to do this we use an efficient MCMC. The spatial Bayesian methods are illustrated on radioactivity counts analyzed by Diggle et al. (1998).

DETERMINING THE OPTIMAL RELEASE TIME FOR SOFTWARE IN THE GEOMETRIC POISSON RELIABILITY MODEL

The geometric Poisson model for software reliability and debugging was introduced by P. B. Moranda to deal with the situation when one is collecting grouped data on faults (say for example the number of faults detected weekly or monthly) in a new piece of software. In this model one assumes that the number of faults or bugs detected in subsequent intervals of equal length of time in testing a piece of software are independent Poisson random variables with respective Poisson parameters which form a decreasing geometric sequence. An important problem in software reliability testing is to decide when to stop testing and release software for commercial use. In this paper we determine the optimal time T* to release software under the geometric Poisson model with a cost structure for repairing or fixing faults by minimizing the expected cost. We consider both the situation where one is interested in the software performing well until some fixed software life-cycle time t, and for the case where we want the software to function well for an additional τ units of (mission) time after release. Solutions are obtained both when parameter estimates are available as well as in the Bayesian framework when prior information on these parameters is at hand.

Remarks on compound Poisson approximation of Gaussian random sequences

Let {X i, i⩾1} be a sequence of m-dependent stationary standard Gaussian random variables and u n, n∈ N some positive constants. In this note we generalise results of Raab (Extremes 1(3) (1999) 29.), who considered compound Poisson approximation for W n=∑ i=1 n 1 {X i>u n} the number of exceedances above the level u n . More precisely, the main result concerns an upper asymptotic bound for the total variational distance d TV (W n, CP(λ ∗)) where CP(λ ∗) = d N 1+2N 2+⋯+rN r, with 2⩽ r⩽2 m and N i = d Poi(λ i), λ i>0 are independent Poisson random variables.

A Markov Renewal Based Model for Wireless Networks

We propose a queueing network model which can be used for the integration of the mobility and teletraffic aspects that are characteristic of wireless networks. In the general case, the model is an open network of infinite server queues where customers arrive according to a non-homogeneous Poisson process. The movement of a customer in the network is described by a Markov renewal process. Moreover, customers have attributes, such as a teletraffic state, that are driven by continuous time Markov chains and, therefore, change as they move through the network. We investigate the transient and limit number of customers in disjoint sets of nodes and attributes. These turn out to be independent Poisson random variables. We also calculate the covariances of the number of customers in two sets of nodes and attributes at different time epochs. Moreover, we conclude that the arrival process per attribute to a node is the sum of independent Poisson cluster processes and derive its univariate probability generating function. In addition, the arrival process to an outside node of the network is a non-homogeneous Poisson process. We illustrate the applications of the queueing network model and the results derived in a particular wireless network.

On the achievability of the Cramer-Rao bound for Poisson distribution

This correspondence examines the Cramer-Rao (CR) bound for data obtained in emission tomography. The likelihood function involved is the combined probability of independent Poisson random variables, the expectation of each being a linear function c/sub i//sup T//spl lambda/ of the parameter vector /spl lambda/. We investigated the achievability of the CR bound in the interior and on the boundary of the domain of the problem. For the former, we found that the CR bound is achievable if and only if the c/sub i/ vectors are obtained from a basis for R/sup N/, by repeating some vectors, multiplied by constant factors. A similar result holds for the boundary case. The practical implication of the achievability condition is that the CR bound is not attainable for typical emission tomographic systems.

Components of the game result in a football league

We assume that the result of a football game depends upon the difference of the strengths of the teams, home-field advantage, random factors and also other components. We describe the goal outcome per game by independent Poisson random variables; we conce

Limit Theorems for the Number of Solutions of a System of Random Equations

We investigate the number and the set structure of the solutions of a consistent system of random equations of the form $$ \varphi_t(x_{s_1(t)},\ldots,x_{s_{d(t)}(t)})=a_t,\quad t=1,\ldots, T, $$ with respect to the variables $x_1,\ldots, x_n\in\{0,\ldots,q-1\}$, $q\ge 2$, where the indices $s_1(t),\ldots,s_{d(t)}(t)$ are selected randomly and independently for different t according to the equiprobable selection procedure without replacement. Conditions are found under which the distribution of the number of solutions of the system converges to the distribution of a random variable of the form $A\cdot 2^{\eta_1}\cdots q^{\eta_{q}-1}$, where A is the order of the group of permutations $g\!: \{0,\ldots,q-1\}{\longleftrightarrow}\{0,\ldots,q-1\}$ satisfying the conditions $\varphi_t(y_1,\ldots y_{d(t)})\equiv\varphi_t(g(y_1)\lz g(y_{d(t)}))$, $t=1,\ldots,T$, and $\eta_1,\ldots,\eta_{q-1}$ are independent Poisson random variables with parameters $\lambda_1,\ldots,\lambda_{q-1}$, respectively. Explicit express...

The problem of low counts in a signal plus noise model

Consider the model X = B + S, where B and S are independent Poisson random variables with means µ and ν, ν is unknown, but µ is known. The model arises in particle physics and some recent articles have suggested conditioning on the observed bound on B; that is, if X = n is observed, then the suggestion is to base inference on the conditional distribution of X given B ≤ n. This conditioning is non-standard in that it does not correspond to a partition of the sample space. It is examined here from the view point of decision theory and shown to lead to admissible formal Bayes procedures. 1. Introduction. In some problems a signal S may be superimposed on a background B, leaving an observed count X = B + S. Here we suppose that B and S are independent Poisson random variables with means µ and ν, so that X has a Poisson distribution with mean θ = µ+ν; ν is regarded as an unknown parameter, but µ is assumed to be known, as might be appropriate if there were historical data on the background. What special techniques, if any, are appropriate if the observed count X is smaller than the expected background? A problem of this nature has arisen recently in physics. The KARMEN 2 Group has been searching for a neutrino oscillation, reported earlier from an experiment at the Los Alamos Neutrino Detector. As of Summer 1998, they had expected to see about 3 ± � 1 background events and at least one signal event, based on the earlier findings (and using rounded values), but had observed nothing. See Zeitnitz et al. (1998). What inference about ν is appropriate here? Can the hypothesis H0 � ν ≥ 1 be rejected? A naive analysis suggests that it can. If µ = 3 is regarded as a known quantity and the hypothesis is rewritten as H0 � θ ≥ 4, then the p value is Pθ� X ≤ 0 �≤ e −4 for θ ≥ 4, and this is less than the usual levels of significance. But this analysis is suspect, because if X = 0, then both B and S must be zero; and P4� S ≤ 0 �= e −1 , which is not less than the usual levels of significance. The second value is obtained by conditioning on the ancillary variable B = 0 and seems right in this context. The problem becomes much more interesting when the observed count is non-zero but smaller than the expected background, since then it is no longer possible to recover the value of B. Roe and Woodroofe (1999) argue that if X = n, then it is appropriate to base inferences on the conditional distribu

ISO* Property of Two-Parameter Compound Poisson Distributions with Applications

The ISO* property of noncentrality parameters is derived for the expected value of an ISO* function of independent nonnegative two-parameter compound Poisson random variables and is then applied to unbiasedness of tests and monotonicity of power functions of tests in an order-restricted hypothesis testing problem for the noncentrality parameter. The ISO* property and the Schur convexity are also studied for a class of two-parameter distributions which has the additive property (i.e., is closed under convolution) and contains the one-parameter family with the semigroup property as a special case.

Limit Theorems on Maximums of Poisson Sequences and Their Applications to Queuing Theory

We consider a sequence of independent Poisson random variables $X_n$ with parameters~$\lambda_n$. Asymptotic properties of maximums $Y_n=\max_{1\le i\le n}X_i$ under the dependence on properties of the sequence $\{\lambda_n\}$ are studied. The results obtained are applied to investigating the maximal number of claims in the system $M|G|\infty$ with parameters dependent on time.

Exact Properties of Demerit Control Charts

A demerit rating system is used to simultaneously monitor counts of several different types of defects in a complex product. The demerit statistic is a linear combination of the counts of these different types of defects. The traditional recommendation is to plot the demerit statistic on a control chart with symmetric 3-sigma control limits. This approach to demerit rating systems is reviewed. An alternative method for determining control limits for the demerit control chart, based on the exact distribution of linear combinations of independent Poisson random variables, is proposed. This method is generalized for use with the exact distribution of linear combinations of independent random variables in a broad family of discrete distributions.

Some properties of a two-parameter family of poisson distributions of orderk

We study the problem of simulating the process of detecting a weighted sum of independent Poisson random variables. We investigate the properties of the resulting compound Poisson distribution: the analytic form of the probability function and recursion formulas for computing it, moments and semi-invariants, asymptotics of the distribution, and recursion relations for the derivatives with respect to the parameters. We give the results of model computations showing the set structure of the distribution. One figure. Bibliography: 8 titles.

A statistical approach to the resolution of point sources

Summary The Rayleigh criterion in optics states that two point sources of equal intensity are 'barely resolved' when the maximum of the diffraction pattern of one source overlaps the first minimum of the diffraction pattern of the second source. Although useful for rough comparisons of optical systems, such a criterion does not take into account the randomness in the detection process and does not tell whether sources can actually be distinguished. We present a statistical approach that addressed these issues. From quantum optics, the photon counts in the pixels are independent Poisson random variables with means that depend on the distance 2theta between the sources. Resolving the sources corresponds to testing H0: theta =0 vs Ha: theta >0, under conditions that make the information number zero at theta =0. We define resolution as the (asymptotic) power function of the likelihood ratio test rather than as a single number. The asymptotic distribution of the test statistic is derived under H0 and under contiguous alternatives. The results are illustrated by an application to a sky survey to detect binary stars using the Hubble space telescope.

Confidence intervals for directly standardized rates: a method based on the gamma distribution.

We offer an approximation to central confidence intervals for directly standardized rates, where we assume that the rates are distributed as a weighted sum of independent Poisson random variables. Like a recent method proposed by Dobson, Kuulasmaa, Eberle and Scherer, our method gives exact intervals whenever the standard population is proportional to the study population. In cases where the two populations differ non-proportionally, we show through simulation that our method is conservative while other methods (the Dobson et al. method and the approximate bootstrap confidence method) can be liberal.

A Simple Method for Generating Correlated Binary Variates

Abstract Correlated binary data are frequently analyzed in studies of repeated measurements, reliability analysis, and others. In such studies correlations among binary variables are usually nonnegative. This article provides a simple algorithm for generating an arbitrary dimensional random vector of non-negatively correlated binary variables. In some frequently encountered situations the algorithm reduces to explicit expressions. The correlated binary variables are generated from correlated Poisson variables. The key idea lies in the property that any Poisson random variable can be expressed as a convolution of other independent Poisson random variables. The binary variables have desired correlations by sharing common independent Poisson variables.

Bayesian reconstruction of PET images: methodology and performance analysis

We describe a practical statistical methodology for the reconstruction of PET images. Our approach is based on a Bayesian formulation of the imaging problem. The data are modelled as independent Poisson random variables and the image is modelled using a Markov random field smoothing prior. We describe a sequence of calibration procedures which are performed before reconstruction: (i) calculation of accurate attenuation correction factors from re-projected Bayesian reconstructions of the transmission image; (ii) estimation of the mean of the randoms component in the data; and (iii) computation of the scatter component in the data using a Klein - Nishina-based scatter estimation method. The Bayesian estimate of the PET image is then reconstructed using a pre-conditioned conjugate gradient method. We performed a quantitation study with a multi-compartment chest phantom in a Siemens/CTI ECAT931 system. Using 40 1 min frames, we computed the ensemble mean and variance over several regions of interest from images reconstructed using the Bayesian and a standard filtered backprojection (FBP) protocol. The values for the region of interest were compared with well counter data for each compartment. These results show that the Bayesian protocol can produce substantial improvements in relative quantitation over the standard FBP protocol, particularly when short transmission scans are used. An example showing the application of the method to a clinical chest study is also given.

Likelihood ratio and a class of strong laws for the sequence of integer-valued random variables

In this paper, the notion of likelihood ratio, as a measure of the deviation between a sequence of integer-valued random variables and an independent random sequence with Poisson distribution is introduced, and a class of strong laws, expressed by inequalities, on certain sets determined by this notion, are obtained. A class of strong laws for a sequence of independent Poisson random variables are special cases of the results of this paper.

On the admissibility of ordered poisson parameter under the entropy loss function

Let X 1, X 2be two independent Poisson random variables with means θ 1and θ 2respectively. Assume 0 ≤ θ 1 ≤ θ 2 ≤ ∞. The problem is estimation of the ordered parameters which has received considerable attention in statistical literature during the last two decades, In this paper the main portion of the study is devoted to the problem of estimating the smallest of the two ordered Poisson means, when it is known which population corresponds to each mean under the entropy loss function. An extension of the problem to the k-ordered Poisson means is also discussed.