Parametric Estimation for Subordinators and Induced OU Processes

Abstract.  Consider a stationary sequence of random variables with infinitely divisible marginal law, characterized by its Lévy density. We analyse the behaviour of a so-called cumulant M-estimator, in case this Lévy density is characterized by a Euclidean (finite dimensional) parameter. Under mild conditions, we prove consistency and asymptotic normality of the estimator. The estimator is considered in the situation where the data are increments of a subordinator as well as the situation where the data consist of a discretely sampled Ornstein–Uhlenbeck (OU) process induced by the subordinator. We illustrate our results for the Gamma-process and the Inverse-Gaussian OU process. For these processes we also explain how the estimator can be computed numerically.     

Likelihood Inference for Unions of Interacting Discs

Abstract. This is probably the first paper which discusses likelihood inference for a random set using a germ‐grain model, where the individual grains are unobservable, edge effects occur and other complications appear. We consider the case where the grains form a disc process modelled by a marked point process, where the germs are the centres and the marks are the associated radii of the discs. We propose to use a recent parametric class of interacting disc process models, where the minimal sufficient statistic depends on various geometric properties of the random set, and the density is specified with respect to a given marked Poisson model (i.e. a Boolean model). We show how edge effects and other complications can be handled by considering a certain conditional likelihood. Our methodology is illustrated by analysing Peter Diggle's heather data set, where we discuss the results of simulation‐based maximum likelihood inference and the effect of specifying different reference Poisson models.     

Extreme Values and Haar Series Estimates of Point Process Boundaries

ABSTRACT.  We present a new method for estimating the edge of a two-dimensional bounded set, given a finite random set of points drawn from the interior. The estimator is based both on Haar series and extreme values of the point process. We give conditions for various kind of convergence and we obtain remarkably different possible limit distributions. We propose a method of reducing the negative bias, illustrated by a simulation.     

Non-parametric weighted tests for independence based on empirical copula process

We propose a class of flexible non-parametric tests for the presence of dependence between components of a random vector based on weighted Cramér–von Mises functionals of the empirical copula process. The weights act as a tuning parameter and are shown to significantly influence the power of the test, making it more sensitive to different types of dependence. Asymptotic properties of the test are stated in the general case, for an arbitrary bounded and integrable weighting function, and computational formulas for a number of weighted statistics are provided. Several issues relating to the choice of the weights are discussed, and a simulation study is conducted to investigate the power of the test under a variety of dependence alternatives. The greatest gain in power is found to occur when weights are set proportional to true deviations from independence copula.     

1961 Section Officers,Committees and Association Representatives

The multiple correlation coefficient, R (or r, in the simple case), is frequently used in evaluating regression models. Statistical significance is the usual criterion for judging R. Various views of its practical significance should also be considered. One practical measure is the percent reduction in the standard deviation of the response variable achieved by the model. A graph giving this as a function of R and percent loss in degrees of freedom is presented. This measure is compared with others which are sometimes appropriate.     

On binomial and circular binomial distributions of orderk forl-overlapping success runs of lengthk

The number ofl-overlapping success runs of lengthk inn trials, which was introduced and studied recently, is presently reconsidered in the Bernoulli case and two exact formulas are derived for its probability distribution function in terms of multinomial and binomial coefficients respectively. A recurrence relation concerning this distribution, as well as its mean, is also obtained. Furthermore, the number ofl-overlapping success runs of lengthk inn Bernoulli trials arranged on a circle is presently considered for the first time and its probability distribution function and mean are derived. Finally, the latter distribution is related to the first, two open problems regarding limiting distributions are stated, and numerical illustrations are given in two tables. All results are new and they unify and extend several results of various authors on binomial and circular binomial distributions of orderk.     

Distribution of a linear function of correlated ordered variables

In this paper, we have considered the problem of finding the distribution of a linear combination of the minimum and the maximum for a general bivariate distribution. The general results are used to obtain the required distribution in the case of bivariate normal, bivariate exponential of Arnold and Strauss, absolutely continuous bivariate exponential distribution of Block and Basu, bivariate exponential distribution of Raftery, Freund's bivariate exponential distribution and Gumbel's bivariate exponential distribution. The distributions of the minimum and maximum are obtained as special cases.     

Robust linear clustering

Summary.  Non-hierarchical clustering methods are frequently based on the idea of forming groups around 'objects'. The main exponent of this class of methods is the k -means method, where these objects are points. However, clusters in a data set may often be due to certain relationships between the measured variables. For instance, we can find linear structures such as straight lines and planes, around which the observations are grouped in a natural way. These structures are not well represented by points. We present a method that searches for linear groups in the presence of outliers. The method is based on the idea of impartial trimming. We search for the 'best' subsample containing a proportion 1− α of the data and the best k affine subspaces fitting to those non-discarded observations by measuring discrepancies through orthogonal distances. The population version of the sample problem is also considered. We prove the existence of solutions for the sample and population problems together with their consistency. A feasible algorithm for solving the sample problem is described as well. Finally, some examples showing how the method proposed works in practice are provided.     

A Third Order Point Process Characteristic

Second order characteristics, in particular Ripley's K -function, are widely used for the statistical analysis of point patterns. We examine a third order analogue, namely the mean number T ( r ) of r -close triples of points per unit area. Equivalently this is the expected number of r -close point pairs in an r -neighbourhood of the typical point. Various estimators for this function are proposed and compared, and we give an explicit formula for the isotropic edge correction. The theoretical value of T seems to be unobtainable for most point process models apart from the homogeneous Poisson process. However, simulation studies show that the function T discriminates well between different types of point processes. In particular it detects a clear difference between the Poisson process and the Baddeley–Silverman cell process whereas the K -functions for these processes coincide.     

Stereological formulas for stationary surface processes

In the paper surface processes are considered, i.e. sets of surfaces distributed at random in the space. For translation invariant surface processes the PALM distribution of the direction of the normal in a typical surface point is defined and related to two roses of intersection. Fibre and point processes are studied which result by intersections with planes and lines respectively. For these processes stereological formulas are proved which connect some quantities with such of the surface process. As a special ease motion invariant, i.e. translation invariant and isotropic, surface processes are treated, Furthermore, weighted surface processes are considered.     

An alternative approach to reliability analysis of cold standby systems

ABSTRACT

An alternative approach is applied for reliability analysis of standby systems on the basis of matrix renewal function. In this regard, a single-server, two identical unit cold standby systems with an imperfect switch is considered as a three-state semi-Markov process. Several important reliability measures such as availability, mean time to failure, expected number of failures, etc., are obtained for general lifetime distributions. Also, the main results have been treated to the case of exponential lifetimes and explicit formulas obtained for this case in addition of some numerical illustrations. This approach can easily be extended to more general standby systems with different configurations.     

Optimal Detection of a Change Point in a Poisson Process for Different Observation Schemes

Abstract.  Change point problems are considered where at some unobservable time the intensity of a point process ( T_n ), n ∈  N , has a jump. For a given reward functional we detect the change point optimally for different information schemes. These schemes differ in the available information. We consider three information levels, namely sequential observation of ( T_n ), ex post decision after observing the point process up to a fixed time t * and a combination of both observation schemes. In all of these cases the detection problem is viewed as an optimal stopping problem which can be solved by deriving a semimartingale representation of the gain process and applying tools from filtering theory.     

A Simple Derivation of r* for Curved Exponential Families

For curved exponential families we consider modified likelihood ratio statistics of the form r_L=r+ log( u/r)/r , where r is the signed root of the likelihood ratio statistic. We are testing a one-dimensional hypothesis, but in order to specify approximate ancillary statistics we consider the test as one in a series of tests. By requiring asymptotic independence and asymptotic normality of the test statistics in a large deviation region there is a particular choice of the statistic u which suggests itself. The derivation of this result is quite simple, only involving a standard saddlepoint approximation followed by a transformation. We give explicit formulas for the statistic u , and include a discussion of the case where some coordinates of the underlying variable are lattice.     

The power functions of the likelihood ratio tests for a simply ordered trend in normal means

Likelihood ratio tests for the homogeneity of k normal means with the alternative restricted by an increasing trend are considered as well as the likelihood ratio tests of the null hypothesis that the means satisfy the trend. While the work is primarily a survey of results concerning the power functions of these tests, the extensions of some results to the case of not necessarily equal sample sizes are presented. For the case of known or unknown population variances, exact expressions are given for the power functions for k=3,4, and approximations are discussed for larger k. The topics of consistency, bias and monotonicity of the power functions are included. Also, Bartholomew's conjectures concerning minimal and maximal powers are investigated, with results of a new numerical study given.     

Matrix-analytic Models and their Analysis

We survey phase-type distributions and Markovian point processes, aspects of how to use such models in applied probability calculations and how to fit them to observed data. A phase-type distribution is defined as the time to absorption in a finite continuous time Markov process with one absorbing state. This class of distributions is dense and contains many standard examples like all combinations of exponential in series/parallel. A Markovian point process is governed by a finite continuous time Markov process (typically ergodic), such that points are generated at a Poisson intensity depending on the underlying state and at transitions; a main special case is a Markov-modulated Poisson process. In both cases, the analytic formulas typically contain matrix-exponentials, and the matrix formalism carried over when the models are used in applied probability calculations as in problems in renewal theory, random walks and queueing. The statistical analysis is typically based upon the EM algorithm, viewing the whole sample path of the background Markov process as the latent variable.     

Average outgoing quality of CSP-C continuous sampling plan under short run production processes

A CSP-C continuous sampling plan is a new single-level continuous sampling procedure developed by Govindaraju & Kandasamy (2000) by incorporating the concept of acceptance number to the CSP-1 plan for the application of continuous production processes. In this new plan, the sampling inspection phase is characterized by a maximum allowable number of non-conforming units, c, and a constant sampling rate, f. Govindaraju & Kandasamy (2000) derived the performance measures such as average outgoing quality (AOQ), average fraction inspected (AFI) etc, of the CSP-C plan using a Markov chain model for long run production processes. Yang (1983) has observed that the AOQ and AFI, being long run average measures, are not satisfactory measures of performance for short run production processes. Hence, formulas are derived in this paper, using the renewal theory approach enabling one to compute AOQ and AFI for both long run and short run production processes. Numerical illustrations are also given. By simulation, the accuracy of the short run measures is studied.     

Dependent models for observations which include angular ones

This paper discusses some stochastic models for dependence of observations which include angular ones. First, we provide a theorem which constructs four-dimensional distributions with specified bivariate marginals on certain manifolds such as two tori, cylinders or discs. Some properties of the submodel of the proposed models are investigated. The theorem is also applicable to the construction of a related Markov process, models for incomplete observations, and distributions with specified marginals on the disc. Second, two maximum entropy distributions on the cylinder are discussed. The circular marginal of each model is distributed as the generalized von Mises distribution which represents a symmetric or asymmetric, unimodal or bimodal shape. The proposed cylindrical model is applied to two data sets.     

ABC Shadow algorithm: a tool for statistical analysis of spatial patterns

This paper presents an original ABC algorithm, ABC Shadow, that can be applied to sample posterior densities that are continuously differentiable. The proposed algorithm solves the main condition to be fulfilled by any ABC algorithm, in order to be useful in practice. This condition requires enough samples in the parameter space region, induced by the observed statistics. The algorithm is tuned on the posterior of a Gaussian model which is entirely known, and then, it is applied for the statistical analysis of several spatial patterns. These patterns are issued or assumed to be outcomes of point processes. The considered models are: Strauss, Candy and area-interaction.     

Random balanced resampling: A new method for estimating variance components in unbalanced designs

Variance components in factorial designs with balanced data are commonly estimated by equating mean squares to expected mean squares. For unbalanced data, the usual extensions of this approach are the Henderson methods, which require formulas that are rather involved. Alternatively, maximum likelihood estimation based on normality has been proposed. Although the algorithm for maximum likelihood is computationally complex, programs exist in some statistical packages. This article introduces a simpler method, that of creating a balanced data set by resampling from the original one. Revised formulas for expected mean squares are presented for the two-way case; they are easily generalized to larger factorial designs. The results of a number of simulation studies indicate that, in certain types of designs, the proposed method has performance advantages over both the Henderson Method I and maximum likelihood estimators.     

Formulas for stationary planar fibre processes III – intersections with fibre systems

In this paper we investigate the distribution of the periodogram, respectively, the periodogram matrix for stationary random sequences. These .distributions are consid¬ered in the case of a fixed frequency as well as in the case of a finite number of frequencies for Gaussian sequences and for sequences of independent random variables. The exact distribution is obtained in the case of a fixed frequence for one-dimensional GAUSsian sequences. Asymptotic expansions, respectively, the rate of convergence to the asymptotic distribution are given in the case mentioned above