Some nonparametric methods for changepoint problems

A general model for changepoint problems is discussed from a nonparametric viewpoint. The test statistics introduced are based on Cramér-von Mises functionals of certain processes and are shown to converge in distribution to corresponding Gaussian functionals (under the assumption of no change in distribution, H₀). We also demonstrate how the distribution of the limiting Gaussian functionals may be tabulated. Finally, properties of the tests under the alternative hypothesis of exactly one changepoint occurring are studied, and some examples are given.     

Computing the probability of hash table/urn overflow

We analyze the probability of a random distribution of n balls into m urns of size b resulting in no overflows. This solves the computational problem associated with a classical combinatorial extreme-value distribution. The problem arose during the analysis of a technique, called perfect hashing, for organizing data in computer files. The results and techniques presented can be used to solve several problems in the analysis of hashing techniques     

Algorithms

Abstract

The main reason for the limited use of multivariate discrete models is the difficulty in calculating the required probabilities. The task is usually undertaken via recursive relationships which become quite computationally demanding for high dimensions and large values. The present paper discusses efficient algorithms that make use of the recurrence relationships in a manner that reduces the computational effort and thus allow for easy and cheap calculation of the probabilities. The most common multivariate discrete distribution, the multivariate Poisson distribution is treated. Real data problems are provided to motivate the use of the proposed strategies. Extensions of our results are discussed. It is shown that probabilities, for a large family of multivariate distributions, can be computed efficiently via our algorithms.     

A MOMENT IDENTITY

A moment identity based on the concept of weighted distributions is presented. The identity has potential applications to problems involving the manipulation of expectations of functions of random variables. Several examples, mainly concerning the evaluation of the risk of shrinkage estimators of several parameters, are given.     

Causal Reasoning from Longitudinal Data*

Abstract. This paper reviews some of the key statistical ideas that are encountered when trying to find empirical support to causal interpretations and conclusions, by applying statistical methods on experimental or observational longitudinal data. In such data, typically a collection of individuals are followed over time, then each one has registered a sequence of covariate measurements along with values of control variables that in the analysis are to be interpreted as causes, and finally the individual outcomes or responses are reported. Particular attention is given to the potentially important problem of confounding. We provide conditions under which, at least in principle, unconfounded estimation of the causal effects can be accomplished. Our approach for dealing with causal problems is entirely probabilistic, and we apply Bayesian ideas and techniques to deal with the corresponding statistical inference. In particular, we use the general framework of marked point processes for setting up the probability models, and consider posterior predictive distributions as providing the natural summary measures for assessing the causal effects. We also draw connections to relevant recent work in this area, notably to Judea Pearl's formulations based on graphical models and his calculus of so‐called do‐probabilities. Two examples illustrating different aspects of causal reasoning are discussed in detail.     

On the robustness of the extreme deviate test for a single multivariate outlier against heavy-tailed distributions

In this paper we assess the sensitivity of the multivariate extreme deviate test for a single multivariate outlier to non-normality in the form of heavy tails. We find that the empirical significance levels can be markedly affected by even modest departures from multivariate normality. The effects are particularly severe when the sample size is large relative to the dimension. Finally, by way of example we demonstrate that certain graphical techniques may prove useful in identifying the source of rejection for the multivariate extreme deviate test.     

Time series analysis of non-Gaussian observations based on state space models from both classical and Bayesian perspectives

The analysis of non-Gaussian time series by using state space models is considered from both classical and Bayesian perspectives. The treatment in both cases is based on simulation using importance sampling and antithetic variables; Markov chain Monte Carlo methods are not employed. Non-Gaussian disturbances for the state equation as well as for the observation equation are considered. Methods for estimating conditional and posterior means of functions of the state vector given the observations, and the mean-square errors of their estimates, are developed. These methods are extended to cover the estimation of conditional and posterior densities and distribution functions. The choice of importance sampling densities and antithetic variables is discussed. The techniques work well in practice and are computationally efficient. Their use is illustrated by applying them to a univariate discrete time series, a series with outliers and a volatility series.     

Robust methods for personal-income distribution models

Statistical problems in modelling personal-income distributions include estimation procedures, testing, and model choice. Typically, the parameters of a given model are estimated by classical procedures such as maximum-likelihood and least-squares estimators. Unfortunately, the classical methods are very sensitive to model deviations such as gross errors in the data, grouping effects, or model misspecifications. These deviations can ruin the values of the estimators and inequality measures and can produce false information about the distribution of the personal income in a country. In this paper we discuss the use of robust techniques for the estimation of income distributions. These methods behave like the classical procedures at the model but are less influenced by model deviations and can be applied to general estimation problems.     

On Calculations Involving the Maximum Cell Frequency

We describe a method of computing the cumulative distribution function of the maximum and minimum cell frequencies in sampling distributions commonly encountered in the analysis of categorical data.The procedure is efficient for exact or approximate calculation in both homogeneous and non-homogeneous cases, is non-recursive, and does not require Dirichlet integrals.Some related statistical problems are also discussed.     

Survival distributions arising from two families and generated by transformations

Two families of distributions are introduced and studied within the framework of parametric survival analysis. The families are derived from a general linear form by specifying a function of the survival function with certain restrictions. Distributions within each family are generated by transformations of the survival time variable subject to certain restrictions. Two specific transformations were selected and, thus, four distributions are identified for further study. The distributions have one scale and two shape parameters and include as special cases the exponential, Weibull, log-logistic and Gompertz distributions. One of the new distributions, the modified Weibull, is studied in some detail.

The distributions are developed with an emphasis on those features that data analysts find especially useful for survivorship studies, A wide variety of hazard shapes are available. The survival, density and hazard functions may be written in simple algebraic forms. Parameter estimation is demonstrated using the least squares and maximum likelihood methods. Graphical techniques to assess goodness of fit are demonstrated. The models may be extended to include concmitant information.     

Inference for Reliability and Stress-Strength for a Scaled Burr Type X Distribution

Inferencefor R=P(Y is considered when Xand Y are independently distributed as scaled Burrtype X random variables. Under this model, exact inference proceduresfor R cannot be found. Hence, based on the expectedFisher information matrix which is derived here, asymptotic inferenceprocedures for R and other general functions ofthe parameters are developed. A bootstrap method to estimatevariance for the maximum likelihood estimators is also discussed.To illustrate these techniques, an example using carbon fiberstrength data is given. Simulations to assess the effectivenessof these techniques, as well as other concerns, are presented.     

Distributions of Runs Revisited

Distributions of runs have important applications in many fields, including biological sequence analysis. The generating function (GF) method provides a unified approach to tackle different run-related problems in multistate trials. By utilizing this method, various run-related distributions are derived in a systematic way for both conditional and unconditional models. The GF approach also naturally yields the asymptotic distributions. For all the distributions considered, the limiting distributions are Gaussian, with mean, variance, and covariance linear functions in the size of the system.     

On approximating point spread distributions

We discuss some properties of the point spread distribution, defined as the distribution of the difference of two independent binomial random variables with the same parameter n including exact and approximate probabilities and related optimization issues. We use various approximation techniques for different distributions, special functions, and analytic, combinatorial and symbolic methods, such as multi-summation techniques. We prove that in case of unequal success rates, if these rates change with their difference kept fixed and small, and n is appropriately bounded, then the point spread distribution only slightly changes for small point differences. We also prove that for equal success rates p, the probability of a tie is minimized if p=1/2. Numerical examples are included for the case with n=12.     

Linear latent structure analysis: Mixture distribution models with linear constraints

A new method for analyzing high-dimensional categorical data, Linear Latent Structure (LLS) analysis, is presented. LLS models belong to the family of latent structure models, which are mixture distribution models constrained to satisfy the local independence assumption. LLS analysis explicitly considers a family of mixed distributions as a linear space, and LLS models are obtained by imposing linear constraints on the mixing distribution.LLS models are identifiable under modest conditions and are consistently estimable. A remarkable feature of LLS analysis is the existence of a high-performance numerical algorithm, which reduces parameter estimation to a sequence of linear algebra problems. Simulation experiments with a prototype of the algorithm demonstrated a good quality of restoration of model parameters.     

Tables for computing shortest confidence intervals involving the f-distribution

Tables of multipliers necessary for computing the shortest confidence interval in situations involving the F-distribution are presented. Such situations are illustrated with several examples.     

Some distributions arising from faulty inspection with multitype defectives,and an application to grading

Continuing their investigation of distributions arising from faulty and "incomplete" inspection from a finite popu-lation (Johnson et al. (1980), Comm. Statist.A9 917-922, Kotz and Johnson, ibid, All, 1997-2016) the authors investigate cases when inspection is not restricted to a single type of defect. Applications to grading problems, when grading is based on sets of defects found by sampling inspections, are indicated.     

UMVU estimation of the reliability function of the generalized life distributions

The problems of estimating the reliability function and P=P_rX > Y are considered for the generalized life distributions. Uniformly minimum variance unbiased estimators (UMVUES) of the powers of the parameter involved in the probabilistic model and the probability density function (pdf) at a specified point are derived. The UMVUE of the pdf is utilized to obtain the UMVUE of the reliability function and ‘P’. Our method of obtaining these estimators is quite simple than the traditional approaches. A theoretical method of studying the behaviour of the hazard-rate is provided.     

Errors in inspection and grading distributional aspects ofscreening and merarchal screening

Continuing the studies of Johnson et al (1980) and Johnson and Kotz (1981), further distributions arising from models of errors in inspection and grading of samples from finite, possibly stratified lots are obtained. Screening, and hierarchal screening forms of inspection are also considered, and the effects of errors on the advantages of these techniques assessed.