Showing papers in "Scandinavian Journal of Statistics in 1997"

Normal Inverse Gaussian Distributions and Stochastic Volatility Modelling

Abstract: The normal inverse Gaussian distribution is defined as a variance-mean mixture of a normal distribution with the inverse Gaussian as the mixing distribution. The distribution determines an homogeneous Levy process, and this process is representable through subordination of Brownian motion by the inverse Gaussian process. The canonical, Levy type, decomposition of the process is determined. As a preparation for developments in the latter part of the paper the connection of the normal inverse Gaussian distribution to the classes of generalized hyperbolic and inverse Gaussian distributions is briefly reviewed. Then a discussion is begun of the potential of the normal inverse Gaussian distribution and Levy process for modelling and analysing statistical data, with particular reference to extensive sets of observations from turbulence and from finance. These areas of application imply a need for extending the inverse Gaussian Levy process so as to accommodate certain, frequently observed, temporal dependence structures. Some extensions, of the stochastic volatility type, are constructed via an observation-driven approach to state space modelling. At the end of the paper generalizations to multivariate settings are indicated.

Estimation of an Ergodic Diffusion from Discrete Observations

Abstract: We consider a one-dimensional diffusion process X, with ergodic property, with drift b(x, ) and diffusion coefficient a(x, U) depending on unknown parameters and U. We are interested in the joint estimation of ( , U). For that purpose, we dispose of a discretized trajectory, observed at n equidistant times t n a ihn ,1 < i < n. We assume that hn! 0 and nhn!1. Under the condition nh p ! 0 for an arbitrary integer p ,w e exhibit a contrast dependent on p which provides us with an asymptotically normal and efficient estimator of ( , U).

Multiple Hypotheses Testing with Weights

Abstract: In this paper we offer a multiplicity of approaches and procedures for multiple testing problems with weights Some rationale for incorporating weights in multiple hypotheses testing are discussed Various type-I error-rates and different possible formulations are considered, for both the intersection hypothesis testing and the multiple hypotheses testing problems An optimal per family weighted error-rate controlling procedure a la Spjotvoll (1972) is obtained This model serves as a vehicle for demonstrating the different implications of the approaches to weighting Alternative approach es to that of Holm (1979) for family-wise error-rate control with weights are discussed, one involving an alternative procedure for family-wise error-rate control, and the other involving the control of a weighted family-wise error-rate Extensions and modifications of the procedures based on Simes (1986) are given These include a test of the overall intersec tion hypothesis with general weights, and weighted sequentially rejective procedures for testing the individual hypotheses The false discovery rate controlling approach and procedure of Benjamini & Hochberg (1995) are extended to allow for different weights

Local Polynomial Estimation of Regression Functions for Mixing Processes

Abstract: Local polynomial fitting has many exciting statistical properties which where established under i.i.d. setting. However, the need for non-linea r time series modeling, constructing predictive intervals, understanding divergence of non-linear time series requires the development of the theory of local polynomial fitting for dependent data. In this paper, we study the problem of estimating conditional mean functions and their derivatives via a local polynomial fit. The functions include conditional moments, conditional distribution as well as conditional density functions. Joint asymptotic normality for derivative estimation is established for both strongly mixing and ρ-mixing processes.

Likelihood Analysis of the I(2) Model

Abstract: The I(2) model is defined as a submodel of the general vector autoregressive model, by two reduced rank conditions. The model describes stochastic processes with stationary second difference. A parametrization is suggested which makes likelihood inference feasible. Consistency of the maximum likelihood estimator is proved, and the asymptotic distribution of the maximum likelihood estimator is given. It is shown that the asymptotic distribution is either Gaussian, mixed Gaussian or, in some cases, even more complicated.

Inference and Missing Data: Asymptotic Results

Abstract: In Rubin (1976) the missing at random (MAR) and missing completely at random (MCAR) conditions are discussed. It is concluded that the MAR condition allows one to ignore the missing data mechanism when doing likelihood or Bayesian inference but also that the stronger MCAR condition is in some sense the weakest generally sufficient condition allowing (conditional) frequentist inference while ignoring the missing data mechanism. In this paper it is shown that (a slightly strengthened version of) the MAR condition is sufficient to yield ordinary large sample results for estimators and test statistics and thus may be used for (asymptotic) frequentist inference.

On the Markov Equivalence of Chain Graphs, Undirected Graphs, and Acyclic Digraphs

Abstract: Graphical Markov models use undirected graphs (UDGs), acyclic directed graphs (ADGs), or (mixed) chain graphs to represent possible dependencies among random variables in a multivariate distribution. Whereas a UDG is uniquely determined by its associated Markov model, this is not true for ADGs or for general chain graphs (which include both UDGs and ADGs as special cases). This paper addresses three questions regarding the equivalence of graphical Markov models: when is a given chain graph Markov equivalent (1) to some UDG? (2) to some (at least one) ADG? (3) to some decomposable UDG? The answers are obtained by means of an extension of Frydenberg’s (1990) elegant graph-theoretic characterization of the Markov equivalence of chain graphs.

Simultaneous Confidence Intervals for the Difference of Two Survival Functions

Abstract: In comparing two treatments with failure time observations, confidence bands for the "difference" of two survival curves provide useful information about a global picture of the treatment difference over time. In this note, we propose a rather simple procedure for constructing such simultaneous confidence intervals. Our technique can also be used in the one-sample case, which has been extensively studied in the literature. In comparing two survival functions S1(.) and S2(.), one usually plots their Kaplan-Meier estimates S2(.) and S,(.), visually examines these curves, and subjectively summarizes how the difference D(.) (= S2(.) - SI (.)) varies over time. In this note, we show a rather simple technique for constructing purely non-parametric confidence bands of D(.). These bands provide an objective way to evaluate the "treatment" difference quantified by D(.) over time. Under the proportional hazards model assumption, asymptotic simultaneous confidence intervals for such a difference have been obtained by Dabrowska et al. (1989). Recently, with the general Cox regression model, confidence bands of the survival function of the patient with a specific set of covariates have been proposed by Lin et al. (1994). For the one-sample case, various confidence bands for the survival function are derived, for example, by Gill (1980), Hall & Wellner (1980), Nair (1984), Csbrgo & Horvath (1986), and Hollander & Peiia (1989). Excellent reviews on this subject are given by Fleming & Harrington (1991, sec. 6.3) and Andersen et al. (1993, sec. IV 1.3). Our new technique can also be used to construct aforementioned confidence bands without referring to any special distribution table.

A Bayesian non-parametric approach to survival analysis using Polya trees

Abstract: This paper presents a Bayesian non-parametric approach to survival analysis based on arbitrarily right censored data. The analysis is based on posterior predictive probabilities using a Polya tree prior distribution on the space of probability measures on [0, ∞). In particular we show that the estimate generalizes the classical Kaplanndash;Meier non-parametric estimator, which is obtained in the limiting case as the weight of prior information tends to zero.

Optimal Change-point Estimation in Inverse Problems

Abstract: We develop a method of estimating a change-point of an otherwise smooth function in the case of indirect noisy observations. As two paradigms we consider deconvolution and non-parametric errors-in-variables regression. In a similar manner to well-established methods for estimating change-points in non-parametric regression, we look essentially at the difference of one-sided kernel estimators. Because of the indirect nature of the observations we employ deconvoluting kernels. We obtain an estimate of the change-point by the extremal point of the differences between these two-sided kernel estimators. We derive rates of convergence for this estimator. They depend on the degree of ill-posedness of the problem, which derives from the smoothness of the error density. Analysing the Hellinger modulus of continuity of the problem we show that these rates are minimax

Infinite Parameter Estimates in Logistic Regression, with Application to Approximate Conditional Inference

Abstract: This paper discusses recovery of information regarding logistic regression parameters in cases when maximum likelihood estimates of some parameters are infinite. An algorithm for detecting such cases and characterizing the divergence of the parameter estimates is presented. A method for fitting the remaining parameters is also presented. All of these methods rely only on sufficient statistics rather than less aggregated quantities, as required for inference according to the method of Kolassa & Tanner (1994). These results are applied to approximate conditional inference via saddlepoint methods. Specifically, the double saddlepoint method of Skovgaard (1987) is adapted to the case when the solution to the saddlepoint equations exists as a point at infinity.

Simple Transformation Techniques for Improved Non‐parametric Regression

Abstract: We propose and investigate two new methods for achieving less bias in non- parametric regression. We show that the new methods have bias of order h4, where h is a smoothing parameter, in contrast to the basic kernel estimator’s order h2. The methods are conceptually very simple. At the first stage, perform an ordinary non-parametric regression on {xi, Yi} to obtain m^(xi) (we use local linear fitting). In the first method, at the second stage, repeat the non-parametric regression but on the transformed dataset {m^(xi, Yi)}, taking the estimator at x to be this second stage estimator at m^(x). In the second, and more appealing, method, again perform non-parametric regression on {m^(xi, Yi)}, but this time make the kernel weights depend on the original x scale rather than using the m^(x) scale. We concentrate more of our effort in this paper on the latter because of its advantages over the former. Our emphasis is largely theoretical, but we also show that the latter method has practical potential through some simulated examples.

A Bounded Derivative Model for Prior Ignorance about a Real-valued Parameter

Abstract: A new method is proposed for drawing coherent statistical inferences about a real-valued parameter in problems where there is little or no prior information. Prior ignorance about the parameter is modelled by the set of all continuous probability density functions for which the derivative of the log-density is bounded by a positive constant. This set is translation-invariant, it contains density functions with a wide variety of shapes and tail behaviour, and it generates prior probabilities that are highly imprecise. Statistical inferences can be calculated by solving a simple type of optimal control problem whose general solution is characterized. Detailed results are given for the problems of calculating posterior upper and lower means, variances, distribution functions and probabilities of intervals. In general, posterior upper and lower expectations are achieved by prior density functions that are piecewise exponential. The results are illustrated by normal and binomial examples

Singly and Doubly Censored Current Status Data: Estimation, Asymptotics and Regression

Abstract: In biostatistical applications interest often focuses on the estimation of the distribution of time between two consecutive events. If the initial event time is observed and the subsequent event time is only known to be larger or smaller than an observed point in time, then the data is described by the well-understood singly censored current status model, also known as interval censored data, case I. Jewell et al. (1994) extended this current status model by allowing the initial time to be unobserved, with its distribution over an observed interval (A, Bl known; the data is referred to as doubly censored current status data. This model has applications in AIDS partner studies. If the initial time is known to be uniformly distributed, the model reduces to a submodel of the current status model with the same asymptotic information bounds as in the current status model, but the distribution of interest is essentially the derivative of the distribution of interest in the current status model. As a consequence the non-parametric maximum likelihood estimator is inconsistent. Moreover, this submodel contains only smooth heavy tailed distributions for which no moments exist. In this paper, we discuss the connection between the singly censored current status model and the doubly censored current status model (for the uniform initial time) in detail and explain the difficulties in estimation which arise in the doubly censored case. We propose a regularized MLE corresponding with the current status model. We prove rate results, efficiency of smooth functionals of the regularized MLE, and present a generally applicable efficient method for estimation of regression parameters, which does not rely on the existence of moments. We also discuss extending these ideas to a non-uniform distribution for the initial time. In many longitudinal studies, interest focuses on the distribution of the length of the interval between two events. This situation is particularly common in epidemiological investigations of the natural history of a disease. Jewell et al. (1994) give two examples that arise from studies of human immunodeficiency virus (HIV) disease. Notationally, let I and J be the chronological times of the two events, respectively, with I - J always. Suppose that, for a given individual, we observe whether either of both of the two events have occurred at a single point in time, B. We assume that only individuals for whom I - B, that is, the first event has occurred by time B, are included in the sample of individuals available for study. We are interested in two forms of data structure; in the first I (- B) is always observed and measurement of J is censored in that we only observe whether J B. For the second situation, measurement of both I and J is limited to observing whether each of these times is greater than or less than B. In both cases we are concerned with non-parametric esti- mation of the distribution G of T = J - I, based on n i.i.d. observations, under the assumption that T is independent of I. The first structure is commonly referred to as (singly censored) current status data. The qualifier "singly censored" serves to differentiate this structure from the second kind discussed in further detail below. Previous work and examples of current status data can be found in Diamond et al. (1986), Jewell & Shiboski (1990), Diamond & McDonald (1991), and Keiding

Testing for Imperfect Debugging in Software Reliability

Abstract: This paper continues the study of the software reliability model of Fakhre- Zakeri & Slud (1995), an "exponential order statistic model" in the sense of Miller (1986) with general mixing distribution, imperfect debugging and large-sample asymptotics reflect- ing increase of the initial number of bugs with software size. The parameters of the model are 0 (proportional to the initial number of bugs in the software), G(., p) (the mixing df, with finite dimensional unknown parameter , for the rates A)i with which the bugs in the software cause observable system failures), and p (the probability with which a detected bug is instantaneously replaced with another bug instead of being removed). Maximum likelihood estimation theory for (0,p,p) is applied to construct a likelihood-based score test for large sample data of the hypothesis of "perfect debugging" (p = 0) vs "imperfect" (p > 0) within the models studied. There are important models (including the Jelinski-Moranda) under which the score statistics with 1/V/i normalization are asymptotically degenerate. These statistics, illustrated on a software reliability data of Musa (1980), can serve nevertheless as important diagnostics for inadequacy of simple models.

A simple derivation of r * for curved exponential families

Abstract: For curved exponential families we consider modified likelihood ratio statistics of the form rL=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.

A Loss Function Model for the Restoration of Grey Level Images

Abstract: Common loss functions used for the restoration of grey scale images include the zero-one loss and the sum of squared errors. The corresponding estimators, the posterior mode and the posterior marginal mean, are optimal Bayes estimators with respect to their way of measuring the loss for different error configurations. However, both these loss functions have a fundamental weakness: the loss does not depend on the spatial structure of the errors. This is important because a systematic structure in the errors can lead to misinterpretation of the estimated image. We propose a new loss function that also penalizes strong local sample covariance in the error and we discuss how the optimal Bayes estimator can be estimated using a two-step Markov chain Monte Carlo and simulated annealing algorithm. We present simulation results for some artificial data which show improvement with respect to small structures in the image.

A Berry–Esséen Bound for M‐estimators

Abstract: We prove a Berry–Esseen bound for general M-estimators under optimal regularity conditions on the score function and the underlying distribution. As an application we obtain Berry–Esseen bounds for the sample median, the Lp-median, p > 1 and Huber's estimator of location

Conditioning of Marked Point Processes within a Bayesian Framework

Abstract: Shale units with low permeability create barriers to fluid flow in a sandstone reservoir. A spatial stochastic model for the location of shale units in a reservoir is defined. The model is based on a marked point process formulation, where the marks are parameterized by random functions for the shape of a shale unit. This extends the traditional formulation in the sense that conditioning on the actual observations of the shale units is allowed in an arbitrary number of wells penetrating the reservoir. The marked point process for the shale units includes spatial interaction of units and allows a random number of units to be present. The model is defined in a Bayesian setting with prior pdfs assigned to size–shape parameters of shale units. The observations of shales in wells are associated with a likelihood function. The posterior pdf of the marked point process can only partially be developed analytically; the final solution must be determined by sampling using the Metropolis–Hastings algorithm. An example is presented, demonstrating the consequences of increasing the number of wells in which observations are made.

Expected Survival in the Cox Model

Abstract: The survival in a group of patients is calculated using a Kaplan-Meier curve and the hypothesis that the survival can be predicted by a specified Cox hazard is studied. This Cox hazard is obtained from a previous study of similar patients. A simple test of the hypothesis is constructed by comparing a suitable average of the individual predicted survival curves with the observed survival. Three averaging procedures are presented; "direct adjusted survival curve", "Bonsel's survival curve" and "expected survival curve". Consistency and asymptotic distribution properties of the comparisons are discussed.

Estimating Mixing Densities in Exponential Family Models for Discrete Variables

Abstract: This paper is concerned with estimating a mixing density gusing a random sample from the mixture distribution f(x)=∫f x | θ)g(θ)dθ where f(· | θ)is a known discrete exponen tial family of density functions. Recently two techniques for estimatingghave been proposed. The first uses Fourier analysis and the method of kernels and the second uses orthogonal polynomials. It is known that the first technique is capable of yielding estimators that achieve (or almost achieve) the minimax convergence rate. We show that this is true for the technique based on orthogonal polynomials as well. The practical implementation of these estimators is also addressed. Computer experiments indicate that the kernel estimators give somewhat disappoint ing finite sample results. However, the orthogonal polynomial estimators appear to do much better. To improve on the finite sample performance of the orthogonal polynomial estimators, a way of estimating the optimal truncation parameter is proposed. The resultant estimators retain the convergence rates of the previous estimators and a Monte Carlo finite sample study reveals that they perform well relative to the ones based on the optimal truncation parameter.

A score test for the multivariate Burr and other Weibull mixture distributions

Abstract: The p-variate Burr distribution has been derived, developed, discussed and deployed by various authors. In this paper a score statistic for testing independence of the components, equivalent to testing for p independent Weibull against a p-variate Burr alternative, is obtained. Its null and non-null properties are investigated with and without nuisance parameters and including the possibility of censoring. Two applications to real data are described. The test is also discussed in the context of other Weibull mixture models.

A comparison of the cost-efficiencies of the sequential, group-sequential, and variable-sample-size-sequential probability ratio tests

Abstract: Wald and Wolfowitz (1948) have shown that the Sequential Probability Ratio Test (SPRT) for deciding between two simple hypotheses is, under very restrictive conditions, optimal in three attractive senses. First, it can be a Bayes-optimal rule. Second, of all level α tests having the same power, the test with the smallest joint-expected number of observations is the SPRT, where this expectation is taken jointly with respect to both data and prior over the two hypotheses. Third, the level α test needing the fewest conditional-expected number of observat ions is the SPRT, where this expectation is now taken with respect to the data conditional on either hypothesis being true. Principal among the strong restrictions is that sampling can proceed only in a one-at-a-time manner. In this paper, we relax some of the conditions and show that there are sequential procedures that strictly dominate the SPRT in all three senses. We conclude that the third type of optimality occurs rarely and that decision-makers are better served by looking for sequential procedures that possess the first two types of optimality. By relaxing the one-at-a-time sampling restriction, we obtain optimal (in the first two senses) variable-s ample-size- sequential probability ratio tests.

A non-parametric test for generalized first-order autoregressive models

Abstract: We derive a non-parametric test for testing the presence of V(Xi,ei) in the non-parametric first-order autoregressive model Xi+1=T(Xi)+V(Xi,ei)+U(Xi)ei+1, where the function T(x) is assumed known. The test is constructed as a functional of a basic process for which we establish a weak invariance principle, under the null hypothesis and under stationarity and mixing assumptions. Bounds for the local and non-local powers are provided under a condition which ensures that the power tends to one as the sample size tends to infinity.The testing procedure can be applied, e.g. to bilinear models, ARCH models, EXPAR models and to some other uncommon models. Our results confirm the robustness of the test constructed in Ngatchou Wandji (1995) and in Diebolt & Ngatchou Wandji (1995).

Bayesian System Reliability Prediction

Abstract: In system reliability studies a common problem is the coherent assessment of system reliability, based on generic database information from components and on failure data from a system and its components in a common, but unknown, environment. A solution to this problem is given.

Estimation of Diffusion Processes by Simulated Moment Methods

Abstract: We consider the parameter estimation of a diffusion process and we suppose that the trend and the diffusion coefficient depend on the parameter θ. The process is observed at time (ti)i=0,...,n with Δ = ti+1−ti fixed and we propose here to estimate θ from simulated moment methods.

Reduced Rank Discrimination

Abstract: In this paper the problem of classifying an individual with p characteristics into one of k multivariate normal distributions with common unknown covariance matrix is considered when the matrix of (k+1) means has a linear structural relationship, that is, it lies in an r-dimensional plane, where r

Testing Covariate Effects in Aalen’s Linear Hazard Model

Abstract: The performance of tests in Aalen’s linear regression model is studied using asymptotic power calculations and stochastic simulation. Aalen’s original least squares test is compared to two modifications: a weighted least squares test with correct weights and a test where the variance is re-estimated under the null hypothesis. The test with re-estimated variance provides the highest power of the tests for the setting of this paper, and the gain is substantial for covariates following a skewed distribution like the exponential. It is further shown that Aalen’s choice for weight function with re-estimated variance is optimal in the one-parameter case against proportional alternatives.

On the Empirical Likelihood Ratio for Smooth Functions of M‐functionals

Abstract: It is known that the empirical likelihood ratio can be used to construct confidence regions for smooth functions of the mean, Frechet differentiable statistical functionals and for a class of M-functionals. In this paper, we argue that this use can be extended to the class of functionals which are smooth functions of M-functionals. In particular, we find the conditions under which the empirical log-likelihood ratio for this kind of functionals admits a χ2 approxima tion. Furthermore, we investigate, by simulation methods, the related approximation error in some contexts of practical interest.

Existence of Maximum Likelihood Estimates for Multi-dimensional Exponential Families

Abstract: This paper deals with the existence of maximum likelihood estimators for multi-dimensional exponential families, including curved exponential families. It first gives an algorithm for determining the MLE from the data. Then it establishes that when the parameter set is either open or relatively closed in the natural parameter set, the MLE of the parameter exists in the sense of Hoffmann-Jorgensen.