Sequential Changepoint Detection in Quality Control and Dynamical SystemsLai, Tze Leung
doi: 10.1111/j.2517-6161.1995.tb02052.xpmid: N/A
After a brief survey of a large variety of sequential detection procedures that are widely scattered in statistical references on quality control and engineering references on fault detection and signal processing, we study some open problems concerning these procedures and introduce a unified theory of sequential changepoint detection. This theory leads to a class of sequential detection rules which are not too demanding in computational and memory requirements for on‐line implementation and yet are nearly optimal under several performance criteria.
Maximum Likelihood Estimation of the Differencing Parameter for Invertible Short and Long Memory Autoregressive Integrated Moving Average ModelsBeran, Jan
doi: 10.1111/j.2517-6161.1995.tb02054.xpmid: N/A
In practical applications of Box‐Jenkins autoregressive integrated moving average (ARIMA) models, the number of times that the observed time series must be differenced to achieve approximate stationarity is usually determined by careful, but mostly informal, analysis of the differenced series. For many time series, some differencing seems appropriate, but taking the first or the second difference may be too strong. As an alternative, Hosking, and Granger and Joyeux proposed the use of fractional differences. For ‐1/2 < d < 1/2, the resulting fractional ARIMA processes are stationary. For 0 < d < 1/2, the correlations are not summable. The parameter d can be estimated, for instance by maximum likelihood. Unfortunately, estimation methods known so far have been restricted to the stationary range ‐1/2 < d < 1/2. In this paper, we show how any real d > ‐1/2 can be estimated by an approximate maximum likelihood method. We thus obtain a unified approach to fitting traditional Box‐Jenkins ARIMA processes as well as stationary and non‐stationary fractional ARIMA processes. A confidence interval for d can be given. Tests, such as for unit roots in the autoregressive parameter or for stationarity, follow immediately. The resulting confidence intervals for the ARMA parameters take into account the additional uncertainty due to estimation of d. A simple algorithm for calculating the estimate of d and the ARMA parameters is given. Simulations and two data examples illustrate the results.
Incorporating Parametric Effects into Functional Principal Components AnalysisSilverman, B. W.
doi: 10.1111/j.2517-6161.1995.tb02055.xpmid: N/A
The ideas of functional principal component analysis are extended to deal with data that are hybrids of ‘functional’ and ‘parametric’ effects. The parametric effects may be more general than just the addition of a multiple of a given function. A detailed development is given in the case of shifts of the time axis for functions observed on a periodic interval, and some remarks are made for the extension to a far more general case. Given data, a Procrustes fitting method can be used to estimate the parametric effects. Several possible ways of treating the estimated parameter values are discussed. The methods are illustrated by reference to temperature data at 35 Canadian weather‐stations.
A Class of Bayesian Models for Optimal ExplorationGlazebrook, K. D.; Boys, R. J.
doi: 10.1111/j.2517-6161.1995.tb02057.xpmid: N/A
Each of several locations contains an unknown number of objects of value. A single search of a location will discover some of these objects which are then removed. Discoveries yield rewards. A ‘distribution of effort’ problem is posed concerning how to explore the locations optimally—i.e. to maximize the expected return from discoveries made. This is formulated as a Bayes sequential decision problem for which index policies are optimal. A natural simple case is one in which, conditionally on the number of undiscovered objects at a location N, the number of discoveries made in a single search is binomial B(N, p) where p is a detection rate. For this case, we can gain considerable insight into how model structure relates to policy structure. The tail behaviour of the priors for the number of objects at each location plays an important role.
Equivalence with Respect to a Control: Stepwise TestsBofinger, Eve; Bofinger, Mark
doi: 10.1111/j.2517-6161.1995.tb02058.xpmid: N/A
The problem of selecting those treatments which are equivalent (or bioequivalent) to a control treatment is investigated by using single‐step, step‐down and step‐up multiple‐testing procedures. For each of the tests considered the null hypothesis is that of non‐equivalence and the hypotheses are rejected in either a single step or a stepwise fashion while controlling the family wise error rate. The single‐step procedure is based on ‘expanded’ confidence intervals, as discussed by Bofinger, the step‐down procedure is based on the method of Naik and the step‐up procedure on the method of Dunnett and Tamhane. Almost all the critical constants used in the procedures are based on tables already published.
The Empirical Process Approach for Semiparametric Two‐Sample Models with Heterogeneous Treatment EffectHsieh, Fushing
doi: 10.1111/j.2517-6161.1995.tb02059.xpmid: N/A
Two types of semiparametric model are proposed for comparing two samples with possibly heterogeneous treatment effects. One model is a two‐sample location‐scale model which assumes that the Q‐Q‐plot of the two distributions involved is linear; the other is the class of two‐sample transformation models which assume parametric forms for a probability plot—the receiver operating characteristic curve. The empirical process approach is used to construct strong approximations for the empirical curves of both types of plot and a convenient generalized least squares (GLS) estimator is derived. The approximate finite sample precision of the estimators can be easily evaluated. Asymptotically, the GLS estimator is shown to be efficient for the first model. A difficulty involved in developing the theory of asymptotic efficiency is discussed for the transformation models. The GLS estimator is shown numerically to approach the Fisher information bound for the proportional hazards model.
Laplace Approximation of High Dimensional IntegralsShun, Zhenming; McCullagh, Peter
doi: 10.1111/j.2517-6161.1995.tb02060.xpmid: N/A
It is shown that the usual Laplace approximation is not a valid asymptotic approximation when the dimension of the integral is comparable with the limiting parameter n. The formal Laplace expansion for multidimensional integrals is given and used to construct asymptotic approximations for high dimensional integrals. One example is considered in which the dimension of the integral is O(n1/2) and the relative error of the unmodified Laplace approximation is O(1). Nevertheless, it is possible to construct a valid asymptotic expansion by regrouping terms in the formal expansion according to asymptotic order in n.