Choosing a Linear Model with a Random Number of Change-Points and Outliers

The problem of determining a normal linear model with possible perturbations, viz. change-points and outliers, is formulated as a problem of testing multiple hypotheses, and a Bayes invariant optimal multi-decision procedure is provided for detecting at most k (k > 1) such perturbations. The asymptotic form of the procedure is a penalized log-likelihood procedure which does not depend on the loss function nor on the prior distribution of the shifts under fairly mild assumptions. The term which penalizes too large a number of changes (or outliers) arises mainly from realistic assumptions about their occurrence. It is different from the term which appears in Akaike‘s or Schwarz‘ criteria, although it is of the same order as the latter. Some concrete numerical examples are analyzed.

[1]  Keith J. Worsley,et al.  On the Likelihood Ratio Test for a Shift in Location of Normal Populations , 1979 .

[2]  Melvin J. Hinich,et al.  A Test for a Shifting Slope Coefficient in a Linear Model , 1970 .

[3]  Tom Leonard.  A Simple Predictive Density Function: Comment , 1982 .

[4]  H. Chernoff,et al.  ESTIMATING THE CURRENT MEAN OF A NORMAL DISTRIBUTION WHICH IS SUBJECTED TO CHANGES IN TIME , 1964 .

[5]  G. Schwarz.  Estimating the Dimension of a Model , 1978 .

[6]  V.K. Jandhyala,et al.  Tests for parameter changes at unknown times in linear regression models , 1991 .

[7]  K. Worsley.  Testing for a Two-Phase Multiple Regression , 1983 .

[8]  D. Hawkins.  Testing a Sequence of Observations for a Shift in Location , 1977 .

[9]  E. S. Page.  A test for a change in a parameter occurring at an unknown point , 1955 .

[10]  John Uebersax,et al.  Statistical Modeling of Expert Ratings on Medical Treatment Appropriateness , 1993 .

[11]  L. I. Pettit,et al.  Bayes Factors for Outlier Models Using the Device of Imaginary Observations , 1992 .

[12]  Michel Lejeune,et al.  A Simple Predictive Density Function , 1982 .

[13]  J. Hartigan,et al.  A Bayesian Analysis for Change Point Problems , 1993 .

[14]  Victor J. Yohai,et al.  A Bivariate Test for the Detection of a Systematic Change in Mean , 1978 .

[15]  L. A. Gardner.  On Detecting Changes in the Mean of Normal Variates , 1969 .

[16]  Testing the means of independent normal random variables , 1993 .

[17]  Gerald S. Rogers,et al.  Mathematical Statistics: A Decision Theoretic Approach , 1967 .

[18]  H. Akaike,et al.  Information Theory and an Extension of the Maximum Likelihood Principle , 1973 .

[19]  D. Hinkley.  Inference in Two-Phase Regression , 1971 .

[20]  Adrian F. M. Smith.  Change-Point problems: approaches and applications , 1980 .

[21]  D. Siegmund,et al.  The likelihood ratio test for a change-point in simple linear regression , 1989 .

[22]  R. Shibata.  An optimal selection of regression variables , 1981 .

[23]  David J. Hand,et al.  A Handbook of Small Data Sets , 1993 .

[24]  N. Kashiwagi.  Bayesian detection of structural changes , 1991 .

[25]  Yi-Ching Yao.  Estimation of a Noisy Discrete-Time Step Function: Bayes and Empirical Bayes Approaches , 1984 .

[26]  A F Smith,et al.  Monitoring renal transplants: an application of the multiprocess Kalman filter. , 1983, Biometrics.

[27]  B. G. Quinn,et al.  The determination of the order of an autoregression , 1979 .

[28]  R. Quandt.  The Estimation of the Parameters of a Linear Regression System Obeying Two Separate Regimes , 1958 .

[29]  A. Raftery,et al.  Analysis of Agricultural Field Trials in the Presence of Outliers and Fertility Jumps , 1994 .

[30]  D. Spiegelhalter,et al.  Bayes Factors and Choice Criteria for Linear Models , 1980 .