Showing papers in "Journal of the royal statistical society series b-methodological in 1978"

Improper Priors, Spline Smoothing and the Problem of Guarding Against Model Errors in Regression

Abstract: SUMMARY Spline and generalized spline smoothing is shown to be equivalent to Bayesian estimation with a partially improper prior. This result supports the idea that spline smoothing is a natural solution to the regression problem when one is given a set of regression functions but one also wants to hedge against the possibility that the true model is not exactly in the span of the given regression functions. A natural measure of the deviation of the true model from the span of the regression functions comes out of the spline theory in a natural way. An appropriate value of this measure can be estimated from the data and used to constrain the estimated model to have the estimated deviation. Some convergence results and computational tricks are also discussed.

Discrete Time Series Generated by Mixtures. I: Correlational and Runs Properties

Abstract: supported by funds provided directly from the Chief of Naval Research under Grant NR-42-284, and the National Science Foundation under Grant NSF-75-02026

Density Estimation, Stochastic Processes and Prior Information

Abstract: SUMMARY A method is proposed for the non-parametric estimation of a probability density, based upon a finite number of observations and prior information about the smoothness of the density. A logistic density transform and a reproducing inner product from the first-order autoregressive stochastic process are employed to represent prior information that the derivative of the transform is unlikely to change radically within small intervals. The posterior estimate of the density possesses a continuous second derivative; it typically satisfies the frequentist property of asymptotic consistency. A direct analogy is demonstrated with a smoothing method for the time-dependent Poisson process; this is similar in spirit to the normal theory Kalman filter. A procedure for grouped observations in a histogram provides an alternative to the histospline method of Boneva, Kendall and Stefanov. Five practical examples are presented, including two investigations of normality, an analysis of pedestrian arrivals at a Pelican crossing and a histogram smoothing method for mine explosions data.

Nearest Neighbour Models in the Analysis of Field Experiments

Abstract: SUMMARY The method of adjusting plot values by covariance on neighbouring plots in randomized field experiments, suggested by Papadakis in 1937, is re-examined theoretically for both one-dimensional and two-dimensional layouts, making use of Markovian and autonormal models. The gain in efficiency over orthodox randomized block analysis can be appreciable when the number of treatments is fairly large, and can be increased by iterating the analysis; this also reduces the discrepancy between the apparent accuracy from the residual sum of squares, and the true accuracy of the treatment comparisons after adjustment. Some illustrative examples are included.

A Quasi‐Bayes Sequential Procedure for Mixtures

Abstract: SUMMARY Coherent Bayes sequential learning and classification procedures are often useless in practice because of ever-increasing computational requirements. On the other hand, computationally feasible procedures may not resemble the coherent solution, nor guarantee consistent learning and classification. In this paper, a particular form of classification problem is considered and a "quasi-Bayes" approximate solution requiring minimal computation is motivated and defined. Convergence properties are established and a numerical illustration provided.

On a Model for Concordance between Judges

Abstract: CONSIDER a number n of judges, each of whom ranks the same set of k objects according to some particular criterion. Assume that each judge ranks (stochastically) independently of the other judges so that we regard the situation as that of n rankings (Daniels, 1950; Kendall, 1970) or of n related samples (Conover, 1971, p. 246). We wish to say something about whether and to what degree the judges act concordantly (or homogeneously) with respect to a particular ranking which is not assumed known beforehand. More particularly we wish to estimate this underlying ranking and also consider a model for the proposed measure of concordance. In so doing we address ourselves to the problem of non-null modelling referred to in Kendall (1970, p. V). To place the suggested model in its proper setting it is appropriate to review briefly the relevant literature on modelling non-null distributions for rankings. These models may be, broadly speaking, divided into three classes: (I) parametric; (II) paired comparison; and (III) sampling. The parametric approach may be further subdivided into the categories: (Ia) multivariate; and (Ib) independent deviations. They may be described as follows:

On Parameter Estimation for Spatial Point Processes

Abstract: is intended to complement K(t) To assess goodness-of-fit, the empirical counterparts k(t) and p(t) calculated from a set of data are compared with those from simulations of the proposed, fully specified model, as suggested by Barnard (1963) Ripley does not discuss the associated problem of parameter estimation, but this can easily be incorporated into the general methodology as follows: let F(t; 0) be any suitable summary of the model with parameter 0, t(t) the empirical counterpart calculated from the data and t(O) some appropriate measure of the discrepancy between F(-) and t(*), for example

A Class of Shrinkage Estimators

Abstract: SUMMARY In this paper we define a class of shrinkage estimators, all of whose members have a mean square error matrix which is less than that of the ordinary least squares estimator by a positive semidefinite matrix if (f-b*)TXTX(8l-b*) < u2. The ridge regression, principal component and minimum conditional mean square error estimators are members of this class.

On the Half‐Sample Method for Goodness‐Of‐Fit

Abstract: SUMMARY Two interesting processes, related to the empirical distribution function, have been pointed out by Rao (1972) and by Durbin (1973). The second, in particular, leads to the half-sample method, an elegant and simple technique for dealing with unknown parameters in goodness-of-fit testing, without the necessity of new tables of percentage points for each distribution tested. To those who have spent some effort in producing such tables, it is natural to wonder whether this effort has been in vain. In this paper we show that the answer is no. continuous distribution F(x; 0), where 0 is a vector of parameters. Some or all of the components of 0 may be unknown, and when they are estimated by efficient estimators, say by maximum likelihood, we write # for the vector 0, with estimated components where necessary. A well-known class of goodness-of-fit tests is based on the EDIF (empirical distri- bution function); the test statistics are measures of the discrepancy y(x) = 1n(Fn(x) - F(x; 0)), for the case where 0 is known. Important statistics are D+ = sup$y(x), D- = sup$(-y(x)), D = max (D+, D-); V = D++ D-, W2 = fy2(x) dF; U2 = f(y(x) j-)2 dF; A2 = fy2(X) w2(x) dF, where F = F(x; 0) and w2(X) = {F(1 -F)}-' and integrals are from - oo to oo. When 0 is known, as above, we refer to the test situation as Case 0. When 0 is replaced by O, the distributions (including asymptotic distributions) of EDIF statistics are greatly changed. When unknown components of 0 are location and scale parameters, the distributions do not depend on the correct values of these parameters but only on the functional form of F. Two important examples are Case 3, where F is the normal distribution with 0(= (u, a2),

Super-slowly Varying Functions in Extreme Value Theory

Abstract: SUMMARY Large-deviation limit theorems for maxima of random samples in the domain of attraction of the extreme value distribution a are studied, motivated by examination of an approximation used in the statistical analysis of large observations. The largedeviation property is shown to be equivalent to a strengthened form of slow variation condition, called here super-slow variation, on the tail of the distribution of a single observation. Various conditions for super-slow variation are discussed and Karamata's representation for slowly varying functions is extended to a wide class of super-slowly varying functions.