Showing papers in "Journal of the royal statistical society series b-methodological in 1972"

Regression Models and Life-Tables

Abstract: The analysis of censored failure times is considered. It is assumed that on each individual arc available values of one or more explanatory variables. The hazard function (age-specific failure rate) is taken to be a function of the explanatory variables and unknown regression coefficients multiplied by an arbitrary and unknown function of time. A conditional likelihood is obtained, leading to inferences about the unknown regression coefficients. Some generalizations are outlined.

Outliers in Time Series

Abstract: THE detection of outliers has mainly been considered for single random samples, although some recent work deals also with standard linear models; see, for example, Anscombe (1960) and Kruskal (1960). Essentially similar problems arise in time series (Burman, 1965) but there seems no published work taking into account correlations between successive observations. In the past, the search for outliers in time series has been based on the assumption that the observations are independently and identically normally distributed. This assumption leads to analyses which will be called random sample procedures. Two types of outlier that may occur in a time series are considered in this paper. A Type I outlier corresponds to the situation in which a gross error of observation or recording error affects a single observation. A Type II outlier corresponds to the situation in which a single "innovation" is extreme. This will affect not only the particular observation but also subsequent observations. For the development of tests and the interpretation of outliers, it is necessary to distinguish among the types of outlier likely to be contained in the process. The present approach is based on four possible formulations of the problem: the outliers are all of Type I; the outliers are all of Type II; the outliers are all of the same type but whether they are of Type I or of Type II is not known; and the outliers are a mixture of the two types. Since more practical solutions than those given by likelihood ratio methods are often obtained from simplifications of likelihood ratio criteria, some simpler criteria are derived. These criteria are of the form /&2a, where A is the estimated error in the observation tested and ^ is the estimated standard error of A. Throughout this paper, trend and seasonal components are assumed either negligible or to have been eliminated. The method adopted to remove these components might affect the results in some way.

Non-Parametric Function Fitting

Abstract: SUMMARY In this note we consider the problem of fitting a general functional relationship between two variables. We require only that the function to be fitted is, in some sense, "smooth", and do not assume that it has a known mathematical form involving only a finite number of unknown parameters.

Components of Cramer-von Mises statistics. I

Abstract: Let F n (x) be the sample distribution function derived from a sample of independent uniform (0, 1) variables. The paper is mainly concerned with the orthogonal representation of the Cramer-von Mises statistic W 2 n in the form Σ ∞ j=1 (jπ) -2 z 2 nj where the z nj are the principal components of $\sqrt n\{F_n(x) - x\}$ . It is shown that the z nj are identically distributed for each n and their significance points are tabulated. Their use for testing goodness of fit is discussed and their asymptotic powers are compared with those of W 2 n , Anderson and Darling's statistic A 2 n and Watson's U 2 n against shifts of mean and variance in a normal distribution. The asymptotic significance points of the residual statistic W 2 n - Σ p j=1 (jπ) -2 z 2 nj are also given for various p. It is shown that the components analogous to z nj for A 2 n are the Legendre polynomial components introduced by Neyman as the basis for his "smooth" test of goodness of fit. The relationship of the components to a Fourier series analysis of F n (x) - x is discussed. An alternative set of components derived from Pyke's modification of the sample distribution function is considered. Tests based on the components z nj are applied to data on coal-mining disasters.

Generalized Least Squares and Maximum Likelihood Estimation of Non‐Linear Functional Relationships

Abstract: SUMMARY A non-linear functional relationship between two mathematical variables is postulated. Departures of observed random variates from the mathematical variable values are assumed to have zero mean and general but known covariance matrix. Assuming normality we obtain by maximum likelihood an expression for the asymptotic covariance matrix of the structural parameter estimates. The expression involves a certain matrix IF. A generalized least squares method is proposed which involves minimizing a quadratic form with the same matrix IF. In the linear case, it is shown that the methods of generalized least squares, maximum likelihood and maximum relative likelihood are equivalent. The least squares procedure is identified with the intuitive approach of Sprent (1966).