Showing papers in "Journal of the royal statistical society series b-methodological in 1976"

The Empirical Distribution Function with Arbitrarily Grouped, Censored, and Truncated Data

Abstract: SUMMARY This paper is concerned with the non-parametric estimation of a distribution function F, when the data are incomplete due to grouping, censoring and/or truncation. Using the idea of self-consistency, a simple algorithm is constructed and shown to converge monotonically to yield a maximum likelihood estimate of F. An application to hypothesis testing is indicated.

A Test for Normality Based on Sample Entropy

Abstract: SUMMARY A test of the composite hypothesis of normality is introduced. The test is based on the property of the normal distribution that its entropy exceeds that of any other distribution with a density that has the same variance. The test statistic is based on a class of estimators of entropy constructed here. The test is shown to be a consistent test of the null hypothesis for all alternatives without a singular continuous part. The power of the test is estimated against several alternatives. It is observed that the test compares favourably with other tests for normality.

Forecasting Transformed Series

Abstract: Suppose that a forecasting model is available for the process Xt but that interest centres on the instantaneous transformation Yt = T(Xt). On the assumption that Xt is Gaussian and stationary, or can be reduced to stationarity by differencing, this paper examines the autocovariance structure of and methods for forecasting the transformed series. The development employs the Hermite polynomial expansion, thus allowing results to be derived for a very general class of instantaneous transformations.

Maximum Likelihood Estimation for Dependent Observations

Abstract: SUMMARY The asymptotic properties of m.l.e. are discussed for generally dependent observations. Conditions are derived for weak consistency and asymptotic Normality of the estimates. We further consider the case where some of the parameters are "transient" in the sense that the accumulated information on them from the sample does not increase indefinitely; then the interest lies in estimating the other parameters consistently. Examples are given, and the work is related to that of Neyman and Scott (1948).

Matrix Weighted Averages and Posterior Bounds

Abstract: SUMMARY Bounds for matrix weighted averages of pairs of vectors are presented. The weight matrices are constrained to certain classes suggested by the Bayesian analysis of the linear regression model and the multivariate normal model. The bounds identify the region within which the posterior location vector must lie if the prior comes from a certain class of priors. WE present in this article a series of results on the behaviour of matrix weighted averages of the form b** = (H + H*)-1 (Hb + H* b*), where b, b* and b** are vectors and where H and H* are square, symmetric, positive semi- definite matrices. These results indicate the extent to which we can generalize to higher dimensions the trivial univariate bound that constrains the scalar b** to lie algebraically between the scalars b and b*. Our interest in the behaviour of matrix weighted averages derives from the fact that two statistical models-the normal linear regression model and the multivariate normal sampling model with normal priors-have posterior means of the location parameters that are matrix weighted averages of a prior location vector and a sample location vector. One or both of the matrices in these averages are arbitrary either because prior distributions are impossible to measure without error or because intended readers may differ in their prior judgments. A Bayesian analysis based on any particular prior distribution will as a result be of little interest. Practical users of the Bayesian tools will necessarily face the difficult reporting problem of characterizing economically the mapping implied by the given data from interesting prior distributions into their respective posterior distributions, thereby servicing a wide readership as well as identifying those features of the prior which critically determine the posterior and which must therefore be measured accurately. One way of characterizing the mapping from priors into posteriors is a local sensitivity analysis that identifies the relative sensitivity of aspects of the posterior distribution to infinitesimal changes in the prior. The feasibility of a local sensitivity analysis is somewhat doubtful since to have great content it will have to be performed for many different prior distributions. Instead, we are suggesting here a global sensitivity analysis that constructs a correspondence between classes of priors and classes of posteriors. We will attempt to answer questions of the form: "If my prior is a member of this class of priors, what can I say about my posterior?" Although we would naturally be interested in both the location and the dispersion of the posterior, we will consider here only the location parameter. We will take the location of the prior as given and will develop a correspondence between classes of prior covariance matrices and regions in the space of the posterior location vector. We will see that a great deal can be said about the posterior location without precisely specifying the prior covariance matrix.

Generation of Factorial Designs

Abstract: JOHN AND DEAN (1975) described a generalized cyclic method for constructing confounded single-replicate factorial arrangements. Unlike many earlier methods, this method is not restricted to treatment factors with prime-power numbers of levels. Nor is it restricted to symmetrical arrangements, i.e. designs in which all factors have the same number of levels. Dean and John (1975) described the construction of asymmetrical arrangements. John and Dean's method appears to be well suited to the production of factorial designs on a computer. Their papers prompt me to compare the generalized cyclic method with a procedure that has been used regularly since 1966 in the routine computer production of designs. This procedure is incorporated in a program called DSIGN, written by Mrs J. Tolmie and used both at Rothamsted and in Edinburgh.t The DSIGN method depends on the specification of required relationships between the levels of different treatment factors. The possibility of constructing designs in this way is, of course, well known. John (1971) gives many examples, particularly of fractional replicates, and numerous references. Possibly the closest antecedent of the DSIGN procedure is the method described by Das (1964) for confounded symmetrical designs in blocks. The DSIGN procedure can be regarded as a generalization of Das's method, giving not only block designs but also designs with more complicated block structure, e.g. rows and columns, split plots. The DSIGN procedure was first developed for construction of symmetrical factorial treatment schemes in rotation experiments (Patterson, 1965). In practice the method has also been found capable of producing many of the most commonly used asymmetrical designs although its mechanism in these applications is less easily understood. The present paper outlines the construction of asymmetrical block designs by the DSIGN method, applies the method to single-replicate designs and considers its relationship with the generalized cyclic method.

A Bayesian Interpretation of Pretesting

Abstract: SUMMARY The usual posterior mean or mode of a k-dimensional regression coefficient is shown to be (1) a weighted average of 2k constrained least-squares points, (2) a weighted average of k +1 "ideal" points and (3) a weighted average of k +1 principal component points. These results are used to interpret pretesting procedures which select a constrained least-squares estimate. For special classes of priors a somewhat distorted Bayesian analysis leads to a posterior measure of location that is an F-weighted average of restricted and unrestricted least squares. A spherical prior can be associated with methods that drop principal components. Multilevel testing is shown to imply a non-traditional prior.

Histosplines with Knots Which are Order Statistics

Abstract: The histospline considered here is a quadratic spline density estimate analogous to the Boneva-Kendall-Stefanov (BKS) histospline, with their equally spaced knots replaced by knots at every knth-order statistic. This estimate can be expected to be relatively more "flexible" where the density is greatest. We obtain "mean square error at a point" convergence rates. The rates obtained are uniform over classes of densities which have first (alternatively second or third) derivatives in a bounded set in Y, If kn is chosen optimally, then this estimate shares the same "near optimal" convergence rates of certain BKS estimates when the knot spacing is chosen optimally, kernel estimates when the scale factor is chosen optimally, and orthogonal series estimates when the length of the series is chosen optimally.