Showing papers in "Journal of the royal statistical society series b-methodological in 1977"
49,597 citations
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TL;DR: In this article, the limiting mean, variance and normality of disarray were established for the set of permutations. But they were not shown to be related to the metric I arising from Kendall's r through the combinatorial inequality I < D < 21.
Abstract: SUMMARY Spearman's measure of disarray D is the sum of the absolute values of the difference between the ranks. We treat D as a metric on the set of permutations. The limiting mean, variance and normality are established. D is shown to be related to the metric I arising from Kendall's r through the combinatorial inequality I< D < 21.
681 citations
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TL;DR: In this paper, the authors considered the problem of estimating an unknown regression function given observations at a fixed set of points, referred to as the Priestley-Chao (PC) estimate, which is restricted only by certain smoothing requirements.
Abstract: AN important statistical problem is the estimation of a regression function g(x) = E(y I x). Typically, g(x) has a specified functional form and parameter estimates are obtained according to certain desirable criteria, such as least squares. Assuming normal errors the investigator can test the appropriateness of the hypothesized model. One may wish, however, to have an estimation technique applicable for an arbitrary g(x). In a recent paper, Priestley and Chao (1972) considered the problem of estimating an unknown regression function g(x) given observations at a fixed set of points. Their estimate, referred to here as the Priestley-Chao (PC) estimate, is nonparametric in the sense that g(x) is restricted only by certain smoothing requirements. It can be viewed as a moving average of sample Y's whose weights are based on a class of kernels suggested by Rosenblatt (1956) and Parzen (1962). These weights are similar to those used in nonparametric density estimation. In this paper, the results of Priestley and Chao will be reviewed and further properties of their estimate considered.
158 citations
TL;DR: In this article, the authors presented optimal incomplete block designs which were constructed utilizing the connection between the concurrence matrix of a design and a regular graph, and conjectured to be optimal.
Abstract: A classical problem in experimental design is the construction of incomplete block designs, i.e., designs that group r replications of each of v treatments into b blocks of size k. This report presents optimal incomplete block designs which were constructed utilizing the connection between the concurrence matrix of a design and a regular graph. The class of regular graph (RG) designs, conjectured to be optimal, is defined. The results of a systematic search of the class of RG designs are given for all cases in which v is less than or equal to 12 and r is less than or equal to 10. In nearly every case, the optimum design with respect to each of the three criteria, D-optimality, A-optimality (identical with) = efficiency and E-optimality, is determined. Many of these designs are new and are tabulated in this report. 2 tables.
149 citations
TL;DR: In this article, the authors proposed a population model for dependence between two angular measurements and a linear observation, which leads to new population and sample measures of dependence in this latter situation and an example relating wind direction to the level of a pollutant.
Abstract: Population models for dependence between two angular measurements and for dependence between an angular and a linear observation are proposed. The method of canonical correlations first leads to new population and sample measures of dependence in this latter situation. An example relating wind direction to the level of a pollutant is given. Next, applied to pairs of angular measurements, the method yields previously proposed sample measures in some special cases and a new sample measure in general.
123 citations
TL;DR: In this paper, the authors consider inference about the parameters of a multivariate linear model, in which the usual assumption of normality for the errors is replaced by a weaker assumption of spherical symmetry, and show that inference about means is identical with that appropriate under normality, being based on a matrix generalization of "studentization".
Abstract: SUMMARY We consider inference about the parameters of a multivariate linear model, in which the usual assumption of normality for the errors is replaced by a weaker assumption of spherical symmetry. Structural distributions and confidence regions are derived, and it is shown that inference about means is identical with that appropriate under normality, being based on a matrix generalization of "Studentization". Some relevant distribution theory is developed, the approach throughout being "densityfree".
98 citations
TL;DR: In this paper, several statistical problems are considered, and they are reduced to problems of estimating functionals of a density, derivatives or both, and a method of finding partial solutions is obtained.
Abstract: SUMMARY Several statistical problems are considered. By reducing them to problems of estimating functionals of a density, derivatives of a density, or both, a method of finding partial solutions is obtained.
TL;DR: In this article, a new family of non-parametric estimators of a smooth regression function, which are shown to have theoretical advantages in small samples over some alternative estimators, is presented.
Abstract: SUMMARY This paper presents a new family of non-parametric estimators of a smooth regression function, which are shown to have theoretical advantages in small samples over some alternative estimators. The new estimator has practical computational advantages over spline functions, while its smoothing properties are very close to the optimal smoothing properties of cubic splines. A simulation experiment demonstrates the remarkable success of the cross-validation technique as a means of determining the appropriate degree of smoothing.
TL;DR: In the univariate case, the distribution of the maximum of a set of independent random variables uniquely determines the distributions of the component random variables under certain conditions as mentioned in this paper, and the identifiability property has been proved for multivariate normal distributions for n = 2 and for every n when all correlations are positive.
Abstract: : The distribution of the maximum of a set of independent random variables uniquely determine the distributions of the component random variables under certain conditions. In the univariate case a sufficient condition is roughly that for every two distinct densities f(x) and g(x) in the family of possible densities f(x)/g(x) approaches 0 or infinity as x approaches infinity. Hence, the distribution of max X(i), i = 1, ..., n, when X(i) has the distribution N mu sub i, sigma squared sub i uniquely determines mu sub i, squared sub i, i=1, ..., n (except for indexing). The identifiability property has been proved for multivariate normal distributions for n = 2 and for every n when all correlations are positive; each component of the vector of maxima consists of the maximum of that component of the n constituent vectors. Inequalities for the probability in the upper right-hand quadrant of the bivariate normal distribution have been developed; these are generalizations of Mills' ratio.
TL;DR: In this paper, the applications of Gabriel's procedures to multivariate linear regression are presented and illustrated as generalizations of Aitkin's technique, which is used in the MANOVA context.
Abstract: SUMMARY Simultaneous procedures for variable selection in multiple linear regression have recently been given by Aitkin. One of these procedures, proposed for the case when the regression equation is to be used for descriptive purposes, is an application of each of a number of simultaneous procedures concerned with the multivariate general linear model and given by Gabriel with applications in the MANOVA context. The applications of Gabriel's procedures to multivariate linear regression are presented here and illustrated as generalizations of Aitkin's technique.
TL;DR: In this article, the authors describe the use of the ratio of order statistics as a measure of time savings, and give results on the distribution of such ratios for several underlying distributions of failure time (uniform, exponential, Pareto).
Abstract: SUMMARY It is often possible to achieve a saving of time in a life testing experiment by allowing right-censoring. This concept is important in clinical trials in which an early decision procedure is used. We describe the use of the ratio of order statistics as a measure of time savings, and give results on the distribution of such ratios. Numerical results are tabled and discussed for several underlying distributions of failure time (uniform, exponential, Pareto). We outline the application of this approach to the evaluation of early stopping procedures, with particular emphasis on analysing the time savings achieved by two-sample rank tests.
TL;DR: In this paper, a model for queue formation and disintegration is proposed for traffic on a road where flow is unimpeded, assuming that vehicles are travelling in random queues and a distribution for queue length is derived having as a single parameter the ratio of the rate of mergings between pairs of queues and rate of overtaking from queues of more than one vehicle.
Abstract: SUMMARY A model for queue formation and disintegration is proposed for traffic on a road where flow is unimpeded. The model assumes that vehicles are travelling in random queues and a distribution for queue length is derived having as a single parameter the ratio of the rate of mergings between pairs of queues and the rate of overtaking from queues of more than one vehicle. The first moment and maximum likelihood estimator associated with the distribution are derived and compared in a simulation study. The distribution is then fitted to data collected in different sites, with varying results. IN the course of developing models for traffic flow along roads a number of distributions have been hypothesized to describe the lengths of queues of vehicles. Several of these distri- butions provide a sufficiently close fit to observed data (Taylor et al., 1974) to serve adequately in describing the data but lack of satisfactory derivation in terms of vehicle behaviour. For instance, the Borel-Tanner distribution (Borel, 1942; Tanner, 1953) was derived by taking a random positioning of vehicles along the road and considering all vehicles within distance, b, of the one ahead as queueing. The queueing vehicles were then moved back so that they were exactly b apart and any new vehicles which were consequently within b of the tail of the queue were treated similarly. The Borel-Tanner distribution provides a good fit to data collected in the field but the derivation in terms of a distance, b, rather than in terms of the rates at which queues break up and merge leaves something to be desired. An alternative approach in terms of vehicle behaviour was made by Miller (1961, 1963a) who considered traffic flow as a queueing process with slow vehicles as the service points. Vehicles were assumed to catch up singly and randomly and the times between consecutive overtakings were assumed to follow an exponential distri- bution. These assumptions produced a geometric distribution whose parameter was p, the ratio of the catching up rate to the overtaking rate. Since the ratio varies with queue speed Miller assigned to it a distribution and chose the beta distribution