Showing papers in "Statistica Neerlandica in 1972"

Note on the generation of most probable frequency distributions

Abstract: Summary The most probable distribution of a stochastic variable is obtained by maximizing the entropy of its distribution under given constraints, by applying Lagrange's procedure. The constraints then determine the type of frequency distribution. The above holds for continuous as well as for discrete distributions. In this note we give a survey of various constraints and the corresponding frequency distributions.

Principal components, analysis of variance and data structure

Abstract: The relation between principal components and analysis of variance is examined. It is shown that the model underlying the extended analysis of variance developed by GOLLOB and MANDEL is useful also as a model for principal component analysis. The elucidation of structure of two-factor data using the new analysis of variance model is illustrated by an example taken from thermodynamics. It has been may good fortune to have spent a full year in close association with Professor HAMAKER at the Technological University of Eindhoven. That year was among the most pleasant and most rewarding of my career. I feel honored to be able to join with Professor HAMAKER'S many friends and colleagues in dedicating this issue of Statistica Neerlandica to him. The method of principal components goes back to ideas proposed by PEARSON as early as 1901 (12) and developed systematically by HOTELLING in 1933 [4]. Since then the method has been applied to numerous sets of data, more particularly in the field of psychology, but also in numerous other areas of research, including the physical sciences [e.g. 1,2, 5, 7, 8, 10, 13, 14, IS, 16). In 1968 and 1969 respectively, GOLLOB (3) and MANDEL (8) proposed, independently of each other, an extension of the analysis of variance approach, which GOLLOB called "Fanova", because it combined features of analysis of variance and of factor analysis. This method, too, had been anticipated by some earlier authors [13, 17). It is immediately apparent that this extension of the analysis of variance involves the same matrix calculations as the method of principal components. The question then arises whether a deeper conceptual relationship exists between the two methods. In this paper this question is examined. The result is not only a positive answer to this question but also a clarification of the method of principal components. The author believes, as a result of this work, that interpretations of principal component analyses found in the literature are sometimes incorrect. We will attempt to show that such misinterpretations are due, in no small measure, to a particular terminology that has acquired common usage in inferences drawn from principal component analysis.

Exact distribution of Durbin's distribution‐free test statistic for Balanced Incomplete Block Designs, and comparison with the Chi‐square and F approximation

Abstract: Summary Durbin's distribution-free rank test can be used to test the null hypothesis that there are no differences among the treatments in a Balanced Incomplete Block Design. Until now only the chi-square and F approximations for the test statistic were known. In this paper the exact distribution has been given for 15 designs and comparison is made with the chi-square and F approximation.

Variables sampling plans for the Poisson and the binomial

Abstract: Variables sampling plans based upon continuous distributions are well known. The usual assumption is that a measurable characteristic associated with a product has a normal distribution, a case which has been treated extensively in the literature. Other continuous distributions, particularly the exponential, have also been used as models. In this paper we discuss variables sampling plans for situations in which the measurable characteristic has either a Poisson or a binomial distribution.

The distribution of random variables reduced modulo a

Abstract: A random variable y reduces a real random variable x to modulo a > 0 if y = x [mod al.The deviation of the distribution of y from the uniform distribution is investigated for an arbitrary distribution of x. The results obtained are applied to random walks reduced modulo ti and to some pre-packing problem. An approximating solution is given. For a particular case the quality of the approximation is investigated.

An asymptotic problem in renewal theory

Abstract: Summary A special problem in renewal theory is considered. The asymptotic behavior of the renewal function was studied by W. L. Smith. Here we show that his result with an exponentially small remainder term follows from a theorem of De Bruijn on Volterra integral equations.

Inventory control with unknown demand distribution : a slow-mover case

Abstract: Summary In this paper some tools will be developed for analysing a simple inventory problem of the slow-mover type with the special feature that the probability distribution of the demand interarrival times is unknown. The role of sufficient statistics in decision making is analysed for an arbitrary optimality criterion and particularly for the Bayesian criterion. A computational approach for the case of a Bayesian criterion is presented together with some numerical results.

The estimation of guaranteed fatigue life under random loading

Abstract: Introduction and Summary One of the pressing problems of mechanical reliability still requiring a satisfactory solution is that of ensuring the guaranteed fatigue life of a component or structure subject to random dynamic loading. In the past, this problem has generally been solved in technical practice by the choice of a sufficiently large “safety factor” when dimensioning critically stressed parts of a complex structure. Application of probability and statistical methods now offers the possibility of developing a theory of reliability of mechanical systems, where the risk of failure can be expressed as a probability, taking into account effects of random loading processes, which characterize either the functioning of the system itself or external environmental operating conditions. In the following paper we describe one method of approach to the solution of this problem. The solution consists of a combination of the Weibull-Freudenthal-Gumbel theory of fatigue estimation using (s, N, P) relations, the Palmgren-Miner hypothesis of linear accumulation of damage and the theory of stationary random processes having a given autocorrelation function or spectral density. Several other stochastic models are discussed in [1]. The subject of this paper was chosen in acknowledgement of the fact that H. C. Hamaker in his applied theoretical work also dealt with a related problem concerning the breaking strength of glass [2].

Probability distributions of two‐component soil samples

Abstract: Summary Mixing of sand layers on top of peat soils is achieved by soil improvement machines. The result of the mixing process is studied by taking samples from a vertical cross-section of the profile. The samples will have various values of the sand to peat volume-ratio. These values can be plotted in emperical cumulative frequency distributions. The distributions for two types of machines were found to be quite different due to differences of mixing intensities. A method is proposed for the evaluation of the mixing effect expressed in the parameters of theoretical probability distributions of two-component soil samples. The probability distribution of the volume-ratios in the sample are derived and their properties are discussed. Moment estimators of the parameters are derived. The theoretical distributions are compared with two experimental results obtained with a mixing rooter and a rotary mixer respectively.

An application of Pitman-efficiency to acceptance sampling procedures*

Abstract: Summary As an exercise the concept of Pitman-efficiency has been applied to the decision problem whether to use acceptance sampling “by attributes” or “by variables”. The Pitman-efficiency has been calculated in the two cases that the variance of the underlying normal distribution is known and that it is unknown. Rather surprisingly the difference between these two cases proves to be considerable, even asymptotically. The asymptotic result is compared with the exact values of the relative efficiency in the case that s is unknown. The asymptotic approximation appears to be rather good. The results derived also help to determine a suitable choice of the null hypothesis in order to increase the Pitman-efficiency.

A comparison of two nonparametric $k$-sample slippage tests

Abstract: Summary The tests which are discussed here are Mosteller's k-sample slippage test for an extreme population [8] and the k-sample slippage analogue of the Wilcoxon two-sample test proposed by Doornbos and Prins [3]. A comparison is made between both tests in respect of their consistency against normal, exponential and uniform shifts and against Lehmann-alternatives. For some small sample sizes the powers of the two tests have been calculated.