van der P Paul Laan

Nonparametric control charts : an overview and some results

We present an overview of the literature on nonparametric or distribution-free control charts for univariate variables data. We highlight various advantages of these charts while pointing out some of the disadvantages of the more traditional, distribution-based control charts. Specific observations are made in the course of review of articles and constructive criticism is offered so that opportunities for further research can be identified. Connections to some areas of active research are made, such as sequential analysis, that are relevant to process control. We hope that this article leads to a wider acceptance of distribution-free control charts among practitioners and serves as an impetus to future research and development in this area.

Imprecise predictive selection based on low structure assumptions

Abstract Based on observations of real-valued random quantities from k⩾2 independent sources, new results are presented on bounds for predictive probabilities for further unknown random quantities from each source. Past and future observations per source are related via the assumption A(n) (Hill, J. Amer. Statist. Assoc. 63 (1968) 677), which is closely linked to finite exchangeability. Attention is particularly focussed on the question: which source will provide the largest next observation, when taking one more observation from each source? Our approach enables a nonparametric predictive comparison between the sources, which might naturally be linked to choices between sources, leading us to suggest our method as an alternative formulation to classical selection approaches and to introduce the term ‘imprecise predictive selection’. We present imprecise predictive selection of a single best source and of a subset of m (m⩽k−1) sources. We consider two cases for selection of a subset, namely a subset that contains the m best sources, and a subset that is just required to include the single best source. We also briefly present a related approach, using imprecise previsions with an interpretation as bounds for expected values for future observations.

Precedence tests and confidence bounds for complete data : an overview and some results

SUMMARY An overview of some nonparametric procedures based on precedence (or exceedance) statistics is given. The procedures include both tests and confidence intervals. In particular, the construction of some simple distribution-free confidence bounds for location difference of two distributions with the same shape is considered and some properties are derived. The asymptotic relative efficiency of an asymptotic form of the corresponding test relative to Wilcoxons two-sample rank sum test and the two-sample Student t-test is given for various cases. Some K-sample problems are discussed where precedence-type tests are useful, along with a review of the literature.

Precedence probability and prediction intervals

Precedence tests are simple yet useful nonparametric tests based on two specified order statistics from two independent random samples or, equivalently, on the count of the number of observations from one of the samples preceding some order statistic of the other sample. The probability that an order statistic from the second sample exceeds an order statistic from the first sample is termed the precedence probability. When the distributions are the same, this probability can be calculated exactly, without any specific knowledge of the underlying common continuous distribution. This fact can be utilized to set up nonparametric prediction intervals in various situations. In this paper, prediction intervals are considered for the number of second-sample observations that exceed a particular order statistic of the first sample. To aid the user, tables are provided for small sample sizes, where exact calculations are most necessary. The same tables can be used to implement a precedence test for small sample sizes.

On Using a Loss Function in Selecting the Best of Two Gamma Populations in Terms of their Scale Parameters

This paper continues the study of the subset selection procedure proposed by van der Laan and van Eeden [1]. In that paper the authors consider a location problem and base their procedure on a continuous loss function. This loss function takes into account the “distance”, in parameter values, between the populations under consideration and the best one among the ones in the selected subset. In defining this “distance”, they incorporate the notion of “e-best” studied by, e.g., Desu [2], Lam [3], van der Laan [4], Gill and Sharma [5] and Gill, Sharma and Misra [6]. As an example of their results, van der Laan and van Eeden [1] consider the case of two normal populations with equal known variances. The present paper develops a similar procedure for scale parameters. The case of two gamma populations with equal known shape parameters is studied in detail.