Estimation in Shewhart control charts

The influence of the estimation of parameters in Shewhart control charts is investigated. It is shown by simulation and asymptotics that (very) large sample sizes are needed to accurately determine control charts if estimators are plugged in. Correction terms are developed to get accurate control limits for common sample sizes in the in-control situation. Simulation and theory show that the new corrections work very well. The performance of the corrected control charts in the out-of-control situation is studied as well. It turns out that the correction terms do not disturb the behavior of the control charts in the out-of-control situation. On the contrary, for moderate sample sizes the corrected control charts remain powerful and therefore, the recommendation to take at least 300 observations can be reduced to 40 observations when corrected control charts are applied

Are estimated control charts in control?

Standard control chart practice assumes normality and uses estimated parameters. Because of the extreme quantiles involved, large relative errors result. Here simple corrections are derived to bring such estimated charts under control. As a criterion, suitable exceedance probabilities are used. \u

From A to Z: asymptotic expansions by van Zwet

Refinements of first order asymptotic results are reviewed, with a number of Ph.D. projects supervised by van Zwet serving as stepping stones. Berry-Esseen bounds and Edgeworth expansions are discussed for R-, L- and U-statistics. After these special classes, the question about a general second order theory for asymptotically normal statistics is addressed. As a final topic, empirical Edgeworth expansions are considere

Efficiency and deficiency considerations in the symmetry problem

Usually, two statistical procedures A and B are compared by means of their asymptotic relative efficiency e. If e= 1, however, it is more informative to compare A and B by means of the concept of deficiency, which was introduced by Hodges and Lehmann [7]. In the present paper we use this concept for the comparison of linear rank tests and parametric tests for the symmetry problem. In this problem, the hypothesis has to be tested that a sample comes from a distribution that is symmetric about zero. The results provide new and strong edivence for the nice performance of linear rank tests for the symmetry problem. The present paper gives a survey of the results obtained by Albers, Bickel and van Zwet [1] and by Albers [2]

Negative Binomial charts for monitoring high-quality processes

Good control charts for high quality processes are often based on the number of successes between failures. Geometric charts are simplest in this respect, but slow in recognizing moderately increased failure rates p. Improvement can be achieved by waiting until r > 1 failures have occurred, i.e. by using negative binomial charts.In this paper we analyze such charts in some detail. On the basis of a fair comparison, we demonstrate how the optimal r is related to the degree of increase of p. As in practice p will usually be unknown, we also analyze the estimated version of the charts. In particular, simple corrections are derived to control the non-negligible effects of this estimation step

From A to Z: Asymptotic expansions by van Zwet

Refinements of first-order asymptotic results are reviewed, with a number of Ph.D. projects supervised by van Zwet serving as stepping stones. Berry-Esseen bounds and Edgeworth expansions are discussed for $R$-, $L$- and $U$-statistics. After these special classes, the question of a general second-order theory for asymptotically normal statistics is addressed. As a final topic, empirical Edgeworth expansions are considered

Power gain by pre-testing?

The aim of this paper is to study whether it is possible to gain power by pre-testing, and to give insight in when this occurs, to what extent, and, at which price. A pre-test procedure consists of a preliminary test which tests a particular property of a given restricted model, followed by a main test for the main hypothesis regarding the parameter of interest. After acceptance by the preliminary test, a basic main test is used in the restricted model. After rejection by the preliminary test, a more general main test is used which does not use prior information about the underlying distribution. The procedure is analyzed in the model against which the preliminary test protects. For classes of tests including the standard first-order optimal tests, a transparent expression is given for the power and size difference of the pre-test procedure compared to the power and (correct) size of the second main test. This expression is based on second-order asymptotics and gives qualitative and quantitative insight in the behaviour of the procedure. It shows that substantial power gain, not merely due to size violation, is possible if the second main test really differs from the basic main test. The smaller the correlation between the two main tests, the larger the power gain