Willem Albers

Are estimated control charts in control

Standard control chart practice typically assumes normality and uses estimated parameters using data from an in-control process. However, because of the extreme quantiles involved, large relative errors will result for common performance characteristics such as the out-of-control signal probability or the average run length. Due to the estimation, such performance characteristics are stochastic and hence the relative errors involved can be analyzed in various ways. To assess the effects of these various ways of estimation, we look at some exceedance probabilities. It is demonstrated how corrections can be derived to bring the estimated false alarm rates close to their nominal values. Exact results are given, followed by simple approximations. The latter reveal the way in which the corrections depend on the underlying parameters, thus allowing a sensible approach in practice. Some illustration is provided, as well as a brief analysis of the out-of-control behavior of the corrected charts.

Empirical non-parametric control charts: estimation effects and corrections.

Owing to the extreme quantiles involved, standard control charts are very sensitive to the effects of parameter estimation and non-normality. More general parametric charts have been devised to deal with the latter complication and corrections have been derived to compensate for the estimation step, both under normal and parametric models. The resulting procedures offer a satisfactory solution over a broad range of underlying distributions. However, situations do occur where even such a large model is inadequate and nothing remains but to consider non-parametric charts. In principle, these form ideal solutions, but the problem is that huge sample sizes are required for the estimation step. Otherwise the resulting stochastic error is so large that the chart is very unstable, a disadvantage that seems to outweigh the advantage of avoiding the model error from the parametric case. Here we analyse under what conditions non-parametric charts actually become feasible alternatives for their parametric counterparts. In particular, corrected versions are suggested for which a possible change point is reached at sample sizes that are markedly less huge (but still larger than the customary range). These corrections serve to control the behaviour during in-control (markedly wrong outcomes of the estimates only occur sufficiently rarely). The price for this protection will clearly be some loss of detection power during out-of-control. A change point comes in view as soon as this loss can be made sufficiently small.

Stop-loss premiums under dependence

Stop-loss premiums are typically calculated under the assumption that the insured lives in the underlying portfolio are independent. Here we study the effects of small departures from this assumption. Using Edgeworth expansions, it is made transparent which configurations of dependence parameters may cause substantial deviations in the stop-loss premiums.

New Corrections for Old Control Charts

ABSTRACT When using standard control charts, typically several parameters need to be estimated. For the usual sample sizes, this is known to affect the performance of the chart. Here we present simple corrections to solve this problem. As a basis, we use existing factors which are widely used for the traditional charts. The advantage of the new proposals is that a clear link is made to the actual performance characteristics of the chart.

Data-Driven Rank Tests for Classes of Tail Alternatives

Tail alternatives describe the occurrence of a nonconstant shift in the two-sample problem with a shift function increasing in the tail. The classes of shift functions can be built up using Legendre polynomials. It is important to choose the number of involved polynomials in the right way. Here this choice is based on the data, using a modification of the Schwarz selection rule. Given the data-driven choice of the model, appropriate rank tests are applied. Simulations show that the new data-driven rank tests work very well. Although other tests for detecting shift alternatives, such as Wilcoxons test, may break down completely for important classes of tail alternatives, the new tests have high and stable power. The new tests also have higher power than data-driven rank tests for the unconstrained two-sample problem. Theoretical support is obtained by proving consistency of the new tests against very large classes of alternatives, including all common tail alternatives. A simple but accurate approximation of the null distribution makes application of the new tests easy.

Asymptotic Relative Efficiencies of Rank Tests for Trend Alternatives

Abstract A number of rank tests and a parametric test are compared in terms of their asymptotic relative efficiencies (AREs) for testing the hypothesis that a sequence of random variables is independent and identically distributed, against the alternative of increasing location trend. The ARE is broken up into two parts in a natural way, corresponding to the choice of regression constants and score generating function in the case of a linear rank test. It turns out that the effect of those choices becomes less important as the trend increases more rapidly.

Combined Rank Tests for Randomly Censored Paired Data

Abstract Many authors have dealt with the problem of extending ordinary two-sample rank tests to cases where censoring occurs. Albers and Akritas (1987) proposed simple tests for this purpose, based on the idea of making separate rankings for uncensored and censored observations and subsequently combining the resulting two rank statistics. Similarly, some papers appeared that studied the problem of censoring for the paired-data case rather than the two-sample case. This article indicates how the approach of Albers and Akritas can be adapted to the paired-data case. The tests involved are not based on the ranks of the differences, but the differences of the ranks in the pooled sample. In this way, use is made of interblock information. The asymptotic distribution of the new statistic is obtained under the null hypothesis and contiguous location alternatives. By way of example, Wilcoxon- and Savage-type scores are introduced that are optimal for logistic location and exponential scale alternatives, respecti...

Accurate Test Limits With Estimated Parameters

Due to measurement errors, producers are typically forced to set test limits well within specification limits. The methods used in practice are rather informal and usually conservative with respect to consumer loss, thus leading to unnecessary loss of yield. We present approximations for test limits that are still relatively easy to evaluate and moreover very accurate. In addition, the analytical tractability of these approximations allows extension to the more realistic case in which parameters are estimated.

The asymptotic behavior of tests for normal means based on a variance pre-test

In comparing two normal means, possible inequality of the corresponding variances plays an important role. Attempts to settle this through a preliminatry F-test present new complications: numerical work from literature indicates that the corresponding two-step procedure performs much less satisfactorily than seems to be taken for granted by most textbooks. In the present paper we demonstrate how second order asymptotics can be applied to obtain simple and transparant approximations to size and power of the resulting procedure. This provides both qualitative insight into the behaviour of this procedure, as well as information on the magnitude of the occurring deviations.

Combined Rank Tests for the Two-Sample Problem with Randomly Censored Data

Abstract In the two-sample problem under random censorship we consider the uncensored observations and the censored ones as two separate groups. For each group a suitable rank statistic is obtained, and these two are then combined to a final one. This idea, which was first employed in Akritas (1983), is shown here to produce tests that (a) closely resemble the ordinary rank tests for the uncensored case and do not require the calculation of the Kaplan-Meier estimator, (b) are comparatively easy to apply and to understand, and (c) allow results on asymptotic normality to follow simply from standard results for the uncensored case. It is shown that the loss due to using two separate rankings rather than one complete ranking is asymptotically negligible. The optimal score function for each of the two separate rank statistics is seen to depend on the censoring distribution. Whereas in Akritas (1983) no assumption on the form of the censoring distribution was made (unrestricted adaptation), here we pursue rest...