W.C.M. Kallenberg

Data-Driven Smooth Tests When the Hypothesis is Composite

In recent years several authors have recommended smooth tests for testing goodness of fit. However, the number of components in the smooth test statistic should be chosen well; otherwise, considerable loss of power may occur. Schwarzs selection rule provides one such good choice. Earlier results on simple null hypotheses are extended here to composite hypotheses, which tend to be of more practical interest. For general composite hypotheses, consistency of the data-driven smooth tests holds at essentially any alternative. Monte Carlo experiments on testing exponentiality and normality show that the data-driven version of Neymans test compares well to other, even specialized, tests over a wide range of alternatives.

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.

Data-Driven Rank Tests for Independence

We introduce new rank tests for testing independence. The new testing procedures are sensitive not only for grade linear correlation, but also for grade correlations of higher-order polynomials. The number of polynomials involved is determined by the data. Model selection is combined with application of the score test in the selected model. Whereas well-known tests as Spearmans test or Hoeffdings test may completely break down for alternatives that are dependent but have low grade linear correlation, the new tests have greater power stability. Monte Carlo results clearly show this behavior. Theoretical support is obtained by proving consistency of the new tests.

Data driven smooth tests for composite hypotheses comparison of powers

Raos score statistic is a standard tool for constructing statistical tests.If departures from the null model are described by some k-dimensional exponential family the resulting score test is called also smooth test or Neymans smooth test with k components. An important practical question in applying a smooth test in the goodness-of-fit problem is how large k should be taken. Since a wrong choice may give a considerable loss of power,it is important to make a careful selection.Renewed research in this area shows that the simple question has no simple deterministic answer. Therefore,edwina introduced,for testing a simple goodness-of-fit hypothesis,a data driven version of Neymans smooth test. First,Schwarzs rule is applied to find a suitable dimension and then the smooth test statistic in the “right” dimension finishes the job. Simulation results and some theoretical considerations show that this data driven version of smooth tests performs well for a wide range of alternatives,and is competitive with ...

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.

Power Approximations to Multinomial Tests of Fit

Abstract Multinomial tests for the fit of iid observations X 1 …, Xn to a specified distribution F are based on the counts Ni of observations falling in k cells E 1, …, Ek that partition the range of the X j . The earliest such test is based on the Pearson (1900) chi-squared statistic: X 2 = Σ k i=1 (Ni – npi )2/npi , where pi = PF (Xj in Ei ) are the cell probabilities under the null hypothesis. A common competing test is the likelihood ratio test based on LR = 2 Σ k i=1 Ni log(Ni/npi ). Cressie and Read (1984) introduced a class of multinomial goodness-of-fit statistics, R λ, based on measures of the divergence between discrete distributions. This class includes both X 2 (when λ = 1) and LR (when λ = 0). All of the R λ have the same chi-squared limiting null distribution. The power of the commonly used members of the class is usually approximated from a noncentral chi-squared distribution that is also the same for all λ. We propose new approximations to the power that vary with the statistic chosen. Bot...

Moderate and Cramér-type large deviation theorems for M-estimators

The known central limit result for broad classes of M-estimators is refined to moderate and large deviation behaviour. The results are applied in relating the local inaccuracy rate and the asymptotic variance of M-estimators in the location and scale problem.

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.

Vanishing shortcoming of data driven neyman's tests

The shortcoming of a test is the difference between the power of the test and the power of the most powerful test. For a large set of alternatives converging to the null hypothesis asymptotic optimality of data driven Neymans tests is shown in terms of vanishing shortcoming when the level of signicance tends to zero. In contrast to classical goodness-of-fit tests data driven Neymans tests are asymptotically efficient in an infinite number of orthogonal directions.