Erhard Reschenhofer

Generalization of the Kolmogorov-Smirnov test

Abstract Because of the low power of the Kolmogorov-Smirnov goodness-of-fit test in case of multimodal alternatives this paper investigates a family of tests indexed by an integer-valued parameter K ≥2. The sensitivity of these tests to multimodal alternatives increases as K increases. The Kolmogorov-Smirnov test is obtained for K =2. The other elements of this family measure local deviations from uniformity in the same way as the Kolmogorov-Smirnov test measures the global deviation. Hence they may be regarded as generalized Kolmogorov-Smirnov tests. Tables of critical values are given and applications to the problem of testing for white noise are discussed. In addition, a further family of related tests is proposed. A Monte Carlo power study compares all the tests proposed in this note with Neymans smooth test and different versions of the length test introduced by Reschenhofer and Bomze ( Biometrika , 78 (1991) 207–216; Biometrika , 79 (1992) 859). A secondary result of this simulation study is that the possibly most widely used version of the length test can break down when a peak is located near the boundary.

Contrasting activity patterns of two related octopus species, Octopus macropus and Octopus vulgaris.

Octopus macropus and Octopus vulgaris have overlapping habitats and are exposed to similar temporal changes. Whereas the former species is described as nocturnal in the field, there are conflicting reports about the activity time of the latter one. To compare activity patterns, the authors tested both species in the laboratory. Octopuses were exposed to a light-dark cycle and held under constant dim light for 7 days each. O. macropus showed nocturnal and light-cued activity. According to casual observations, O. vulgaris started out nocturnal but had switched to mostly diurnal when the experiment began. Individual variation of its activity was found. The different activity patterns of O. macropus and O. vulgaris might reflect their lifestyles, the latter species being more generalist.

Prediction with vague prior knowledge

Recent results (Kass and Wasserman, 1995, Reschenhofer, 1995) concerning the approximation of posterior probabilities are used to develop further the Bayesian prediction criterion proposed by San Martini and Spezzaferri (1984).

Robust tests of the random walk hypothesis

In this article, simple tests of the random walk hypothesis are proposed that are robust against various kinds of conditional heteroskedasticity, non-stationarities, calendar effects and non-synchronous trading effects. In contrast, conventional tests are usually only robust against conditional heteroskedasticity. The robustness of the tests proposed in this paper is based on the fact that they examine the four popular summary measures (open, close, high, low) for each trading day separately. The results of a simulation study show that the tests are also quite robust against certain intraday anomalies like increased volatility at the beginning and at the end of the trading session. There is also evidence that the tests are robust against asymmetries in the returns.

The Bimodality Principle

In statistics courses, students often find it difficult to understand the concept of a statistical test. An aggravating aspect of this problem is the seeming arbitrariness in the selection of the level of significance. In most hypothesis-testing exercises with a fixed level of significance, the students are just asked to choose the 5% level, and no explanation for this particular choice is given. This article tries to make this arbitrary choice more appealing by providing a nice geometric interpretation of approximate 5% hypothesis tests for means. Usually, we want to know not only whether an observed deviation from the null hypothesis is statistically significant, but also whether it is of practical relevance. We can use the same geometrical approach that we use to illustrate hypothesis tests to distinguish qualitatively between small and large deviations.

IMPROVED ESTIMATION OF THE EXPECTED KULLBACK LEIBLER DISCREPANCY IN CASE OF MISSPECIFICATION

In case of misspecification, the Akaike information criterion (AIC; Akaike, 1973, in Petrov & Csaki, eds., Second International Symposium on Information Theory, pp. 267–281. Budapest: Akademia Kiado) is an asymptotically biased estimator of the expected Kullback–Leibler discrepancy. This paper gives simple expressions for the bias that can be used to construct improved estimators. However, for the examples that are considered in detail it turns out that model selection procedures based on such improved estimators are nearly equivalent to model selection procedures based on severely biased estimators.

Modelling exchange rates: long-run dependence versus conditional heteroscedasticity

Indications for two different features not captured by low-order linear time-series models can be found in day-to-day changes of exchange rates: long memory and conditional heteroscedasticity. These characteristics have inspired the development of ARFIMA and GARCH models. By means of Monte Carlo simulation, it is demonstrated that either of the two features stands a non-negligible chance of being detected spuriously in the presence of the other. A table of explicit empirical small-sample quantiles for identification of long-memory structures in the presence of GARCH effects is included.

Analysis of Austrian Stocks: Testing for Stability and Randomness

This paper is concerned with subjecting two popular assumptions about the behavior of stock market prices to empirical tests: first, the random walk hypothesis developed by Bachelier (1900), Osborne (1959), and Mandelbrot (1963); second, the stable distributions hypothesis by Mandelbrot (1963) and Fama (1965). For this purpose, ten time series from the Vienna Stock Exchange were used. The first hypothesis was tested using both non-parametric and parametric methods. To obtain evidence with regard to the seond hypothesis, a graphical procedure and statistical estimation on the basis of the empirical characteristic function were applied. On analysis of our data, it turned out that, at least for the time period under consideration (1985–1990), severe doubts are cast on the above assumptions.

Estimation of the fractionally differencing parameter with the R/S method

Extensive computer experiments are performed to examine the finite sample behavior of the R/S analysis in case of long-term dependence. Our simulation study differs from former studies both in its design and in its size. In addition, we used a method for obtaining finite sequences of a long-term dependent process, which exactly reproduces the correlation structure of the underlying process. The results obtained show that the R/S analysis does not allow reliable estimation of the fractionally differencing parameter.

Testing for Cycles in Multiple Time Series

In practice, it is often impossible to assess the validity of the smoothness assumptions crucial to standard tests for singularities in the spectrum. We therefore propose new tests which are completely insensitive to sharp peaks in the absolutely continuous part of the spectrum. Using Neyman Pearson tests of Bayesian mixtures we first derive admissible tests under simplifying assumptions and then show that under realistic assumptions our test statistics remain the same. The tests are designed to have high power especially against alternatives containing oscillations which are positively correlated with each other. Motivated by a biological dataset with non-sinusoidal oscillations, we finally extend our approach by including higher harmonics.