The Consistency and Robustness of Modified Cramér–Von Mises and Kolmogorov–Cramér Estimators

This article focuses on the minimum distance estimators under two newly introduced modifications of Cramér–von Mises distance. The generalized power form of Cramér–von Mises distance is defined together with the so-called Kolmogorov–Cramér distance which includes both standard Kolmogorov and Cramér–von Mises distances as limiting special cases. We prove the consistency of Kolmogorov-Cramér estimators in the (expected) L₁-norm by direct technique employing domination relations between statistical distances. In our numerical simulation we illustrate the quality of consistency property for sample sizes of the most practical range from n = 10 to n = 500. We study dependence of consistency in L₁-norm on ?-contamination neighborhood of the true model and further the robustness of these two newly defined estimators for normal families and contaminated samples. Numerical simulations are used to compare statistical properties of the minimum Kolmogorov–Cramér, generalized Cramér–von Mises, standard Kolmogorov, and Cramér–von Mises distance estimators of the normal family scale parameter. We deal with the corresponding order of consistency and robustness. The resulting graphs are presented and discussed for the cases of the contaminated and uncontaminated pseudo-random samples.     

Test for model selection using Cramér–von Mises distance in a fixed design regression setting

In this paper a test for model selection is proposed which extends the usual goodness-of-fit test in several ways. It is assumed that the underlying distribution H depends on a covariate value in a fixed design setting. Secondly, instead of one parametric class we consider two competing classes one of which may contain the underlying distribution. The test allows to select one of two equally treated model classes which fits the underlying distribution better. To define the distance of distributions various measures are available. Here the Cramér-von Mises has been chosen. The null hypothesis that both parametric classes have the same distance to the underlying distribution H can be checked by means of a test statistic, the asymptotic properties of which are shown under a set of suitable conditions. The performance of the test is demonstrated by Monte Carlo simulations. Finally, the procedure is applied to a data set from an endurance test on electric motors.     

Asymptotic Properties of Error Density Estimator in Regression Model Under α-Mixing Assumptions

This paper aims at working out economic groupscreening plans to sort out defective items from a population which consists of tems with unequal a-priori probabilities of being defective. It is shown that in the case of group-screening from a population with unequal a-priori probabilities of factors being defective, the number of obseruations needed on the average is considerably smaller than that required in the case of a population with factors having the same a-priori probability of being defective. Tables at the end give some group-screening plans as illustrations.     

On Cramér–von Mises type test based on local time of switching diffusion process

We consider a Cramér–von Mises type test for hypothesis that the observed diffusion process has sign-type trend coefficient based on empirical density function. It is shown that the limit distribution of the proposed test statistic is defined by the integral type functional of continuous Gaussian process. We provide the Karhunen–Loève expansion of the corresponding limiting process. Approximation of the threshold is given through the representation for the limit statistic.     

Semiparametric model average prediction in panel data analysis

Forecasting in economic data analysis is dominated by linear prediction methods where the predicted values are calculated from a fitted linear regression model. With multiple predictor variables, multivariate nonparametric models were proposed in the literature. However, empirical studies indicate the prediction performance of multi-dimensional nonparametric models may be unsatisfactory. We propose a new semiparametric model average prediction (SMAP) approach to analyse panel data and investigate its prediction performance with numerical examples. Estimation of individual covariate effect only requires univariate smoothing and thus may be more stable than previous multivariate smoothing approaches. The estimation of optimal weight parameters incorporates the longitudinal correlation and the asymptotic properties of the estimated results are carefully studied in this paper.     

Testing the equality among distribution functions from independent and right censored samples via Cramér–von Mises criterion

The traditional Cramér–von Mises criterion is used in order to develop a test to compare the equality of the underlying lifetime distributions in the presence of independent censoring times. Its asymptotic distribution is proved and a resampling plan, which is valid for unbalanced data situations, is proposed. Its statistical power is studied and compared with commonly used linear rank tests by Monte Carlo simulations and a real data analysis is also considered. It is observed that the new test is clearly more powerful than the traditional ones when there exists no uniform dominance among involved distributions and in the presence of late differences. Its statistical power is also good in the other considered scenarios.     

Nonexponential Applications of a Global Cramèr-Rao Inequality

The paper is concerned with an application of the information inequality for the Bayes risk (global Cramèr-Rao inequality) to nonexponential estimation problems. A new methodology of proving minimaxity is presented by considering the example of estimating the scale or location parameter under one-sided truncation of the parameter space.     

Cramér-Rao lower bounds for estimators based on censored data

The Cramér-Rao lower bounds for the variances of unbiased estimators based on censored data are given. Useful techniques of evaluation are then derived for these lower bounds. Examples are given to illustrate these techniques. Small-sample comparisons are made between the resulting lower bounds, the variances of the best linear unbiased estimators, and the variances of unbiased esti-mators which are based on the maximum likelihood estimators.     

A Nonstationary Cylinder–Based Model Describing Group Dispersal in a Fragmented Habitat

□ A doubly nonstationary cylinder-based model is built to describe the dispersal of a population from a point source. In this model, each cylinder represents a fraction of the population, i.e., a group. Two contexts are considered: The dispersal can occur in a uniform habitat or in a fragmented habitat described by a conditional Boolean model. After the construction of the models, we investigate their properties: the first and second order moments, the probability that the population vanishes, and the distribution of the spatial extent of the population.     

Asymptotic normality of Huber–Dutter estimators in a linear model with AR(1) processes

The paper studies a linear regression model with first order autoregressive (AR(1)) processes. The Huber–Dutter (HD) estimators of unknown parameters are given, and the asymptotic normality of the HD estimators is investigated. An example is presented to illustrate the proposed method.     

Semiparametric inference for open populations using the Jolly–Seber model: a penalized spline approach

When there are frequent capture occasions, both semiparametric and nonparametric estimators for the size of an open population have been proposed using kernel smoothing methods. While kernel smoothing methods are mathematically tractable, fitting them to data is computationally intensive. Here, we use smoothing splines in the form of P-splines to provide an alternate less computationally intensive method of fitting these models to capture–recapture data from open populations with frequent capture occasions. We fit the model to capture data collected over 64 occasions and model the population size as a function of time, seasonal effects and an environmental covariate. A small simulation study is also conducted to examine the performance of the estimators and their standard errors.     

Asymptotic normal tests for integration in panels with cross-dependent units

The asymptotically normal, regression-based LM integration test is adapted for panels with correlated units. The N different units may be integrated of different (fractional) orders under the null hypothesis. The paper first reviews conditions under which the test statistic is asymptotically (as T→∞) normal in a single unit. Then we adopt the framework of seemingly unrelated regression [SUR] for cross-correlated panels, and discuss a panel test statistic based on the feasible generalized least squares [GLS] estimator, which follows a χ ²(N) distribution. Third, a more powerful statistic is obtained by working under the assumption of equal deviations from the respective null in all units. Fourth, feasible GLS requires inversion of sample covariance matrices typically imposing T>N; in addition we discuss alternative covariance matrix estimators for T<N. The usefulness of our results is assessed in Monte Carlo experimentation.     

Asymptotic efficiency of estimation in the partial Koziol–Green model

In this paper, the convolution theorem and the minimax theorem for estimating the survival function in the partial Koziol–Green model (PKG) are presented. The result indicates that the partial Abdushukurov–Cheng–Lin (ACL) estimator in the PKG model is asymptotically efficient in the sense of being the least dispersed regular estimator. Consequently, the calculation shows that the ACL estimator in the KG model is also asymptotically efficient.     

Preisindikatoren für Wohnimmobilien in Deutschland

Zusammenfassung: Vermögenspreise im Allgemeinen und Immobilienpreise im Besonderen gewannen in den zurückliegenden Jahren mehr und mehr an Bedeutung. Während sie in den späten 80er Jahren (nach dem Börsencrash im Herbst 1987) und im vergangenen Jahrzehnt vornehmlich unter dem Schlagwort asset-price inflation/deflation betrachtet wurden, stehen neuerdings die Tragfähigkeit und Bestandsfestigkeit der Finanzsysteme im Vordergrund. In den Ausführungen geht es vor allem um die Frage, warum, seit wann und aufgrund welcher Grunddaten die Deutsche Bundesbank auf diesem Gebiet der Preisstatistik tätig geworden ist. Dabei wird nicht nur auf das hohe Maß an Unsicherheit in den vorgelegten Angaben hingewiesen, sondern auch der Second–Best–Charakter der Berechnungen hervorgehoben.

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Summary: Asset prices in general and property prices in particular have gained increasing importance in recent years. Whereas in the late 1980s (after the stock market crash in autumn 1987) and in the last decade these prices mainly came under the heading of asset-price inflation/deflation, the focus has recently shifted to sustainable and viable financial systems. The notes primarily explain why the Bundesbank is involved in this area of price statistics, when this involvement began and what underlying data the Bundesbank uses. At the same time, they not only indicate the large degree of uncertainty in the reported data but also highlight the second-best nature of the calculations.

*Vortrag anlässlich der 9. Konferenz Messen der Teuerung am 17./18. Juni 2004 in Marburg. Der Verfasser gibt seine persönliche Auffassung wieder, die nicht unbedingt mit derjenigen der Deutschen Bundesbank übereinstimmen muss.