Hajo Holzmann

Hidden Markov models for circular and linear-circular time series

We introduce a new class of circular time series based on hidden Markov models. These are compared with existing models, their properties are outlined and issues relating to parameter estimation are discussed. The new models conveniently describe multi-modal circular time series as dependent mixtures of circular distributions. Two examples from biology and meteorology are used to illustrate the theory. Finally, we introduce a hidden Markov model for bivariate linear-circular time series and use it to describe larval movement of the fly Drosophila.

Statistical inference for inverse problems

In this paper we study statistical inference for certain inverse problems. We go beyond mere estimation purposes and review and develop the construction of confidence intervals and confidence bands in some inverse problems, including deconvolution and the backward heat equation. Further, we discuss the construction of certain hypothesis tests, in particular concerning the number of local maxima of the unknown function. The methods are illustrated in a case study, where we analyze the distribution of heliocentric escape velocities of galaxies in the Centaurus galaxy cluster, and provide statistical evidence for its bimodality.

The role of the information set for forecasting—with applications to risk management

Predictions are issued on the basis of certain information. If the forecasting mechanisms are correctly specified, a larger amount of available information should lead to better forecasts. For point forecasts, we show how the effect of increasing the information set can be quantified by using strictly consistent scoring functions, where it results in smaller average scores. Further, we show that the classical Diebold-Mariano test, based on strictly consistent scoring functions and asymptotically ideal forecasts, is a consistent test for the effect of an increase in a sequence of information sets on

Testing for Image Symmetries—With Application to Confocal Microscopy

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Peak vs Components

-step point forecasts. For the value at risk (VaR), we show that the average score, which corresponds to the average quantile risk, directly relates to the expected shortfall. Thus, increasing the information set will result in VaR forecasts which lead on average to smaller expected shortfalls. We illustrate our results in simulations and applications to stock returns for unconditional versus conditional risk management as well as univariate modeling of portfolio returns versus multivariate modeling of individual risk factors. The role of the information set for evaluating probabilistic forecasts by using strictly proper scoring rules is also discussed.

Testing parametric assumptions on band- or time-limited signals under noise

Statistical tests are introduced for checking whether an image function f(x,y) defined on the unit disc D={(x,y): x 2+y 2 les 1} is invariant under certain symmetry transformations of D , given that discrete and noisy data are observed. Invariance under reflections or under rotations by rational angles is considered, as well as rotational invariance. These symmetry relations can be naturally expressed as restrictions for the Zernike moments of f(x,y). Therefore, the test statistics are based on the L 2 distance between Zernike series estimates of the image function itself and its version obtained after applying the symmetry transformation. The asymptotic distribution of the test statistics under both the hypothesis of symmetry as well as under fixed alternatives is derived. Furthermore, the quality of the asymptotic approximations via simulation studies is investigated. The usefulness of our theory is verified by examining an important problem in confocal microscopy, i.e., possible imprecise alignments in the optical path of the microscope are investigated. For optical systems with rotational symmetry, the theoretical point-spread function (PSF) is reflection symmetric with respect to two orthogonal axes, and rotationally invariant if the detector plane matches the optical plane of the microscope. The tests are used to investigate whether the required symmetries can indeed be detected in the empirical PSF.

THE CENTRAL LIMIT THEOREM FOR STATIONARY MARKOV CHAINS UNDER INVARIANT SPLITTINGS

We analyze the cross‐national distribution of gross domestic product (GDP) per capita and its evolution from 1970 to 2009. We argue that peaks are not a suitable measure for distinct convergence clubs/equilibria in the cross‐country distribution of GDP per capita, because the number of peaks is not invariant under non‐linear strictly monotonic transformations of the data such as the logarithmic transformation. Instead, we model the distribution as a finite mixture, and determine its number of components via statistical testing. We find that the number of components in the cross‐country distribution changes from three to two in the mid 1990s.

DEMAND ANALYSIS AS AN ILL-POSED INVERSE PROBLEM WITH SEMIPARAMETRIC SPECIFICATION

This paper considers the problem of testing parametric assumptions on signals f from which only noisy observations y/sub k/=f(/spl tau/k)+/spl epsi//sub k/ are available, and where the signal is assumed to be either band-limited or time-limited. To this end, the signal is reconstructed by an estimator based on the Whittaker-Shannon (WS) sampling theorem with oversampling. As test statistic, the minimal L/sub 2/ distance between the estimated signal and the parametric model is used. To construct appropriate tests, the asymptotic distribution of the test statistic is derived both under the hypothesis of the validity of the parametric model and under fixed local alternatives. As a byproduct, the asymptotic distribution of the integrated square error of the estimator is computed, which is of interest by itself, e.g., for the analysis of a cross-validated bandwidth selector.

Semiparametric location mixtures with distinct components

The central limit theorem (CLT) for stationary ergodic Markov chains is investigated. We give a short survey of related results on the CLT for general (not necessarily Harris recurrent) chains and formulate a new sufficient condition for its validity. Furthermore, Markov operators are considered which admit invariant orthogonal splittings of the space of square-integrable functions. We show how conditions for the CLT can be improved if this additional structure is taken into account. Finally we give examples of this situation, namely endomorphisms of compact Abelian groups and random walks on compact homogeneous spaces.

Confidence bands for inverse regression models

In this paper we are concerned with analyzing the behavior of a semiparametric estimator that corrects for endogeneity in a nonparametric regression by assuming mean independence of residuals from instruments only. Because it is common in many applications, we focus on the case where endogenous regressors and additional instruments are jointly normal, conditional on exogenous regressors. This leads to a severely ill-posed inverse problem. In this setup, we show first how to test for conditional normality. More importantly, we then establish how to exploit this knowledge when constructing an estimator, and we derive the large sample behavior of such an estimator. In addition, in a Monte Carlo experiment we analyze its finite sample behavior. Our application comes from consumer demand. We obtain new and interesting findings that highlight both the advantages and the difficulties of an approach that leads to ill-posed inverse problems. Finally, we discuss the somewhat problematic relationship between endogenous nonparametric regression models and the recently emphasized issue of unobserved heterogeneity in structural models.