## Report in Wirtschaftsmathematik (WIMA Report)

- 148
- Monitoring time series based on estimating functions (2014)
- A large class of estimators including maximum likelihood, least squares and M-estimators are based on estimating functions. In sequential change point detection related monitoring functions can be used to monitor new incoming observations based on an initial estimator, which is computationally efficient because possible numeric optimization is restricted to the initial estimation. In this work, we give general regularity conditions under which we derive the asymptotic null behavior of the corresponding tests in addition to their behavior under alternatives, where conditions become particularly simple for sufficiently smooth estimating and monitoring functions. These regularity conditions unify and even extend a large amount of existing procedures in the literature, while they also allow us to derive monitoring schemes in time series that have not yet been considered in the literature including non-linear autoregressive time series and certain count time series such as binary or Poisson autoregressive models. We do not assume that the estimating and monitoring function are equal or even of the same dimension, allowing for example to combine a non-robust but more precise initial estimator with a robust monitoring scheme. Some simulations and data examples illustrate the usefulness of the described procedures.

- 149
- Edgeworth expansions for lattice triangular arrays (2014)
- Edgeworth expansions have been introduced as a generalization of the central limit theorem and allow to investigate the convergence properties of sums of i.i.d. random variables. We consider triangular arrays of lattice random vectors and obtain a valid Edgeworth expansion for this case. The presented results can be used, for example, to study the convergence behavior of lattice models.

- 146
- Maximum Likelihood Estimators for Multivariate Hidden Markov Mixture Models (2013)
- In this paper we consider a multivariate switching model, with constant states means and covariances. In this model, the switching mechanism between the basic states of the observed time series is controlled by a hidden Markov chain. As illustration, under Gaussian assumption on the innovations and some rather simple conditions, we prove the consistency and asymptotic normality of the maximum likelihood estimates of the model parameters.

- 147
- On the Generality of the Greedy Algorithm for Solving Matroid Base Problems (2013)
- It is well known that the greedy algorithm solves matroid base problems for all linear cost functions and is, in fact, correct if and only if the underlying combinatorial structure of the problem is a matroid. Moreover, the algorithm can be applied to problems with sum, bottleneck, algebraic sum or \(k\)-sum objective functions.

- 139a
- Asymptotic Order of the Parallel Volume Difference (2012)
- In this paper we investigate the asymptotic behaviour of the parallel volume of fixed non-convex bodies in Minkowski spaces as the distance \(r\) tends to infinity. We will show that the difference of the parallel volume of the convex hull of a body and the parallel volume of the body itself can at most have order \(r^{d-2}\) in a \(d\)-dimensional space. Then we will show that in Euclidean spaces this difference can at most have order \(r^{d-3}\). These results have several applications, e.g. we will use them to compute the derivative of \(f_\mu(rK)\) in \(r = 0\), where \(f_\mu\) is the Wills functional or a similar functional, \(K\) is a body and \(rK\) is the Minkowski-product of \(r\) and \(K\). Finally we present applications concerning Brownian paths and Boolean models and derive new proofs for formulae for intrinsic volumes.

- 144
- A limitation of the estimation of intrinsic volumes via pixel configuration counts (2012)
- It is often helpful to compute the intrinsic volumes of a set of which only a pixel image is observed. A computational efficient approach, which is suggested by several authors and used in practice, is to approximate the intrinsic volumes by a linear functional of the pixel configuration histogram. Here we want to examine, whether there is an optimal way of choosing this linear functional, where we will use a quite natural optimality criterion that has already been applied successfully for the estimation of the surface area. We will see that for intrinsic volumes other than volume or surface area this optimality criterion cannot be used, since estimators which ignore the data and return constant values are optimal w.r.t. this criterion. This shows that one has to be very careful, when intrinsic volumes are approximated by a linear functional of the pixel configuration histogram.

- 133
- On a Cardinality Constrained Multicriteria Knapsack Problem (2011)
- We consider a variant of a knapsack problem with a fixed cardinality constraint. There are three objective functions to be optimized: one real-valued and two integer-valued objectives. We show that this problem can be solved efficiently by a local search. The algorithm utilizes connectedness of a subset of feasible solutions and has optimal run-time.

- 137
- Testing for parameter stability in nonlinear autoregressive models (2011)
- In this paper we develop testing procedures for the detection of structural changes in nonlinear autoregressive processes. For the detection procedure we model the regression function by a single layer feedforward neural network. We show that CUSUM-type tests based on cumulative sums of estimated residuals, that have been intensively studied for linear regression, can be extended to this case. The limit distribution under the null hypothesis is obtained, which is needed to construct asymptotic tests. For a large class of alternatives it is shown that the tests have asymptotic power one. In this case, we obtain a consistent change-point estimator which is related to the test statistics. Power and size are further investigated in a small simulation study with a particular emphasis on situations where the model is misspecified, i.e. the data is not generated by a neural network but some other regression function. As illustration, an application on the Nile data set as well as S&P log-returns is given.

- 138
- A uniform central limit theorem for neural network based autoregressive processes with applications to change-point analysis (2011)
- We consider an autoregressive process with a nonlinear regression function that is modeled by a feedforward neural network. We derive a uniform central limit theorem which is useful in the context of change-point analysis. We propose a test for a change in the autoregression function which - by the uniform central limit theorem - has asymptotic power one for a large class of alternatives including local alternatives.

- 139
- Asymptotic order of the parallel volume difference (2011)
- In this paper we continue the investigation of the asymptotic behavior of the parallel volume in Minkowski spaces as the distance tends to infinity that was started in [13]. We will show that the difference of the parallel volume of the convex hull of a body and the parallel volume of the body itself can at most have order \(r^{d-2}\) in dimension \(d\). Then we will show that in the Euclidean case this difference can at most have order \(r^{d-3}\). We will also examine the asymptotic behavior of the derivative of this difference as the distance tends to infinity. After this we will compute the derivative of \(f_\mu (rK)\) in \(r\), where \(f_\mu\) is the Wills functional or a similar functional, \(K\) is a fixed body and \(rK\) is the Minkowski-product of \(r\) and \(K\). Finally we will use these results to examine Brownian paths and Boolean models and derive new proofs for formulae for intrinsic volumes.