Peter Spreij

A kernel type nonparametric density estimator for decompounding

Abstract Given a sample from a discretely observed compound Poisson process, we consider estimation of the density of the jump sizes. We propose a kernel type nonparametric density estimator and study its asymptotic properties. An order bound for the bias and an asymptotic expansion of the variance of the estimator are given. Pointwise weak consistency and asymptotic normality are established. The results show that, asymptotically, the estimator behaves very much like an ordinary kernel estimator. Keywords: asymptotic normality; consistency; decompounding; kernel estimation Full-text: Access by subscription (subscriber: Univ Biblio SZ (UVA))

Approximation of stationary processes by hidden Markov models

Stochastic realization is still an open problem for the class of hidden Markov models (HMM): given the law Q of an HMM find a finite parametric description of it. Fifty years after the introduction of HMMs, no computationally effective realization algorithm has been proposed. In this paper we direct our attention to an approximate version of the stochastic realization problem for HMMs. We aim at the realization of an HMM of assigned complexity (number of states of the underlying Markov chain) which best approximates, in Kullback Leibler divergence rate, a given stationary law Q. In the special case of Q being the law of an HMM this corresponds to solving the approximate realization problem for HMMs. In general there is no closed form expression of the Kullback Leibler divergence rate, therefore we replace it, as approximation criterion, with the informational divergence between the Hankel matrices of the processes. This not only has the advantage of being easy to compute, while providing a good approximation of the divergence rate, but also makes the problem amenable to the use of nonnegative matrix factorization (NMF) techniques. We propose a three step algorithm, based on the NMF, which realizes an optimal HMM. The viability of the algorithm as a practical tool is tested on a few examples of HMM order reduction.

On Fisher's information matrix of an ARMAX process and Sylvester's resultant matrices

We establish a relation between Fishers information matrix of a stationary autoregressive moving average process, with an exogenous component, and two Sylvesters resultant matrices.

Tail Behavior of Credit Loss Distributions for General Latent Factor Models

Using a limiting approach to portfolio credit risk, we obtain analytic expressions for the tail behavior of credit losses. To capture the co‐movements in defaults over time, we assume that defaults are triggered by a general, possibly non‐linear, factor model involving both systematic and idiosyncratic risk factors. The model encompasses default mechanisms in popular models of portfolio credit risk, such as CreditMetrics and CreditRisk+. We show how the tail characteristics of portfolio credit losses depend directly upon the factor models functional form and the tail properties of the models risk factors. In many cases the credit loss distribution has a polynomial (rather than exponential) tail. This feature is robust to changes in tail characteristics of the underlying risk factors. Finally, we show that the interaction between portfolio quality and credit loss tail behavior is strikingly different between the CreditMetrics and CreditRisk+ approach to modeling portfolio credit risk. Correspondence to: alucas@econ.vu.nl, pieter.klaassen@nl.abnamro.com, spreij@wins.uva.nl, or s.straetmans@berfin.unimaas.nl.

Recursive approximate maximum likelihood estimation for a class of counting process models

In this paper we present a recursive algorithm that produces estimators of an unknown parameter that occurs in the intensity of a counting process. The estimators can be considered as approximations of the maximum likelihood estimator. We prove consistency of the estimators and derive their asymptotic distribution by using Lyapunov functions and weak convergence for martingales. The conditions that we impose in order to prove our results are similar to those in papers on (quasi) least squares estimation.

Spectral characterization of the optional quadratic variation process

In this paper we show how the periodogram of a semimartingale can be used to characterize the optional quadratic variation process.

Deconvolution for an atomic distribution

Let X1, . . . ,Xn be i.i.d. observations, where Xi = Yi + σZi and Yi and Zi are independent. Assume that unobservable Ys are distributed as a random variable UV, where U and V are independent, U has a Bernoulli distribution with probability of zero equal to p and V has a distribution function F with density f. Furthermore, let the random variables Zi have the standard normal distribution and let σ > 0. Based on a sample X1, . . . ,Xn, we consider the problem of estimation of the density f and the probability p. We propose a kernel type deconvolution estimator for f and derive its asymptotic normality at a fixed point. A consistent estimator for p is given as well. Our results demonstrate that our estimator behaves very much like the kernel type deconvolution estimator in the classical deconvolution problem.

On Stein's equation, Vandermonde matrices and Fisher's information matrix of time series processes. I. The autoregressive moving average process

Abstract This paper introduces several forms of relationships between Fishers information matrix of an autoregressive-moving average or ARMA process and the solution of a corresponding Stein equation. Fishers information matrix consists of blocks associated with the autoregressive and moving average parameters. An interconnection with a solution of Steins equation is set forth for the block case as well as for Fishers information matrix as a global matrix involving all parameter blocks. Both cases have their importance for the interpretation of the estimated parameters. The cases of distinct and multiple eigenvalues are addressed. The obtained links involve equations with left and right inverses, these can be expressed in terms of the inverse of appropriate Vandermonde matrices. A condition is set forth for establishing an equality between Fishers information matrix and a solution to Steins equation. Two examples are presented for illustrating some of the results obtained. The global and off-diagonal block case with distinct and multiple roots, respectively, are considered.

Markov-modulated Ornstein-Uhlenbeck processes

Abstract In this paper we consider an Ornstein–Uhlenbeck (OU) process (M(t))t≥0 whose parameters are determined by an external Markov process (X(t))t≥0 on a finite state space {1, . . ., d}; this process is usually referred to as Markov-modulated Ornstein–Uhlenbeck. We use stochastic integration theory to determine explicit expressions for the mean and variance of M(t). Then we establish a system of partial differential equations (PDEs) for the Laplace transform of M(t) and the state X(t) of the background process, jointly for time epochs t = t1, . . ., tK. Then we use this PDE to set up a recursion that yields all moments of M(t) and its stationary counterpart; we also find an expression for the covariance between M(t) and M(t + u). We then establish a functional central limit theorem for M(t) for the situation that certain parameters of the underlying OU processes are scaled, in combination with the modulating Markov process being accelerated; interestingly, specific scalings lead to drastically different limiting processes. We conclude the paper by considering the situation of a single Markov process modulating multiple OU processes.

On Fisher's information matrix of an ARMA process.

In this paper we study the Fisher information matrix for a stationary ARMA process with the aid of Sylvester’s resultant matrix. Some properties are explained via realizations in state space form of the derivates of the white noise process with respect to the parameters.