Volume 21, Issue 1 - 1

A process X is observed continuously in time; it behaves like Brownian motion with drift, which changes from zero to a known constant ϑ>0 at some time τ that is not directly observable. It is important to detect this change when it happens, and we attempt to do so by selecting a stopping rule T * that minimizes the “expected miss” E | T −τ| over all stopping rules T . Assuming that τ has an exponential distribution with known parameter λ>0 and is independent of the driving Brownian motion, we show that the optimal rule T * is to declare that the change has occurred, at the first time t for which . Here, with Λ=2λ/ϑ 2 , the constant p * is uniquely determined in (½,1) by the equation .

Regression analysis of a response variable Y requires careful selection of explanatory variables. The quality of a set of explanatory features X =( X (1) ,..., X ( d ) ) can be measured in terms of the minimum mean squared error . This paper investigates methods for estimating L * from i.i.d. data. No estimate can converge rapidly for all distributions of ( X , Y ). For Lipschitz continuous regression function E { Y | X = x }, two estimators for L * are discussed: fitting a regression estimate to a subset of the data and assessing its mean residual sum of squares on the remaining samples, and a nearest neighbor cross-validation type estimate.

The present paper deals with the problem of parameter estimation for nonlinear stochastic differential equations with solution tending to infinity with time. It is shown that if the trend coefficient is asymptotically linear (like that of an Ornstein-Uhlenbeck process), then the maximum likelihood and trajectory fitting estimators are consistent and asymptotically mixing normal. That is, these estimators behave similar as in the case of a non-ergodic Ornstein-Uhlenbeck process.

In the framework of general semimartingale models we provide limit theorems for variational sums including the p -th power variation, i.e. the sum of p -th absolute powers of increments of a process. This gives new insight in the use of quadratic and realised power variation as an estimate for the integrated volatility in finance. It also provides a criterion to decide from high frequency data, whether a jump component should be included in the model. Furthermore, results on the asymptotic behaviour of integrals with respect to Lévy processes, estimates for integrals with respect to Lévy measures and non-parametric estimation for Lévy processes will be derived and viewed in the framework of variational sums.

The problem of estimating a mean vector for spherically symmetric distributions with the quadratic loss function is considered. A robust generalized Bayes estimator improving on the James-Stein estimator is given.

In this paper we consider a general control scheme. The control statistic Z t is equal to an arbitrary weighted sum of the past observations X t ,..., X 1 . This approach covers most of the applied control schemes like for instance moving average, EWMA and ARMA(1,1) charts. The process { X t } is assumed to be a stationary Gaussian process. The aim of the work is to analyze the behaviour of the tail probability of the run length N =inf{ t ∈ℕ: Z t − E ( Z t )> c √{Var( Z t )}} with respect to the autocorrelation of { X t }. It is shown under which conditions on the weights and on the autocorrelations of { X t } the correlation between Z t and Z t − i is a nondecreasing function in the autocorrelations of the observed process. Using this result it can be proved that the probability of a false alarm is a nondecreasing function of the autocorrelations of { X t }, too. The weight conditions are verified for several well-known charts.