Volume 22, Issue 3 - 3

We give a precise statement and proof of a projection formula due to J. Robins for computing influence functions in models where the observation is a random process that is stopped by a coarsening-at-random mechanism. This projection formula is important for constructing estimating equations. We illustrate this by computing the efficient influence functions in the current status model with time-dependent covariates.

Generalizing MSE-optimality on 1/√ n -shrinking neighborhoods of contamination type, we determine the robust influence curve that minimizes maximum asymptotic risk, where risk may be any convex and isotone function G of L 2 - and L ∞ -norms. The solutions necessarily minimize the trace of the covariance subject to an upper bound on the sup-norm, and also include an implicit equation for the optimal bound. For parameter dimension p  = 1, also the asymptotic minimax problem for neighborhoods of total variational type is solved. In technical respects, general risk may be reduced to MSE by weighting bias suitably. In case p  = 1, the result covers L q -risks, q  ∈ [1,∞), confidence intervals of minimal length, and over-/undershooting probabilities. In case p  > 1, in addition to the L ∞ -norm, a solution for coordinatewise norms is given (relevant for total variation, p  > 1). Passing to the least favorable contamination radius as in [RKR01], we obtain that for a large class of risks, the radius-minimax procedure does not depend on the function G .

In this note we revisit Merton’s optimal portfolio selection problem in an exponential Lévy market, for an agent acting with a canonical utility function, the logarithm. As we show, the explicit optimal portfolios exhibit features similar to some of the pathological examples given by Kramkov and Schachermayer where certain discounted asset processes fail to be martingales. In our examples, these pathologies are seen to arise from natural shortselling and borrowing constraints imposed by the logarithmic utility.

The experiments of nonparametric regression with equidistant design points and Gaussian white noise are considered. Brown and Low have proven asymptotic equivalence of these models under a quite general smoothness assumption on the parameter space of regression functions. In the present paper we focus on periodic Sobolev classes. We prove asymptotic equivalence of nonparametric regression and white noise with a construction different to Brown and Low. Whereas their original method cannot give a better rate than n −1/2 for the smoothness classes under consideration, even if the underlying function class is actually smoother than just Lipschitz, in the present work a rate of convergence n −β+1/2 for the delta-distance over a Sobolev class with any smoothness index β > 1/2 is derived. Furthermore, the results are constructive and therefore lead to a simple transfer of decision procedures.

The paper concerns with parameter estimation in a heteroscedastic random coefficient autoregressive (RCA) model of the form X t = b t X t −1 + Y t . A conditionally weighted least squares (CWLS) estimator of β =  Eb t is studied. Its strong consistency and asymptotic normality are proved. For this purpose theory of near-epoch dependent (NED) processes is used. Consistency results are also obtained in case that variances both of the random parameter and heterogeneous errors are unknown and have to be estimated. Some simulations are presented to support the theory.