Chris A. J. Klaassen

Efficient and adaptive estimation for semiparametric models

Introduction.- Asymptotic Inference for (Finite-Dimensional) Parametric Models.- Information Bounds for Euclidean Parameters in Infinite-Dimensional Models.- Euclidean Parameters: Further Examples.- Information Bounds for Infinite-Dimensional Parameters.- Infinite-Dimensional Parameters: Further Examples: Construction of Examples.

Adaptive estimation in time series models

In a framework particularly suited for many time-series models we obtain a LAN result under quite natural and economical conditions. This enables us to construct adaptive estimators for (part of) the Euclidean parameter in these semiparametric models. Special attention is directed to group models in time series with the important subclass of models with time varying location and scale. Our set-up is confronted with the existing literature and, as examples, we reconsider linear regression and ARMA, TAR and ARCH models.

√n-consistent parameter estimation for systems of ordinary differential equations: bypassing numerical integration via smoothing

We consider the problem of parameter estimation for a system of ordinary differential equations from noisy observations on a solution of the system. In case the system is nonlinear, as it typically is in practical applications, an analytic solution to it usually does not exist. Consequently, straightforward estimation methods like the ordinary least squares method depend on repetitive use of numerical integration in order to determine the solution of the system for each of the parameter values considered, and to find subsequently the parameter estimate that minimises the objective function. This induces a huge computational load to such estimation methods. We study the consistency of an alternative estimator that is defined as a minimiser of an appropriate distance between a nonparametrically estimated derivative of the solution and the righthand side of the system applied to a nonparametrically estimated solution. This smooth and match estimator (SME) bypasses numerical integration altogether and reduces the amount of computational time drastically compared to ordinary least squares. Moreover, we show that under suitable regularity conditions this smooth and match estimation procedure leads to a √ n-consistent estimator of the parameter of interest.

Squared skewness minus kurtosis bounded by 186/125 for unimodal distributions

The sharp inequality for squared skewness minus kurtosis is derived for the class of unimodal distributions.

Survival analysis under cross sectional sampling: length bias and multiplicative censoring

Consider a parametric, nonparametric or semiparametric model for survival times. Interest is in estimation of Euclidean and Banach parameters for these models. However, not the survival times themselves will be observed, since this might be quite time consuming. Instead, cross-sectional sampling is applied: at some point in time one identifies a random sample from the population under study and one registers the survival time up to this time-point. Typically, the resulting reduced survival times do not have the same distributions as the true survival times. On the one hand, longer survival times have a higher probability to be sampled than smaller ones. On the other hand, the observed survival times have been censored multiplicatively. The length bias and multiplicative censoring properties of cross-sectional sampling will be discussed and reviewed as well as estimation in the resulting parametric, nonparametric, and semiparametric models.

Efficient Estimation of Banach Parameters in Semiparametric Models

Abtract: Consider a semiparametric model with a Euclidean parameter and an infinite dimensional parameter, to be called Banach parameter. Assume:- There exists an efficient estimator of the Euclidean parameter.- When the value of the Euclidean parameter is known, there exists an estimator of the Banach parameter, which depends on this value and is efficient within this restricted model.Substituting the efficient estimator of the Euclidean parameter for the value of this parameter in the estimator of the Banach parameter, one obtains an efficient estimator of the Banach parameter for the full semiparametric model with the Euclidean parameter unknown. This heredity property of efficiency completes estimation in semiparametric models in which the Euclidean parameter has been estimated efficiently. Typically, estimation of both the Euclidean and the Banach parameter is necessary in order to describe the random phenomenon under study to sufficient extent. Since efficient estimators are asymptotically linear the above substitution method is a particular case of substituting asymptotically linear estimators of a Euclidean parameter into estimators that are asymptotically linear themselves and that depend on this Euclidean parameter. This more general substitution case is studied for its own sake as well, and a heredity property for asymptotic linearity is proved.

Optimized appointment scheduling

In service systems, in order to balance the server’s idle times and the customers’ waiting times, one may fix the arrival times of the customers beforehand in an appointment schedule. We propose a procedure for determining appointment schedules in such a D/G/1-type of system by sequentially minimizing the per-customer expected loss. Our approach provides schedules for any convex loss function; for the practically relevant cases of the quadratic and absolute value loss functions appealing closed-form results are derived. Importantly, our approach does not impose any conditions on the service time distribution; it is even allowed that the customers’ service times have different distributions.

Credit in Acceptance Sampling on Attributes

Credit is introduced in acceptance sampling on attributes and a credit-based acceptance sampling system is developed that is very easy to apply in practice. The credit of a producer is defined as the total number of items accepted since the last rejection. In my sampling system, the sample size for a lot depends via a simple function on the lot size, the credit, and the chosen guaranteed upper limit on the outgoing quality and will be much smaller than in isolated lot inspection. My credit-based acceptance sampling system also yields a simple continuous sampling plan.

Optimal rate of direct estimators in systems of ordinary differential equations linear in functions of the parameters

Many processes in biology, chemistry, physics, medicine, and engineering are modeled by a system of differential equations. Such a system is usually characterized via unknown parameters and estimating their ‘true’ value is thus required. In this paper we focus on the quite common systems for which the derivatives of the states may be written as sums of products of a function of the states and a function of the parameters. For such a system linear in functions of the unknown parameters we present a necessary and sufficient condition for identifiability of the parameters. We develop an estimation approach that bypasses the heavy computational burden of numerical integration and avoids the estimation of system states derivatives, drawbacks from which many classic estimation methods suffer. We also suggest an experimental design for which smoothing can be circumvented. The optimal rate of the proposed estimators, i.e., their √n-consistency, is proved and simulation results illustrate their excellent finite sample performance and compare it to other estimation approaches.

Consistent Estimation of the Structural Distribution Function

Motivated by problems in linguistics we consider a multinomial random vector for which the number of cells N is not much smaller than the sum of the cell frequencies, i.e. the sample size n. The distribution function of the uniform distribution on the set of all cell probabilities multiplied by N is called the structural distribution function of the cell probabilities. Conditions are given that guarantee that the structural distribution function can be estimated consistently as n increases indefinitely although n/N does not. The natural estimator is inconsistent and we prove consistency of essentially two alternative estimators.