The paper deals with a class of discrete-time stochastic control processes under a discounted optimality criterion with random discount rate, and possibly unbounded costs. The state process { x_{t} } and the discount process { \alpha_{t} } evolve according to the coupled difference equations x_{t+1}=F(x_{t},\alpha _{t},a_{t},\xi _{t}), \alpha_{t+1}=G(\alpha _{t},\eta _{t}) where the state and discount disturbance processes { \xi _{t} } and { \eta _{t} } are sequences of i.i.d. random variables with densities \rho^{\xi } and \rho^{\eta } respectively. The main objective is to introduce approximation algorithms of the optimal cost function that lead up to construction of optimal or nearly optimal policies in the cases when the densities \rho ^{\xi } and \rho ^{\eta } are either known or unknown. In the latter case, we combine suitable estimation methods with control procedures to construct an asymptotically discounted optimal policy.
The absolute population density of adult Pardosa agrestis (Westring, 1862), the dominant epigeic spider species in many arable lands in Central Europe, was quantified in two alfalfa fields using a multiple mark-recapture method. The resulting density estimates are presented together with catch data from simultaneously performed suction sampling and pitfall trapping. Two week long mark-recapture surveys were conducted in August 1995 and 1996 using grids of 11 × 11 live-catching pitfall traps covering a square area of 400 m2 in the first, and 900 m2 in the second survey. The trap checking and marking procedure, using individual codes, was done daily. Over 5,000 spiders were marked in the two surveys. The number of spiders caught varied greatly between days. The recapture rate was also variable, ranging between 5-19%. Recaptured animals moved considerable daily distances, marked animals left the area of the trapping grid within a few days. Population sizes were estimated for 3 day time windows during which period the population could be considered closed. Spider densities were calculated from the estimated abundances and the capture area, which consisted of the grid area plus a boundary strip calculated from the estimated movement ranges of spiders. The resulting densities were close to 2 males and 1 female per m2 in the first, and 4.5 males and 4.5 females per m2 in the second experimental site. Suction sampling caught very few adult individuals and gave unsatisfactory data for statistical comparison. Live catching pitfall trap catches did not correlate with the derived mark-recapture density estimates across the short time windows, but for the entire trapping sessions the catches were consistent with the density estimates.
We consider, in the framework of multidimensional observations, nonparametric functional estimators, which include, as special cases, the Akaike-Parzen-Rosenblatt kernel density estimators ([1], [18], [20]), and the Nadaraya-Watson kernel regression estimators ([16], [22]). We evaluate the sup-norm, over a given set <span class="tex"><b>I</b></span>, of the difference between the estimator and a non-random functional centering factor (which reduces to the estimator mean for kernel density estimation). We show that, under suitable general conditions, this random quantity is consistently estimated by the sup-norm over <span class="tex"><b>I</b></span> of the difference between the original estimator and a bootstrapped version of this estimator. This provides a simple and flexible way to evaluate the estimator accuracy, through a single bootstrap. The present work generalizes former results of Deheuvels and Derzko [4], given in the setup of density estimation in <b>R</b>.