In order to develop a general criterion for proving strong consistency of estimators in Statistics of stochastic processes, we study an extension, to the continuous-time case, of the strong law of large numbers for discrete time square integrable martingales (e.g. Neveu, 1965, 1972). Applications to estimation in diffusion models are given.
In the present work, we briefly analyze the development of the mathematical theory of records. We first consider applications associated with records. We then view distributional and limit results for record values and times. We further present methods of generation of continuous records. In the end of this work, we discuss some tests based on records.
Let be the empirical distribution function (df) pertaining to independent random variables with continuous df . We investigate the minimizing point of the empirical process , where is another df which differs from . If and are locally Hölder-continuous of order at a point our main result states that converges in distribution. The limit variable is the almost sure unique minimizing point of a two-sided time-transformed homogeneous Poisson-process with a drift. The time-transformation...
Let Fn be the empirical distribution function (df) pertaining to independent random variables with continuous df F. We investigate the minimizing point of the empirical process Fn - F0, where F0 is another df which differs from F. If F and F0 are locally Hölder-continuous of order α at a point τ our main result states that converges in distribution. The limit variable is the almost sure unique minimizing point of a two-sided time-transformed homogeneous Poisson-process with a drift. The time-transformation...
An asymptotic local power of Pearson chi-squared tests is considered, based on convex mixtures of the null densities with fixed alternative densities when the mixtures tend to the null densities for sample sizes This local power is used to compare the tests with fixed partitions of the observation space of small partition sizes with the tests whose partitions depend on and the partition sizes tend to infinity for . New conditions are presented under which it is asymptotically optimal...
We study sample-based estimates of the expectation of the function produced by the empirical minimization algorithm. We investigate the extent to which one can estimate the rate of convergence of the empirical minimizer in a data dependent manner. We establish three main results. First, we provide an algorithm that upper bounds the expectation of the empirical minimizer in a completely data-dependent manner. This bound is based on a structural result due to Bartlett and Mendelson, which relates...
We study the performance of empirical risk minimization (ERM), with respect to the quadratic risk, in the context of convex aggregation, in which one wants to construct a procedure whose risk is as close as possible to the best function in the convex hull of an arbitrary finite class . We show that ERM performed in the convex hull of is an optimal aggregation procedure for the convex aggregation problem. We also show that if this procedure is used for the problem of model selection aggregation,...
The main goal of supervised learning is to construct a function from labeled training data which assigns arbitrary new data points to one of the labels. Classification tasks may be solved by using some measures of data point centrality with respect to the labeled groups considered. Such a measure of centrality is called data depth. In this paper, we investigate conditions under which depth-based classifiers for directional data are optimal. We show that such classifiers are equivalent to the Bayes...
2000 Mathematics Subject Classification: 62G07, 62L20.Tsybakov [31] introduced the method of stochastic approximation to construct a recursive estimator of the location q of the mode of a probability density. The aim of this paper is to provide a companion algorithm to Tsybakov's algorithm, which allows to simultaneously recursively approximate the size m of the mode. We provide a precise study of the joint weak convergence rate of both estimators. Moreover, we introduce the averaging principle...
For the Bickel-Rosenblatt goodness-of-fit test with fixed bandwidth studied by Fan (1998) we derive its Bahadur exact slopes in a neighbourhood of a simple hypothesis f = f0 and we use them to get a better understanding on the role played by the smoothing parameter in the detection of departures from the null hypothesis. When f0 is an univariate normal distribution and we take for kernel the standard normal density function, we compute these slopes for a set of Edgeworth alternatives which give...