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We propose a new statistical method for obtaining information about particle size distributions from sectional data without specific assumptions about particle shape. The method utilizes recent advances in local stereology. We show how to estimate separately from sectional data the variance due to the local stereological estimation procedure and the variance due to the variability of particle sizes in the population. Methods for judging the difference between the distribution of estimated particle sizes and the distribution of true particle sizes are also provided.

Abstract. Several convenient methods for calculation of fractional absolute moments are given with application to heavy tailed distributions. We use techniques of fractional differentiation to obtain formulae for E[|X − µ| γ ] with 1 < γ < 2 and µ ∈ R, in terms of Laplace transform or characteristic function. The main focus is on heavy tailed distributions, several examples are given with analytical expressions of fractional absolute moments. As applications, we calculate the fractional moment errors for both prediction and parameter estimation problems.

Abstract. We study the following problem: How to verify Brillinger-mixing of stationary point processes in R d by imposing conditions on a suitable mixing coefficient? For this, we define an absolute regularity (or β-mixing) coefficient for point processes and derive an explicit condition in terms of this coefficient which implies finite total variation of the kth-order reduced factorial cumulant measure of the point process for fixed k ≥ 2. To prove this, we introduce higher-order covariance measures and use Statulevičius' representation formula for mixed cumulants in case of random (counting) measures. To illustrate our results, we consider some Brillinger-mixing point processes occurring in stochastic geometry.

A new statistical method for the estimation of the response latency is proposed. When spontaneous discharge is present, the first spike after the stimulus application may be caused by either the stimulus itself, or it may appear due to the prevailing spontaneous activity. Therefore, an appropriate method to deduce the response latency from the time to the first spike after the stimulus is needed. We develop a nonparametric estimator of the response latency based on repeated stimulations. A simulation study is provided to show how the estimator behaves with an increasing number of observations and for different rates of spontaneous and evoked spikes. Our nonparametric approach requires very few assumptions. For comparison, we also consider a parametric model. The proposed probabilistic model can be used for both single and parallel neuronal spike trains. In the case of simultaneously recorded spike trains in several neurons, the estimators of joint distribution and correlations of response latencies are also introduced. Real data from inferior colliculus auditory neurons obtained from a multielectrode probe are studied to demonstrate the statistical estimators of response latencies and their correlations in space.