In this paper we introduce Lévy-driven Cox point processes (LCPs) as Cox point processes with driving intensity function defined by a kernel smoothing of a Lévy basis (an independently scattered, infinitely divisible random measure). We also consider log Lévy-driven Cox point processes (LLCPs) with equal to the exponential of such a kernel smoothing. Special cases are shot noise Cox processes, log Gaussian Cox processes, and log shot noise Cox processes. We study the theoretical properties of Lévy-based Cox processes, including moment properties described by nth-order product densities, mixing properties, specification of inhomogeneity, and spatio-temporal extensions.

In the present paper, we propose a Palm likelihood approach as a general estimating principle for stationary point processes in R d for which the density of the second-order factorial moment measure is available in closed form. Examples of such point processes include the Neyman-Scott processes and the log Gaussian Cox processes. The computations involved in determining the Palm likelihood estimator are simple. Conditions are provided under which the Palm likelihood estimator is consistent and asymptotically normally distributed. R 0 [ĝ c (u) − g c (u; θ)] 2 du

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We investigate a class of kernel estimators σ 2 n of the asymptotic variance σ 2 of a d-dimensional stationary point process Ψ = i≥1 δ X i which can be observed in a cubic sampling windowand its existence is guaranteed whenever the corresponding reduced covariance measure γ (2) red (·) has finite total variation. Depending on the rate of decay (polynomially or exponentially) of the total variation of γ (2) red (·) outside of an expanding ball centered at the origin, we determine optimal bandwidths b n (up to a constant) minimizing the mean squared error of σ 2 n . The case when γ (2) red (·) has bounded support is of particular interest. Further we suggest an isotropised estimator σ 2 n suitable for motion-invariant point processes and compare its properties with σ 2 . Our theoretical results are illustrated and supported by a simulation study which compares the (relative) mean squared errors of σ 2 for planar Poisson, Poisson cluster, and hard-core point processes and for various values of n b n .

In this paper we introduce Lévy-driven Cox point processes (LCPs) as Cox point processes with driving intensity function Λ defined by a kernel smoothing of a Lévy basis (an independently scattered, infinitely divisible random measure). We also consider log Lévy-driven Cox point processes (LLCPs) with Λ equal to the exponential of such a kernel smoothing. Special cases are shot noise Cox processes, log Gaussian Cox processes, and log shot noise Cox processes. We study the theoretical properties of Lévy-based Cox processes, including moment properties described by nth-order product densities, mixing properties, specification of inhomogeneity, and spatio-temporal extensions.

In the present paper we develop several two-step estimation procedures for inhomogeneous shot-noise Cox processes. The intensity function is parametrized by the inhomogeneity parameters while the pair-correlation function is parametrized by the interaction parameters. The suggested procedures are based on a combination of Poisson likelihood estimation of the inhomogeneity parameters in the first step and an adaptation of a method from the homogeneous case for estimation of the interaction parameters in the second step. The adapted methods, based on minimum contrast estimation, composite likelihood and Palm likelihood, are compared both theoretically and by means of a simulation study. Two-step estimation with Palm likelihood has not been considered before. Asymptotic normality of the two-step estimator with Palm likelihood is proved.

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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.