Confidence intervals for the difference of two binomial proportions are well known, however, confidence intervals for the weighted sum of two binomial proportions are less studied. We develop and compare seven methods for constructing confidence intervals for the weighted sum of two independent binomial proportions. The interval estimates are constructed by inverting the Wald test, the score test and the Likelihood ratio test. The weights can be negative, so our results generalize those for the difference between two independent proportions. We provide a numerical study that shows that these confidence intervals based on large-sample approximations perform very well, even when a relatively small amount of data is available. The intervals based on the inversion of the score test showed the best performance. Finally, we show that as for the difference of two binomial proportions, adding four pseudo-outcomes to the Wald interval for the weighted sum of two binomial proportions improves its coverage significantly, and we provide a justification for this correction.

Mandatory emission trading schemes are being established around the world. Participants of such market schemes are always exposed to risks. This leads to the creation of an accompanying market for emission-linked derivatives. To evaluate the fair prices of such financial products, one needs appropriate models for the evolution of the underlying assets, emission allowance certificates. In this paper, we discuss continuous time diffusion and jump-diffusion models, the latter enabling one to model information shocks that cause jumps in allowance prices. We show that the resulting martingale dynamics can be described in terms of non-linear partial differential and integro-differential equations and use a finite difference method to investigate numerical properties of their discretizations. The results are illustrated by a small numerical study.

For a probability distribution P on an at most countable alphabet A, this article gives finite sample bounds for the expected occupancy counts EK n,r and probabilities EM n,r . Both upper and lower bounds are given in terms of the counting function ν of P . Special attention is given to the case where ν is bounded by a regularly varying function. In this case, it is shown that our general results lead to an optimal-rate control of the expected occupancy counts and probabilities with explicit constants. Our results are also put in perspective with Turing's formula and recent concentration bounds to deduce bounds in probability. At the end of the paper, we discuss an extension of the occupancy problem to arbitrary distributions in a metric space.

The paper deals with nonlinear Poisson neuron network models with bounded memory dynamics, which can include both Hebbian learning mechanisms and refractory periods. The state of the network is described by the times elapsed since its neurons fired within the post-synaptic transfer kernel memory span, and the current strengths of synaptic connections, the state spaces of our models being hierarchies of finitedimensional components. We prove the ergodicity of the stochastic processes describing the behaviour of the networks, establish the existence of continuously differentiable stationary distribution densities (with respect to the Lebesgue measures of corresponding dimensionality) on the components of the state space, and find upper bounds for them. For the density components, we derive a system of differential equations that can be solved in a few simplest cases only. Approaches to approximate computation of the stationary density are discussed. One approach is to reduce the dimensionality of the problem by modifying the network so that each neuron cannot fire if the number of spikes it emitted within the post-synaptic transfer kernel memory span reaches a given threshold. We show that the stationary distribution of this 'truncated' network converges to that of the unrestricted network as the threshold increases, and that the convergence is at a superexponential rate. A complementary approach uses discrete Markov chain approximations to the network process.The anatomy of a neuron involves three distinct parts with different electrical activity functions: dendrites that form a tree and contain post-synaptic receptors (inputs), the cell body (soma) that integrates the input currents coming from the dendrites, and a long-limbed axon that terminates with pre-synaptic buttons (outputs). A typical feature of the neuronal electrical activity is the propagation of membrane depolarisation. The membrane of a resting neuron is polarised. Brief high-amplitude depolarisations that propagate from the soma along the axon are called action potentials (or spikes) and have a characteristic shape. When a spike reaches an axonal termination that 'connects' to a post-synaptic neuron, neurotransmitters are released into the extra cellular space and excite receptors on the post-synaptic neuron (usually on dendrites). This generates a local variation of the membrane potential in that neuron, which propagates towards the soma. The soma can be seen as a spatiotemporal integrator of these post-synaptic potentials (PSPs) to generate an output spike. The soma potential often remains close to the resting value for a few milliseconds after firing an action potential, which is referred to as the refractory period (during which the neuron cannot fire again). These basic elements of the neuronal information processing actually depend upon many different mechanisms at the molecular level, such as ionic concentrations, density of ion channels, axonal myelination, and types of neurotransmitters (for a review, we refer the reader to [1]).There exis...

We introduce a new method for improving the coverage accuracy of confidence intervals for means of lattice distributions. The technique can be applied very generally to enhance existing approaches, although we consider it in greatest detail in the context of estimating a binomial proportion or a Poisson mean, where it is particularly effective.The method is motivated by a simple theoretical result, which shows that, by splitting the original sample of size n into two parts, of sizes n 1 and n 2 D n n 1 , and basing the confidence procedure on the average of the means of these two subsamples, the highly oscillatory behaviour of coverage error, as a function of n, is largely removed. Perhaps surprisingly, this approach does not increase confidence interval width; usually the width is slightly reduced. Contrary to what might be expected, our new method performs well when it is used to modify confidence intervals based on existing techniques that already perform very well-it typically improves significantly their coverage accuracy. Each application of the split sample method to an existing confidence interval procedure results in a new technique.