A functional limit theorem for general shot noise processes

By a general shot noise process we mean a shot noise process in which the counting process of shots is arbitrary locally finite. Assuming that the counting process of shots satisfies a functional limit theorem in the Skorokhod space with a locally H\"{o}lder continuous Gaussian limit process and that the response function is regularly varying at infinity we prove that the corresponding general shot noise process satisfies a similar functional limit theorem with a different limit process and different normalization and centering functions. For instance, if the limit process for the counting process of shots is a Brownian motion, then the limit process for the general shot noise process is a Riemann-Liouville process. We specialize our result for five particular counting processes. Also, we investigate H\"{o}lder continuity of the limit processes for general shot noise processes.Comment: 15 pages, submitted to a journa

Digital frequency counter permits readout without disturbing counting process

Digital frequency counter system enables readout accurately at one-second intervals without interrupting or disturbing the counting process. The system incorporates a master counter and a slave counter with novel logic interconnections. The counter can be readily adapted to provide frequency readouts at 0.1 second intervals

A class of CTRWs: Compound fractional Poisson processes

This chapter is an attempt to present a mathematical theory of compound fractional Poisson processes. The chapter begins with the characterization of a well-known L\'evy process: The compound Poisson process. The semi-Markov extension of the compound Poisson process naturally leads to the compound fractional Poisson process, where the Poisson counting process is replaced by the Mittag-Leffler counting process also known as fractional Poisson process. This process is no longer Markovian and L\'evy. However, several analytical results are available and some of them are discussed here. The functional limit of the compound Poisson process is an $\alpha$-stable L\'evy process, whereas in the case of the compound fractional Poisson process, one gets an $\alpha$-stable L\'evy process subordinated to the fractional Poisson process.Comment: 23 pages. To be published in a World Scientific book edited by Ralf Metzle

Counting Process Based Dimension Reduction Methods for Censored Outcomes

We propose a class of dimension reduction methods for right censored survival data using a counting process representation of the failure process. Semiparametric estimating equations are constructed to estimate the dimension reduction subspace for the failure time model. The proposed method addresses two fundamental limitations of existing approaches. First, using the counting process formulation, it does not require any estimation of the censoring distribution to compensate the bias in estimating the dimension reduction subspace. Second, the nonparametric part in the estimating equations is adaptive to the structural dimension, hence the approach circumvents the curse of dimensionality. Asymptotic normality is established for the obtained estimators. We further propose a computationally efficient approach that simplifies the estimation equation formulations and requires only a singular value decomposition to estimate the dimension reduction subspace. Numerical studies suggest that our new approaches exhibit significantly improved performance for estimating the true dimension reduction subspace. We further conduct a real data analysis on a skin cutaneous melanoma dataset from The Cancer Genome Atlas. The proposed method is implemented in the R package "orthoDr".Comment: First versio

A likelihood ratio test for stationarity of rating transitions

For a time-continuous discrete-state Markov process as model for rating transitions, we study the time-stationarity by means of a likelihood ratio test. For multiple Markov process data from a multiplicative intensity model, maximum likelihood parameter estimates can be represented as martingale transform of the processes counting transitions between the rating states. As a consequence, the profile partial likelihood ratio is asymptotically X-2-distributed. An internal rating data set reveals highly significant instationarity. --Stationarity,Multiple Markov process,Counting process,Likelihood ratio,Panel data

A "winner" under any voting rule ? An experiment on the single transferable vote

In this paper, we expose the results of a voting experiment realised in 2007, during the French Presidential election. This experiment aimed at confronting the Single Transferable vote (SVT) procedure to two criteria : simplicity and the selection of a Condorcet-winner. Building on our electoral sample's preferences, we show that this voting procedure can design a different winner, depending on the vote counting process. With the vote counting process advocated by Hare, the winner is Nicolas Sarkozy, while the Coombs vote counting process has François Bayrou as winner. For these two vote counting processes, the details of the experiment are the same and it is shown that the simplicity criterion is respected. However, with regard to the Condorcet-winner criterion, the Coombs methods is the only one to elect the Condorcet-winner, i.e. François Bayrou.Field experiments, elections, Single Transferable Vote, voting system, Condorcet Winner.