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Register a new userGeneralization of the fractional Poisson distribution
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Added by
Richard Herrmann
Publication date
2015
fields
Mathematical Statistics
and research's language is
English
Authors
Richard Herrmann
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A generalization of the Poisson distribution based on the generalized Mittag-Leffler function $E_{alpha, beta}(lambda)$ is proposed and the raw moments are calculated algebraically in terms of Bell polynomials. It is demonstrated, that the proposed distribution function contains the standard fractional Poisson distribution as a subset. A possible interpretation of the additional parameter $beta$ is suggested.
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Let $b(x)$ be the probability that a sum of independent Bernoulli random variables with parameters $p_1, p_2, p_3, ldots in [0,1)$ equals $x$, where $lambda := p_1 + p_2 + p_3 + cdots$ is finite. We prove two inequalities for the maximal ratio $b(x)/pi_lambda(x)$, where $pi_lambda$ is the weight function of the Poisson distribution with parameter $lambda$.
In this paper, we deal with the problem of calibrating thresholding rules in the setting of Poisson intensity estimation. By using sharp concentration inequalities, oracle inequalities are derived and we establish the optimality of our estimate up to a logarithmic term. This result is proved under mild assumptions and we do not impose any condition on the support of the signal to be estimated. Our procedure is based on data-driven thresholds. As usual, they depend on a threshold parameter $gamma$ whose optimal value is hard to estimate from the data. Our main concern is to provide some theoretical and numerical results to handle this issue. In particular, we establish the existence of a minimal threshold parameter from the theoretical point of view: taking $gamma<1$ deteriorates oracle performances of our procedure. In the same spirit, we establish the existence of a maximal threshold parameter and our theoretical results point out the optimal range $gammain[1,12]$. Then, we lead a numerical study that shows that choosing $gamma$ larger than 1 but close to 1 is a fairly good choice. Finally, we compare our procedure with classical ones revealing the harmful role of the support of functions when estimated by classical procedures.
The purpose of this paper is to estimate the intensity of a Poisson process $N$ by using thresholding rules. In this paper, the intensity, defined as the derivative of the mean measure of $N$ with respect to $ndx$ where $n$ is a fixed parameter, is assumed to be non-compactly supported. The estimator $tilde{f}_{n,gamma}$ based on random thresholds is proved to achieve the same performance as the oracle estimator up to a possible logarithmic term. Then, minimax properties of $tilde{f}_{n,gamma}$ on Besov spaces ${cal B}^{ensuremath alpha}_{p,q}$ are established. Under mild assumptions, we prove that $$sup_{fin B^{ensuremath alpha}_{p,q}cap ensuremath mathbb {L}_{infty}} ensuremath mathbb {E}(ensuremath | | tilde{f}_{n,gamma}-f| |_2^2)leq C(frac{log n}{n})^{frac{ensuremath alpha}{ensuremath alpha+{1/2}+({1/2}-frac{1}{p})_+}}$$ and the lower bound of the minimax risk for ${cal B}^{ensuremath alpha}_{p,q}cap ensuremath mathbb {L}_{infty}$ coincides with the previous upper bound up to the logarithmic term. This new result has two consequences. First, it establishes that the minimax rate of Besov spaces ${cal B}^{ensuremath alpha}_{p,q}$ with $pleq 2$ when non compactly supported functions are considered is the same as for compactly supported functions up to a logarithmic term. When $p>2$, the rate exponent, which depends on $p$, deteriorates when $p$ increases, which means that the support plays a harmful role in this case. Furthermore, $tilde{f}_{n,gamma}$ is adaptive minimax up to a logarithmic term.
Several representations of the exact cdf of the sum of squares of n independent gamma-distributed random variables Xi are given, in particular by a series of gamma distribution functions. Using a characterization of the gamma distribution by Laha, an expansion of the exact distribution of the sample variance is derived by a Taylor series approach with the former distribution as its leading term. In particular for integer orders alpha some further series are provided, including a convex combination of gamma distributions for alpha = 1 and nearly of this type for alpha > 1. Furthermore, some representations of the distribution of the angle Phi between (X1,...,Xn) and (1,...,1) are given by orthogonal series. All these series are based on the same sequence of easily computed moments of cos(Phi).
Herein, we review the properties of the Amoroso distribution, the natural unification of the gamma and extreme value distribution families. Over 50 distinct, named distributions (and twice as many synonyms) occur as special cases or limiting forms. Consequently, this single simple functional form encapsulates and systematizes an extensive menagerie of interesting and common probability distributions.
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