Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe density ratio of Poisson binomial versus Poisson distributions
68
0
0.0
(
0
)
Added by
Lutz Duembgen
Publication date
2019
fields
Mathematical Statistics
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
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 $lambda$.
rate research
Read More
In this paper, we have developed a new class of sampling schemes for estimating parameters of binomial and Poisson distributions. Without any information of the unknown parameters, our sampling schemes rigorously guarantee prescribed levels of precision and confidence.
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.
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.
We are interested in estimating the location of what we call smooth change-point from $n$ independent observations of an inhomogeneous Poisson process. The smooth change-point is a transition of the intensity function of the process from one level to another which happens smoothly, but over such a small interval, that its length $delta_n$ is considered to be decreasing to $0$ as $nto+infty$. We show that if $delta_n$ goes to zero slower than $1/n$, our model is locally asymptotically normal (with a rather unusual rate $sqrt{delta_n/n}$), and the maximum likelihood and Bayesian estimators are consistent, asymptotically normal and asymptotically efficient. If, on the contrary, $delta_n$ goes to zero faster than $1/n$, our model is non-regular and behaves like a change-point model. More precisely, in this case we show that the Bayesian estimators are consistent, converge at rate $1/n$, have non-Gaussian limit distributions and are asymptotically efficient. All these results are obtained using the likelihood ratio analysis method of Ibragimov and Khasminskii, which equally yields the convergence of polynomial moments of the considered estimators. However, in order to study the maximum likelihood estimator in the case where $delta_n$ goes to zero faster than $1/n$, this method cannot be applied using the usual topologies of convergence in functional spaces. So, this study should go through the use of an alternative topology and will be considered in a future work.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
