scholarly journals Empirical cumulant function based parameter estimation in stable laws

2019 ◽  
Vol 22 (2) ◽  
pp. 311-338 ◽  
Author(s):  
Annika Krutto
Keyword(s):  
Closed Form ◽  
Selection Rule ◽  
Empirical Characteristic Function ◽  
Challenging Problem ◽  
Positive Real ◽  
Infinitely Divisible ◽  
Asymptotically Normal ◽  
Cumulant Function ◽  
Independent Identically Distributed ◽  
Infinite Moments

Stable distributions are a subclass of infinitely divisible distributions that form the only family of possible limiting distributions for sums of independent identically distributed random variables. A challenging problem is estimating their parameters because many have densities with no explicit form and infinite moments. To address this problem, a class of closed-form estimators, called cumulant estimators, has been introduced. Cumulant estimators are derived from the logarithm of empirical characteristic function at two arbitrary distinct positive real arguments. This paper extends cumulant estimators in two directions: (i) it is proved that they are asymptotically normal and (ii) a sample based rule for selecting the two arguments is proposed. Extensive simulations show that under the provided selection rule, the closed-form cumulant estimators generally outperform the well-known algorithmic methods.

scholarly journals Closed-Form Asymptotic Sampling Distributions under the Coalescent with Recombination for an Arbitrary Number of Loci

2012 ◽  
Vol 44 (2) ◽  
pp. 391-407 ◽  
Author(s):  
Anand Bhaskar ◽  
Yun S. Song
Keyword(s):  
Asymptotic Expansion ◽  
Closed Form ◽  
Recombination Rate ◽  
Functional Form ◽  
Asymptotic Series ◽  
Sampling Distribution ◽  
Challenging Problem ◽  
Sampling Distributions ◽  
Asymptotic Sampling ◽  
New Framework

Obtaining a closed-form sampling distribution for the coalescent with recombination is a challenging problem. In the case of two loci, a new framework based on an asymptotic series has recently been developed to derive closed-form results when the recombination rate is moderate to large. In this paper, an arbitrary number of loci is considered and combinatorial approaches are employed to find closed-form expressions for the first couple of terms in an asymptotic expansion of the multi-locus sampling distribution. These expressions are universal in the sense that their functional form in terms of the marginal one-locus distributions applies to all finite- and infinite-alleles models of mutation.

Linear estimation of self-similar processes via Lamperti's transformation

2000 ◽  
Vol 37 (2) ◽  
pp. 429-452 ◽  
Author(s):  
Carl J. Nuzman ◽  
H. Vincent Poor
Keyword(s):  
Brownian Motion ◽  
Closed Form ◽  
Real Line ◽  
Continuous Time ◽  
Stationary Processes ◽  
Linear Estimation ◽  
Positive Real ◽  
Self Similar ◽  
Infinite Intervals ◽  
Positive Real Line

Lamperti's transformation, an isometry between self-similar and stationary processes, is used to solve some problems of linear estimation of continuous-time, self-similar processes. These problems include causal whitening and innovations representations on the positive real line, as well as prediction from certain finite and semi-infinite intervals. The method is applied to the specific case of fractional Brownian motion (FBM), yielding alternate derivations of known prediction results, along with some novel whitening and interpolation formulae. Some associated insights into the problem of discrete prediction are also explored. Closed-form expressions for the spectra and spectral factorization of the stationary processes associated with the FBM are obtained as part of this development.

X-parameters®-based closed-form expressions for evaluating power-dependent fundamental negative and positive real impedance boundaries in oscillator design

2012 ◽  
Vol 6 (8) ◽  
pp. 835 ◽  
Author(s):  
A.M. Pelaez-Perez ◽  
S. Woodington ◽  
J.I. Alonso ◽  
M. Fernandez-Barciela ◽  
P.J. Tasker
Keyword(s):  
Closed Form ◽  
Positive Real

scholarly journals Closed-Form Asymptotic Sampling Distributions under the Coalescent with Recombination for an Arbitrary Number of Loci

2012 ◽  
Vol 44 (02) ◽  
pp. 391-407 ◽  
Author(s):  
Anand Bhaskar ◽  
Yun S. Song
Keyword(s):  
Asymptotic Expansion ◽  
Closed Form ◽  
Recombination Rate ◽  
Functional Form ◽  
Asymptotic Series ◽  
Sampling Distribution ◽  
Challenging Problem ◽  
Sampling Distributions ◽  
Asymptotic Sampling ◽  
New Framework

Obtaining a closed-form sampling distribution for the coalescent with recombination is a challenging problem. In the case of two loci, a new framework based on an asymptotic series has recently been developed to derive closed-form results when the recombination rate is moderate to large. In this paper, an arbitrary number of loci is considered and combinatorial approaches are employed to find closed-form expressions for the first couple of terms in an asymptotic expansion of the multi-locus sampling distribution. These expressions are universal in the sense that their functional form in terms of the marginal one-locus distributions applies to all finite- and infinite-alleles models of mutation.

scholarly journals Odds Theorem with Multiple Selection Chances

2010 ◽  
Vol 47 (04) ◽  
pp. 1093-1104 ◽  
Author(s):  
Katsunori Ano ◽  
Hideo Kakinuma ◽  
Naoto Miyoshi
Keyword(s):  
Finite Number ◽  
Closed Form ◽  
Selection Rule ◽  
Random Variables ◽  
Secretary Problem ◽  
Optimal Selection ◽  
Maximum Probability ◽  
Optimal Rule ◽  
Interesting Implication

We study the multi-selection version of the so-called odds theorem by Bruss (2000). We observe a finite number of independent 0/1 (failure/success) random variables sequentially and want to select the last success. We derive the optimal selection rule when m (≥ 1) selection chances are given and find that the optimal rule has the form of a combination of multiple odds-sums. We provide a formula for computing the maximum probability of selecting the last success when we have m selection chances and also provide closed-form formulae for m = 2 and 3. For m = 2, we further give the bounds for the maximum probability of selecting the last success and derive its limit as the number of observations goes to ∞. An interesting implication of our result is that the limit of the maximum probability of selecting the last success for m = 2 is consistent with the corresponding limit for the classical secretary problem with two selection chances.

Linear estimation of self-similar processes via Lamperti's transformation

2000 ◽  
Vol 37 (02) ◽  
pp. 429-452 ◽  
Author(s):  
Carl J. Nuzman ◽  
H. Vincent Poor
Keyword(s):  
Brownian Motion ◽  
Closed Form ◽  
Real Line ◽  
Continuous Time ◽  
Stationary Processes ◽  
Linear Estimation ◽  
Positive Real ◽  
Self Similar ◽  
Infinite Intervals ◽  
Positive Real Line

Lamperti's transformation, an isometry between self-similar and stationary processes, is used to solve some problems of linear estimation of continuous-time, self-similar processes. These problems include causal whitening and innovations representations on the positive real line, as well as prediction from certain finite and semi-infinite intervals. The method is applied to the specific case of fractional Brownian motion (FBM), yielding alternate derivations of known prediction results, along with some novel whitening and interpolation formulae. Some associated insights into the problem of discrete prediction are also explored. Closed-form expressions for the spectra and spectral factorization of the stationary processes associated with the FBM are obtained as part of this development.

scholarly journals On subclasses of infinitely divisible distributions on R related to hitting time distributions of 1-dimensional generalized diffusion processes

1992 ◽  
Vol 127 ◽  
pp. 175-200 ◽  
Author(s):  
Makoto Yamazato
Keyword(s):  
Laplace Transform ◽  
Diffusion Processes ◽  
Hitting Time ◽  
Infinitely Divisible Distributions ◽  
Strict Sense ◽  
Real Numbers ◽  
Positive Real ◽  
Infinitely Divisible

A distribution μ on R+ = [0, ∞) is said to be a distribution if there are an increasing (in the strict sense) sequence of positive real numbers such that, for each j = 0, …, m, there is at least one ak satisfying bj < ak < b+1 and theLaplace transform of μ is represented as

scholarly journals Discrete Scan Statistics Generated by Exchangeable Binary Trials

2010 ◽  
Vol 47 (4) ◽  
pp. 1084-1092 ◽  
Author(s):  
Serkan Eryilmaz
Keyword(s):  
Closed Form ◽  
Random Variables ◽  
Random Variable ◽  
Scan Statistics ◽  
Bernoulli Trials ◽  
Scan Statistic ◽  
Binary Trials ◽  
Special Case ◽  
Independent Identically Distributed

Let {Xi}i=1n be a sequence of random variables with two possible outcomes, denoted 0 and 1. Define a random variable Sn,m to be the maximum number of 1s within any m consecutive trials in {Xi}i=1n. The random variable Sn,m is called a discrete scan statistic and has applications in many areas. In this paper we evaluate the distribution of discrete scan statistics when {Xi}i=1n consists of exchangeable binary trials. We provide simple closed-form expressions for both conditional and unconditional distributions of Sn,m for 2m ≥ n. These results are also new for independent, identically distributed Bernoulli trials, which are a special case of exchangeable trials.

scholarly journals The permanental process

2006 ◽  
Vol 38 (4) ◽  
pp. 873-888 ◽  
Author(s):  
Peter McCullagh ◽  
Jesper Møller
Keyword(s):  
Closed Form ◽  
Large Class ◽  
Point Processes ◽  
Infinitely Divisible ◽  
Determinantal Processes ◽  
Cox Processes

We extend the boson process first to a large class of Cox processes and second to an even larger class of infinitely divisible point processes. Density and moment results are studied in detail. These results are obtained in closed form as weighted permanents, so the extension is called a permanental process. Temporal extensions and a particularly tractable case of the permanental process are also studied. Extensions of the fermion process along similar lines, leading to so-called determinantal processes, are discussed.

scholarly journals Closed-Form Solution of a Rational Difference Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Tarek F. Ibrahim ◽  
Abdul Qadeer Khan ◽  
Burak Oğul ◽  
Dağistan Şimşek
Keyword(s):  
Difference Equation ◽  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Real Numbers ◽  
Positive Real ◽  
Rational Difference Equation ◽  
The Difference

In this paper, we study the solution of the difference equation Ω m + 1 = Ω m − 7 q + 6 / 1 + ∏ t = 0 5 Ω m − q + 1 t − q , where the initials are positive real numbers.

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