scholarly journals On powers of likelihood functions of random walks on ℤͩ

2018 ◽  
Vol 50 (A) ◽  
pp. 63-66
Author(s):  
Krishna B. Athreya
Keyword(s):  
Random Walks ◽  
Likelihood Function ◽  
Random Vectors ◽  
Open Problems ◽  
Likelihood Functions ◽  
Independent Identically Distributed

Abstract Let {Xi}i≥1 be independent, identically distributed random vectors in ℤd,d≥1. Let LLn(x)≡ℙ(Sn=x),n≥1,x∈ℤd, be the likelihood function for Sn=∑i=1nXi. For integers j≥2 and n≥1, let an(j)≡∑x∈ℤd(Ln(x))j. We show that if X1-X2 has a nondegenerate aperiodic distribution in ℤd and 𝔼(∥X1∥2)>∞, then limn→∞n(j-1)d∕2an(j)≡a(j,d) exists and 0<a(j,d)<∞. Some extensions and open problems are also outlined.

scholarly journals The asymptotic expansions of the distribution function and of density function for the sums of independent identically distributed random vectors

1968 ◽  
Vol 8 (3) ◽  
pp. 405-422
Author(s):  
A. Bikelis
Keyword(s):  
Distribution Function ◽  
Density Function ◽  
Asymptotic Expansions ◽  
Random Vectors ◽  
Independent Identically Distributed

The abstracts (in two languages) can be found in the pdf file of the article. Original author name(s) and title in Russian and Lithuanian: А. Бикялис. Асимптотические разложения для плотностей и распределений сумм независимых одинаково распределенных случайных векторов A. Bikelis. Nepriklausomų vienodai pasiskirsčiusių atsitiktinių vektorių sumų tankių ir pasiskirstymo funkcijų asimptotiniai išdėstymai

scholarly journals The estimation of the uncertainty associated with rating curves of the river Ivinhema in the state of Paraná/Brazil

2018 ◽  
Vol 40 ◽  
pp. 06029
Author(s):  
Luiz Henrique Maldonado ◽  
Daniel Firmo Kazay ◽  
Elio Emanuel Romero Lopez
Keyword(s):  
Bayesian Inference ◽  
Likelihood Function ◽  
The State ◽  
Likelihood Estimator ◽  
Likelihood Functions ◽  
Important Impact ◽  
Rating Curves

The estimation of the uncertainty associated with stage-discharge relations is a challenge to the hydrologists. Bayesian inference with likelihood estimator is a promissory approach. The choice of the likelihood function has an important impact on the capability of the model to represent the residues. This paper aims evaluate two likelihood functions with DREAM algorithm to estimate specific non-unique stage-discharge rating curves: normal likelihood function and Laplace likelihood function. The result of BaRatin is also discussed. The MCMC of the DREAM and the BaRatin algorithm have been compared and its results seem consistent for the studied case. The Laplace likelihood function presented as good results as normal likelihood function for the residues. Other gauging stations should be evaluated to attend more general conclusions.

Objective Inference for Climate Parameters: Bayesian, Transformation-of-Variables, and Profile Likelihood Approaches

2014 ◽  
Vol 27 (19) ◽  
pp. 7270-7284 ◽  
Author(s):  
Nicholas Lewis
Keyword(s):  
Climate Sensitivity ◽  
Bayesian Methods ◽  
Likelihood Function ◽  
Profile Likelihood ◽  
Likelihood Method ◽  
Jeffreys Prior ◽  
Bayesian Approaches ◽  
Noninformative Priors ◽  
Likelihood Functions ◽  
Transformation Of Variables

Abstract Insight is provided into the use of objective-Bayesian methods for estimating climate sensitivity by considering their relationship to transformations of variables in the context of a simple case considered in a previous study, and some misunderstandings about Bayesian inference are discussed. A simple model in which climate sensitivity (S) and effective ocean heat diffusivity (Kυ) are the only parameters varied is used, with twentieth-century warming attributable to greenhouse gases (AW) and effective ocean heat capacity (HC) being the only data-based observables. Probability density functions (PDFs) for AW and HC are readily derived that represent valid independent objective-Bayesian posterior PDFs, provided the error distribution assumptions involved in their construction are justified. Using them, a standard transformation of variables provides an objective joint posterior PDF for S and Kυ; integrating out Kυ gives a marginal PDF for S. Close parametric approximations to the PDFs for AW and HC are obtained, enabling derivation of likelihood functions and related noninformative priors that give rise to the objective posterior PDFs that were computed initially. Bayes’s theorem is applied to the derived AW and HC likelihood functions, demonstrating the effect of differing prior distributions on PDFs for S. Use of the noninformative Jeffreys prior produces an identical PDF to that derived using the transformation-of-variables approach. It is shown that similar inference for S to that based on these two alternative objective-Bayesian approaches is obtained using a profile likelihood method on the derived joint likelihood function for AW and HC.

Limiting diffusion for random walks with drift conditioned to stay positive

1978 ◽  
Vol 15 (02) ◽  
pp. 280-291 ◽  
Author(s):  
Peichuen Kao
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Random Function ◽  
Random Variables ◽  
Principal Result ◽  
Hitting Time ◽  
Brownian Excursion ◽  
Norming Constant ◽  
Finite Dimensional ◽  
Independent Identically Distributed

Let {ξ k : k ≧ 1} be a sequence of independent, identically distributed random variables with E{ξ 1} = μ ≠ 0. Form the random walk {S n : n ≧ 0} by setting S 0, S n = ξ 1 + ξ 2 + ··· + ξ n , n ≧ 1. Define the random function Xn by setting where α is a norming constant. Let N denote the hitting time of the set (–∞, 0] by the random walk. The principal result in this paper is to show (under appropriate conditions on the distribution of ξ 1) that the finite-dimensional distributions of Xn , conditioned on n &lt; N &lt; ∞ converge to those of the Brownian excursion process.

scholarly journals Improved Approximation of the Sum of Random Vectors by the Skew Normal Distribution

2014 ◽  
Vol 51 (2) ◽  
pp. 466-482 ◽  
Author(s):  
Marcus C. Christiansen ◽  
Nicola Loperfido
Keyword(s):  
Normal Distribution ◽  
Normal Approximation ◽  
Distribution Functions ◽  
Skew Normal Distribution ◽  
Random Vectors ◽  
Multivariate Distributions ◽  
Uniform Distance ◽  
Special Case ◽  
Independent Identically Distributed ◽  
Skew Normal

We study the properties of the multivariate skew normal distribution as an approximation to the distribution of the sum of n independent, identically distributed random vectors. More precisely, we establish conditions ensuring that the uniform distance between the two distribution functions converges to 0 at a rate of n-2/3. The advantage over the corresponding normal approximation is particularly relevant when the summands are skewed and n is small, as illustrated for the special case of exponentially distributed random variables. Applications to some well-known multivariate distributions are also discussed.

scholarly journals Coping with translational non-crystallographic symmetry in molecular replacement

2014 ◽  
Vol 70 (a1) ◽  
pp. C319-C319
Author(s):  
Randy Read ◽  
Paul Adams ◽  
Airlie McCoy
Keyword(s):  
Diffraction Pattern ◽  
Likelihood Function ◽  
Molecular Replacement ◽  
Asymmetric Unit ◽  
Space Group ◽  
New Targets ◽  
Likelihood Functions ◽  
Crystallographic Symmetry ◽  
Modified Likelihood ◽  
Statistical Effects

In translational noncrystallographic symmetry (tNCS), two or more copies of a component are present in a similar orientation in the asymmetric unit of the crystal. This causes systematic modulations of the intensities in the diffraction pattern, leading to problems with methods that assume, either implicitly or explicitly, that the distribution of intensities is a function only of resolution. To characterize the statistical effects of tNCS accurately, it is necessary to determine the translation relating the copies, any small rotational differences in their orientations, and the size of random coordinate differences caused by conformational differences. An algorithm has been developed to estimate these parameters and refine their values against a likelihood function. By accounting for the statistical effects of tNCS, it is possible to unmask the competing statistical effects of twinning and tNCS and to more robustly assess the crystal for the presence of twinning. Modified likelihood functions that account for the statistical effects of tNCS have been developed for use in molecular replacement and implemented in Phaser. With the use of these new targets, it is now possible to solve structures that eluded earlier versions of the program. Pseudosymmetry and space group ambiguities often accompany tNCS, but the new version of Phaser is less likely to fall into the traps that these set.

Improved Bayesian Update Method on Flaw Distributions Reflecting Non-Destructive Inspection Result

2020 ◽  
Author(s):  
Jinya Katsuyama ◽  
Yuhei Miyamoto ◽  
Kai Lu ◽  
Akihiro Mano ◽  
Yinsheng Li
Keyword(s):  
Likelihood Function ◽  
Bayesian Updating ◽  
Pressure Vessels ◽  
Irradiation Embrittlement ◽  
Failure Frequency ◽  
Likelihood Functions ◽  
Pressurized Thermal Shock ◽  
Bayesian Update ◽  
Non Destructive ◽  
Reactor Pressure

Abstract We have developed a probabilistic fracture mechanics (PFM) analysis code named PASCAL4 for evaluating the failure frequency of reactor pressure vessels (RPVs) through consideration of neutron irradiation embrittlement and transients such as pressurized thermal shock events. It is well-known that flaw distributions, including flaw size and density, have an important role in the failure frequency calculations of a PFM analysis. NUREG-2163 report provides a methodology to obtain much more realistic flaw distributions based on a Bayesian updating approach by reflecting the non-destructive inspection (NDI) results, which is applicable for case when there are flaw indications through NDI. There may, however, be no flaw indications resulting after inspection of some RPVs. Therefore, we proposed likelihood functions applicable for both cases when flaws are detected and when there is no flaw indication as the NDI results. In the Bayesian updating method, the likelihood functions were applied to independently acquire the posterior distributions of flaw depth and density using the same NDI results. In this study, we further improve the likelihood functions to enable them to update flaw depth and density simultaneously. Based on this improved likelihood function, several application examples are presented where the flaw distributions are estimated by reflecting the NDI results through Bayesian update. In addition, PFM analyses are also performed considering those estimated flaw distributions. All the results indicate that the improved likelihood functions are useful for estimating flaw distributions.

scholarly journals Choosing various likelihood functions on uncertainty assessment in groundwater simulation-optimization model

2020 ◽  
Vol 20 (2) ◽  
pp. 737-750 ◽  
Author(s):  
Khadije Norouzi Khatiri ◽  
Mohammad Hossein Niksokhan ◽  
Amin Sarang
Keyword(s):  
Optimization Model ◽  
Likelihood Function ◽  
Simulation Optimization ◽  
Minimum Mean Square Error ◽  
Error Variance ◽  
Uncertainty Assessment ◽  
Likelihood Functions ◽  
Function Selection ◽  
The Relationship ◽  
Residual Errors

Abstract The main goal in this research is study of impacts of various likelihood functions on DREAM(zs) (Differential Evolution Adaptive Metropolis) method results in simulation-optimization model of aquifer. In this study, DREAM(zs) algorithm has been developed to study aquifer simulation-optimization model uncertainties. DREAM(zs) is used to investigate uncertainty of parameters of the simulation-optimization model in Isfahan-Barkhar aquifer, Isfehan province, Iran. This study is carried out on an aquifer simulation model of MODFLOW that is coupled with MOPSO (multi-objective particle swarm optimization) optimization. Three likelihood functions, L1, L2, and L3, are considered as informal and the remaining (L4 and L5) are represented as formal categories. Likelihood function L1 is Nash–Sutcliffe efficiency and L2 is based on minimum mean square error. L3 uses estimation of model error variance and L4 focuses on the relationship between the traditional least squares fitting and the Bayesian inference. In likelihood function L5 the serial dependence of residual errors is calculated using a first-order autoregressive model of the residuals. Results suggested that the parameters sensitivity depend on the likelihood function selection, and sensitivity of all parameters is not similar in different likelihood functions. MOPSO algorithm outputs indicated that likelihood function No. 5 has a higher speed in reaching convergence and this function also showed that objective functions had a better performance compared to the other likelihood functions.

On the existence of submultiplicative moments for the stationary distributions of some Markovian random walks

1999 ◽  
Vol 36 (1) ◽  
pp. 78-85 ◽  
Author(s):  
M. S. Sgibnev
Keyword(s):  
Markov Chains ◽  
Random Walks ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Stationary Distributions ◽  
Necessary And Sufficient ◽  
Submultiplicative Function ◽  
Independent Identically Distributed

This paper is concerned with submultiplicative moments for the stationary distributions π of some Markov chains taking values in ℝ+ or ℝ which are closely related to the random walks generated by sequences of independent identically distributed random variables. Necessary and sufficient conditions are given for ∫φ(x)π(dx) < ∞, where φ(x) is a submultiplicative function, i.e. φ(0) = 1 and φ(x+y) ≤ φ(x)φ(y) for all x, y.

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