scholarly journals The tenets of indirect inference in Bayesian models

2021 ◽  
Author(s):  
Dmytro Perepolkin ◽  
Benjamin Goodrich ◽  
Ullrika Sahlin
Keyword(s):  
Cumulative Distribution Function ◽  
Probability Distributions ◽  
Bayesian Models ◽  
Quantile Function ◽  
Cumulative Distribution ◽  
Likelihood Method ◽  
Indirect Inference ◽  
Root Finding ◽  
Sampling Distributions ◽  
Definition Of

This paper extends the application of Bayesian inference to probability distributions defined in terms of its quantile function. We describe the method of *indirect likelihood* to be used in the Bayesian models with sampling distributions which lack an explicit cumulative distribution function. We provide examples and demonstrate the equivalence of the "quantile-based" (indirect) likelihood to the conventional "density-defined" (direct) likelihood. We consider practical aspects of the numerical inversion of quantile function by root-finding required by the indirect likelihood method. In particular, we consider a problem of ensuring the validity of an arbitrary quantile function with the help of Chebyshev polynomials and provide useful tips and implementation of these algorithms in Stan and R. We also extend the same method to propose the definition of an *indirect prior* and discuss the situations where it can be useful

scholarly journals Transmuted Probability Distributions: A Review

2020 ◽  
pp. 83-94
Author(s):  
Md. Mahabubur Rahman ◽  
Bander Al-Zahrani ◽  
Saman Hanif Shahbaz ◽  
Muhammad Qaiser Shahbaz
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Probability Distributions ◽  
Quantile Function ◽  
Cumulative Distribution ◽  
Functional Composition ◽  
Inverse Cumulative Distribution Function ◽  
Family Of Distributions

Transmutation is the functional composition of the cumulative distribution function (cdf) of one distribution with the inverse cumulative distribution function (quantile function) of another. Shaw and Buckley(2007), first apply this concept and introduced quadratic transmuted family of distributions. In this article, we have presented a review about the transmuted families of distributions. We have also listed the transmuted distributions, available in the literature along with some concluding remarks.

DISSONANCE — A MEASURE OF VARIABILITY FOR ORDINAL RANDOM VARIABLES

2001 ◽  
Vol 09 (01) ◽  
pp. 39-53 ◽  
Author(s):  
RONALD R. YAGER
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Probability Distributions ◽  
Random Variables ◽  
Cumulative Distribution ◽  
General Properties

We look at the issue of obtaining a variance like measure associated with probability distributions over ordinal sets. We call these dissonance measures. We specify some general properties desired in these dissonance measures. The centrality of the cumulative distribution function in formulating the concept of dissonance is pointed out. We introduce some specific examples of measures of dissonance.

scholarly journals On the Analysis of New COVID-19 Cases in Pakistan Using an Exponentiated Version of the M Family of Distributions

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 953
Author(s):  
Rashad A. R. Bantan ◽  
Christophe Chesneau ◽  
Farrukh Jamal ◽  
Mohammed Elgarhy
Keyword(s):  
Statistical Models ◽  
Cumulative Distribution Function ◽  
Cumulative Distribution ◽  
Exponential Distributions ◽  
New Family ◽  
Proposed Model ◽  
Complete Study ◽  
Probability And Statistics ◽  
Definition Of ◽  
Analyze Data

This paper develops the exponentiated Mfamily of continuous distributions, aiming to provide new statistical models for data fitting purposes. It stands out from the other families, as it depends on two baseline distributions, with the use of ratio and power transforms in the definition of the main cumulative distribution function. Thanks to the joint action of the possibly different baseline distributions, flexible statistical models can be created, motivating a complete study in this regard. Thus, we discuss the theoretical properties of the new family, with emphasis on those of potential interest to the overall probability and statistics. Then, a new three-parameter lifetime distribution is derived, with the choices of the inverse exponential and exponential distributions as baselines. After pointing out the great flexibility of the related model, we apply it to analyze an actual dataset of current interest: the daily COVID-19 cases observed in Pakistan from 21 March to 29 May 2020 (inclusive). As notable results, we demonstrate that the proposed model is the best among the 15 top ranked models in the literature, including the inverse exponential and exponential models, several modern extensions of them depending on more parameters, and the “unexponentiated” version of the proposed model as well. As future perspectives, the proposed model can be of interest to analyze data on COVID-19 cases in other countries, for possible comparison studies.

scholarly journals A Load-Share Reliability Model under the Changeable Piecewise Smooth Load

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Sergey V. Gurov ◽  
Lev V. Utkin
Keyword(s):  
Cumulative Distribution Function ◽  
Probability Distributions ◽  
Continuous Functions ◽  
Piecewise Smooth ◽  
Cumulative Distribution ◽  
Residual Lifetime ◽  
Time To Failure ◽  
Reliability Model ◽  
Piecewise Constant ◽  
Proposed Model

A new load-share reliability model of systems under the changeable load is proposed in the paper. It is assumed that the load is a piecewise smooth function which can be regarded as an extension of the piecewise constant and continuous functions. The condition of the residual lifetime conservation, which means continuity of a cumulative distribution function of time to failure, is accepted in the proposed model. A general algorithm for computing reliability measures is provided. Simple expressions for determining the survivor functions under assumption of the Weibull probability distribution of time to failure are given. Various numerical examples illustrate the proposed model by different forms of the system load and different probability distributions of time to failure.

scholarly journals Certain parameterized inequalities arising from fractional integral operators with exponential kernels

Filomat ◽  
2021 ◽  
Vol 35 (5) ◽  
pp. 1707-1724
Author(s):  
Zhengrong Yuan ◽  
Taichun Zhou ◽  
Qiang Zhang ◽  
Tingsong Du
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Fractional Integral ◽  
Integral Operators ◽  
Integral Inequalities ◽  
Cumulative Distribution ◽  
Digamma Function ◽  
Fractional Integral Operators ◽  
Definition Of ◽  
Fractional Type

We utilize the definition of a fractional integral operators, which was presented by Ahmad et al., to investigate a general fractional-type identity with a parameter. We establish certain parameterized fractional integral inequalities based on this identity, and provide two examples to illustrate the obtained results. Also, these results derived in this paper are applied to the estimations of q-digamma function, divergence measures and cumulative distribution function, respectively.

scholarly journals Bin-Free Fitting of Probability Distributions Emphasizing DNA Histograms

2017 ◽  
Author(s):  
Nash Rochman
Keyword(s):  
Cumulative Distribution Function ◽  
Probability Distributions ◽  
Cumulative Distribution ◽  
Data Set ◽  
Cell Synchronization ◽  
Approximate Probability ◽  
A Cell ◽  
The Right ◽  
Probability Density Function Pdf ◽  
Dna Histograms

AbstractIt is often challenging to find the right bin size when constructing a histogram to represent a noisy experimental data set. This problem is frequently faced when assessing whether a cell synchronization experiment was successful or not. In this case the goal is to determine whether the DNA content is best represented by a unimodal, indicating successful synchronization, or bimodal, indicating unsuccessful synchronization, distribution. This choice of bin size can greatly affect the interpretation of the results; however, it can be avoided by fitting the data to a cumulative distribution function (CDF). Fitting data to a CDF removes the need for bin size selection. The sorted data can also be used to reconstruct an approximate probability density function (PDF) without selecting a bin size. A simple CDF-based approach is presented and the benefits and drawbacks relative to usual methods are discussed.

scholarly journals The Metalog Distributions

2016 ◽  
Vol 13 (4) ◽  
pp. 243-277 ◽  
Author(s):  
Thomas W. Keelin
Keyword(s):  
Decision Analysis ◽  
Cumulative Distribution Function ◽  
Probability Distributions ◽  
Probability Density Functions ◽  
Cumulative Distribution ◽  
Density Functions ◽  
Creative Commons ◽  
Discrete Methods ◽  
Actual Decision ◽  
New System

The metalog distributions constitute a new system of continuous univariate probability distributions designed for flexibility, simplicity, and ease/speed of use in practice. The system is comprised of unbounded, semibounded, and bounded distributions, each of which offers nearly unlimited shape flexibility compared to previous systems of distributions. Explicit shape-flexibility comparisons are provided. Unlike other distributions that require nonlinear optimization for parameter estimation, the metalog quantile functions and probability density functions have simple closed-form expressions that are quantile parameterized linearly by cumulative-distribution-function data. Applications in fish biology and hydrology show how metalogs may aid data and distribution research by imposing fewer shape constraints than other commonly used distributions. Applications in decision analysis show how the metalog system can be specified with three assessed quantiles, how it facilities Monte Carlo simulation, and how applying it aided an actual decision that would have been made wrongly based on commonly used discrete methods. This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. You are free to download this work and share with others for any purpose, except commercially, if you distribute your contributions under the same license as the original, and you must attribute this work as “Decision Analysis. Copyright © 2016 The Author(s). https://doi.org/10.1287/deca.2016.0338 , used under a Creative Commons Attribution License: https://creativecommons.org/licenses/by-nc-sa/4.0/ .”

scholarly journals A Comparative Study on Computation of Cumulative Distribution Function in Predicting Time of Failure of Engineering Systems

2019 ◽  
Vol 11 (1) ◽  
Author(s):  
Gina Katherine Sierra Paez ◽  
Matthew Daigle ◽  
Kai Goebel
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Degradation Process ◽  
Cumulative Distribution ◽  
System State ◽  
Underlying Assumption ◽  
Hazard Zone ◽  
The Difference ◽  
Definition Of ◽  
Time Of Failure

Estimating accurate Time-of-Failure (ToF) of a system is key in making the decisions that impact operational safety and optimize cost. In this context, it is interesting to note that different approaches have been explored to tackle the problem of estimating ToF. The difference is in part characterized by different definitions of the hazard zones. The conventional definition for the cumulative distribution function (CDF) calculation is assumed to have well-defined hazard zones, that is, hazard zones defined as a function of the system state trajectory. An alternate method suggests the use of hazard zones defined as a function of the system state at time , instead of hazard zones defined as a function of system state up to and including time k (Acuña and Orchard 2018, 2017). This paper explores these differences and their impact on ToF estimation. Results for the conventional CDF definition indicated that, (i) the cumulative distribution function is always an increasing function of time, even when realizations of the degradation process are not monotonic, (ii) the sum of all probabilities is always 1 and does not need to be normalized, and (iii) all probabilities are positive and less than or equal to 1. Similar results are not observed for CDF calculation with hazard zones defined as a function only of the system state at time k. Results for ToF estimation using Acuña's definition differ, suggesting that there is an underlying assumption of independence in the hazard zone definition.  Therefore, we present an alternate definition of hazard zone which guarantees the properties of a well-defined CDF with a more straightforward ToF definition.

scholarly journals Deterministic Sampling from Univariate Normal Distributions with Sierpinski Space-Filling Curves

2021 ◽  
Author(s):  
Hime Oliveira
Keyword(s):  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Random Number ◽  
Probability Distributions ◽  
Cumulative Distribution ◽  
Standard Normal Distribution ◽  
Space Filling ◽  
Space Filling Curves ◽  
Deterministic Sampling ◽  
Standard Normal

This work addresses the problem of sampling from Gaussian probability distributions by means of uniform samples obtained deterministically and directly from space-filling curves (SFCs), a purely topological concept. To that end, the well-known inverse cumulative distribution function method is used, with the help of the probit function,which is the inverse of the cumulative distribution function of the standard normal distribution. Mainly due to the central limit theorem, the Gaussian distribution plays a fundamental role in probability theory and related areas, and that is why it has been chosen to be studied in the present paper. Numerical distributions (histograms) obtained with the proposed method, and in several levels of granularity, are compared to the theoretical normal PDF, along with other already established sampling methods, all using the cited probit function. Final results are validated with the Kullback-Leibler and two other divergence measures, and it will be possible to draw conclusions about the adequacy of the presented paradigm. As is amply known, the generation of uniform random numbers is a deterministic simulation of randomness using numerical operations. That said, sequences resulting from this kind of procedure are not truly random. Even so, and to be coherent with the literature, the expression ”random number” will be used along the text to mean ”pseudo-random number”.

scholarly journals A New Approach to Assess Wind Potential

2021 ◽  
Keyword(s):  
Distribution Function ◽  
Maximum Likelihood ◽  
Weibull Distribution ◽  
Wind Power ◽  
Cumulative Distribution Function ◽  
Maximum Likelihood Method ◽  
New Method ◽  
Cumulative Distribution ◽  
Likelihood Method ◽  
Wind Distribution

<p>Weibull Cumulative Distribution Function (C.D.F.) has been employed to assess and compare wind potentials of two wind stations Europlatform and Stavenisse of The Netherland. Weibull distribution has been used for accurate estimation of wind energy potential for a long time. The Weibull distribution with two parameters is suitable for modeling wind data if wind distribution is unimodal. Whereas wind distribution is generally unimodal, random weather changes can make the distribution bimodal. It is always desirable to find a method that accurately represents actual statistical data. Some well-known statistical methods are Method of Moment (MoM), Linear Least Square Method (LLSM), Maximum Likelihood Method (M.L.M.), Modified Maximum Likelihood Method (MMLM), Energy Pattern Factor Method (EPFM), and Empirical Method (E.M.), etc. All these methods employ Probability Density Function (PDF) of Weibull distribution, except LLSM, which uses Cumulative Distribution Function (C.D.F.). In this communication, we are presenting a newly proposed method of evaluating Weibull parameters. Unlike most methods, this new method employs a cumulative distribution function. A MATLAB® GUI-based simulation is developed to estimate Weibull parameters using the C.D.F. approach. It is found that the Mean Square Error (M.S.E.) is the lowest when using the new method. The new method, therefore, estimates wind power density with reasonable accuracy. Wind Power (W.P.) is estimated by considering four different Wind Turbine (W.T.) models for two sites, and maximum W.P. is found using Evance R9000.</p>

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