A Comparative Study on the Double Prior for Reliability Kumaraswamy Distribution with Numerical Solution

This work, deals with Kumaraswamy distribution. Kumaraswamy (1976, 1978) showed well known probability distribution functions such as the normal, beta and log-normal but in (1980) Kumaraswamy developed a more general probability density function for double bounded random processes, which is known as Kumaraswamy’s distribution. Classical maximum likelihood and Bayes methods estimator are used to estimate the unknown shape parameter (b). Reliability function are obtained using symmetric loss functions by using three types of informative priors two single priors and one double prior. In addition, a comparison is made for the performance of these estimators with respect to the numerical solution which are found using expansion method. The results showed that the reliability estimator under Rn and R3 is the best.

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APPROXIMATING PROBABILITY DISTRIBUTION FUNCTIONS WITH FEW MOMENTS

Latin American Applied Research - An international journal ◽

10.52292/j.laar.2020.132 ◽

2019 ◽

Vol 50 (1) ◽

pp. 21-25

Author(s):

Emilio Wille

Keyword(s):

Probability Distribution ◽

Statistical Data ◽

Piecewise Linear ◽

Distribution Functions ◽

Theoretical Simulation ◽

Product Of Random Variables ◽

Probability Distribution Functions ◽

Piecewise Linear Approximations ◽

Log Normal ◽

Good Agreement

A procedure is presented for approximating a given probability distribution function or statistical data considering a subset of their moments.This is done by a method of fitting moments of a piecewise linear functionto the moments of the known data. The approach has many advantages over popular approximation approaches. The procedure is demonstrated with commonly used cdfs (Exponential, Gamma, Log-Normal, Normal) andmore difficult problems involving sum and product of random variables,obtaining good agreement between the theoretical/simulation curves and the piecewise linear approximations.

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Estimating the Reliability Function of some Stress- Strength Models for the Generalized Inverted Kumaraswamy Distribution

Iraqi Journal of Science ◽

10.24996/ijs.2021.62.1.23 ◽

2021 ◽

pp. 240-251

Author(s):

Hakeem Hussain Hamad ◽

Nada Sabah Karam

Keyword(s):

Maximum Likelihood ◽

Shape Parameter ◽

Maximum Likelihood Estimators ◽

Reliability Function ◽

Least Square ◽

Mean Square ◽

Kumaraswamy Distribution ◽

Scale Parameters ◽

Strength Models ◽

Best Estimator

This paper discusses reliability of the stress-strength model. The reliability functions 𝑅1 and 𝑅2 were obtained for a component which has an independent strength and is exposed to two and three stresses, respectively. We used the generalized inverted Kumaraswamy distribution GIKD with unknown shape parameter as well as known shape and scale parameters. The parameters were estimated from the stress- strength models, while the reliabilities 𝑅1, 𝑅2 were estimated by three methods, namely the Maximum Likelihood, Least Square, and Regression. A numerical simulation study a comparison between the three estimators by mean square error is performed. It is found that best estimator between the three estimators is Maximum likelihood estimators.

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Comparing Different Estimators for the shape Parameter and the Reliability function of Kumaraswamy Distribution

Journal of Economics and Administrative Sciences ◽

10.33095/jeas.v25i116.1795 ◽

2019 ◽

Vol 25 (116) ◽

pp. 199-225

Author(s):

Jinan Abbas Naser Al-Obedy

Keyword(s):

Shape Parameter ◽

Reliability Function ◽

Bayes Estimation ◽

Likelihood Method ◽

Prior Distributions ◽

Squared Error Loss Function ◽

Kumaraswamy Distribution ◽

Chi Squared ◽

Two Parameters ◽

Mean Square Errors

In this paper, we used maximum likelihood method and the Bayesian method to estimate the shape parameter (θ), and reliability function (R(t)) of the Kumaraswamy distribution with two parameters l , θ (under assuming the exponential distribution, Chi-squared distribution and Erlang-2 type distribution as prior distributions), in addition to that we used method of moments for estimating the parameters of the prior distributions. Bayes estimators derived under the squared error loss function. We conduct simulation study, to compare the performance for each estimator, several values of the shape parameter (θ) from Kumaraswamy distribution for data-generating, for different samples sizes (small, medium, and large). Simulation results have shown that the Best method is the Bayes estimation according to the smallest values of mean square errors(MSE) for all samples sizes (n).

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Frequency analysis of low flows in intermittent and non-intermittent rivers from hydrological basins in Turkey

Water Science & Technology Water Supply ◽

10.2166/ws.2018.051 ◽

2018 ◽

Vol 19 (1) ◽

pp. 30-39 ◽

Cited By ~ 8

Author(s):

Ebru Eris ◽

Hafzullah Aksoy ◽

Bihrat Onoz ◽

Mahmut Cetin ◽

Mehmet Ishak Yuce ◽

...

Keyword(s):

Distribution Function ◽

Probability Distribution ◽

Probability Distribution Function ◽

Distribution Functions ◽

Low Flow ◽

Probability Distribution Functions ◽

Low Flows ◽

Intermittent Rivers ◽

Log Normal ◽

Best Fit

Abstract This study attempts to find out the best-fit probability distribution function to low flows using the up-to-date data of intermittent and non-intermittent rivers in four hydrological basins from different regions in Turkey. Frequency analysis of D = 1-, 7-, 14-, 30-, 90- and 273-day low flows calculated from the daily flow time series of each stream gauge was performed. Weibull (W2), Gamma (G2), Generalized Extreme Value (GEV) and Log-Normal (LN2) are selected among the 2-parameter probability distribution functions together with the Weibull (W3), Gamma (G3) and Log-Normal (LN3) from the 3-parameter probability distribution function family. Selected probability distribution functions are checked for their suitability to fit each D-day low flow sequence. LN3 mostly conforms to low flows by being the best-fit among the selected probability distribution functions in three out of four hydrological basins while W3 fits low flows in one basin. With the use of the best-fit probability distribution function, the low flow-duration-frequency curves are determined, which have the ability to provide the end-users with any D-day low flow discharge of any given return period.

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Probability distribution functions of gas surface density in M 33

Astronomy and Astrophysics ◽

10.1051/0004-6361/201833266 ◽

2018 ◽

Vol 617 ◽

pp. A125 ◽

Cited By ~ 8

Author(s):

Edvige Corbelli ◽

Bruce G. Elmegreen ◽

Jonathan Braine ◽

David Thilker

Keyword(s):

Star Formation ◽

Probability Distribution ◽

Power Law ◽

Distribution Functions ◽

Morphological Properties ◽

Molecular Gas ◽

Star Forming ◽

Normal Functions ◽

Probability Distribution Functions ◽

Log Normal

Aims. We examine the interstellar medium (ISM) of M 33 to unveil fingerprints of self-gravitating gas clouds throughout the star-forming disk. Methods. The probability distribution functions (PDFs) for atomic, molecular, and total gas surface densities are determined at a resolution of about 50 pc over regions that share coherent morphological properties and considering cloud samples at different evolutionary stages in the star formation cycle. Results. Most of the total gas PDFs are well fit by log-normal functions whose width decreases radially outward. Because the HI velocity dispersion is approximately constant throughout the disk, the decrease in PDF width is consistent with a lower Mach number for the turbulent ISM at large galactocentric radii where a higher fraction of HI is in the warm phase. The atomic gas is found mostly at face-on column densities below NHlim = 2.5 × 1021 cm−2, with small radial variations of NHlim. The molecular gas PDFs do not show strong deviations from log-normal functions in the central region where molecular fractions are high. Here the high pressure and rate of star formation shapes the PDF as a log-normal function, dispersing self-gravitating complexes with intense feedback at all column densities that are spatially resolved. Power-law PDFs for the molecules are found near and above NHlim, in the southern spiral arm and in a continuous dense filament extending at larger galactocentric radii. In the filament nearly half of the molecular gas departs from a log-normal PDF, and power laws are also observed in pre-star-forming molecular complexes. The slope of the power law is between −1 and −2. This slope, combined with maps showing where the different parts of the power law PDFs come from, suggests a power-law stratification of the density within molecular cloud complexes, in agreement with the dominance of self-gravity.

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Extended two-dimensional belief function based on divergence measurement

Journal of Intelligent & Fuzzy Systems ◽

10.3233/jifs-201727 ◽

2020 ◽

pp. 1-8

Author(s):

Jianping Fan ◽

Jing Wang ◽

Meiqin Wu

Keyword(s):

Belief Function ◽

Distribution Functions ◽

Divergence Measure ◽

Uncertain Information ◽

Belief Functions ◽

Two Dimensional ◽

Probability Distribution Functions ◽

Basic Probability ◽

The Difference ◽

Supporting Degree

The two-dimensional belief function (TDBF = (mA, mB)) uses a pair of ordered basic probability distribution functions to describe and process uncertain information. Among them, mB includes support degree, non-support degree and reliability unmeasured degree of mA. So it is more abundant and reasonable than the traditional discount coefficient and expresses the evaluation value of experts. However, only considering that the expert’s assessment is single and one-sided, we also need to consider the influence between the belief function itself. The difference in belief function can measure the difference between two belief functions, based on which the supporting degree, non-supporting degree and unmeasured degree of reliability of the evidence are calculated. Based on the divergence measure of belief function, this paper proposes an extended two-dimensional belief function, which can solve some evidence conflict problems and is more objective and better solve a class of problems that TDBF cannot handle. Finally, numerical examples illustrate its effectiveness and rationality.

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Development and Evaluation of a Fluctuating Plume Model for Odor Impact Assessment

Applied Sciences ◽

10.3390/app11083310 ◽

2021 ◽

Vol 11 (8) ◽

pp. 3310

Author(s):

Marzio Invernizzi ◽

Federica Capra ◽

Roberto Sozzi ◽

Laura Capelli ◽

Selena Sironi

Keyword(s):

Experimental Data ◽

Distribution Functions ◽

Plume Model ◽

Probability Distribution Functions ◽

Weibull Pdf ◽

Fluctuating Plume ◽

Concentration Statistics ◽

Modified Weibull ◽

Instantaneous Concentration ◽

Mean Concentration

For environmental odor nuisance, it is extremely important to identify the instantaneous concentration statistics. In this work, a Fluctuating Plume Model for different statistical moments is proposed. It provides data in terms of mean concentrations, variance, and intensity of concentration. The 90th percentile peak-to-mean factor, R90, was tested here by comparing it with the experimental results (Uttenweiler field experiment), considering different Probability Distribution Functions (PDFs): Gamma and the Modified Weibull. Seventy-two percent of the simulated mean concentration values fell within a factor 2 compared to the experimental ones: the model was judged acceptable. Both the modelled results for standard deviation, σC, and concentration intensity, Ic, overestimate the experimental data. This evidence can be due to the non-ideality of the measurement system. The propagation of those errors to the estimation of R90 is complex, but the ranges covered are quite repeatable: the obtained values are 1–3 for the Gamma, 1.5–4 for Modified Weibull PDF, and experimental ones from 1.4 to 3.6.

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