Review of Probability Distributions

2018 ◽  
pp. 25-73
Keyword(s):  
Confidence Intervals ◽  
Goodness Of Fit ◽  
Probability Distributions ◽  
Discrete Distributions ◽  
Continuous Distributions ◽  
R Language ◽  
Goodness Of Fit Tests ◽  
Anderson Darling ◽  
Analysis And Study ◽  
Waiting Lines

In Chapter 2, probability distributions are presented; the distributions exposed are those with more relation to the analysis and study of waiting lines; discrete distributions: binomial, geometric, Poisson; continuous distributions: uniform, exponential, erlang, and normal. Confidence intervals are calculated for some of the parameters of the distributions. A brief example of the generation of pseudorandom exponential times using a spreadsheet is presented. The chapter closes with the goodness-of-fit tests of probability distributions, especially the Anderson-Darling test. The statistical language of programming R is used in the exercises performed. Several codes are proposed in R Language to perform calculations automatically.

scholarly journals Estimating the Best-Fitted Probability Distribution for Monthly Maximum Temperature at the Sylhet Station in Bangladesh

2021 ◽  
Vol 2 (2) ◽  
pp. 60-67
Author(s):  
Rashidul Hasan Rashidul Hasan
Keyword(s):  
Probability Distribution ◽  
Goodness Of Fit ◽  
Probability Distributions ◽  
Probability Model ◽  
Temperature Data ◽  
Maximum Temperature ◽  
Chi Square ◽  
Goodness Of Fit Tests ◽  
Anderson Darling ◽  
Log Normal

The estimation of a suitable probability model depends mainly on the features of available temperature data at a particular place. As a result, existing probability distributions must be evaluated to establish an appropriate probability model that can deliver precise temperature estimation. The study intended to estimate the best-fitted probability model for the monthly maximum temperature at the Sylhet station in Bangladesh from January 2002 to December 2012 using several statistical analyses. Ten continuous probability distributions such as Exponential, Gamma, Log-Gamma, Beta, Normal, Log-Normal, Erlang, Power Function, Rayleigh, and Weibull distributions were fitted for these tasks using the maximum likelihood technique. To determine the model’s fit to the temperature data, several goodness-of-fit tests were applied, including the Kolmogorov-Smirnov test, Anderson-Darling test, and Chi-square test. The Beta distribution is found to be the best-fitted probability distribution based on the largest overall score derived from three specified goodness-of-fit tests for the monthly maximum temperature data at the Sylhet station.

scholarly journals Stochastic Generation of Low Stream Flow Data of Iokastis Stream, Kavala City, NE Greece

Proceedings ◽  
2018 ◽  
Vol 2 (11) ◽  
pp. 579
Author(s):  
Thomas Papalaskaris ◽  
Theologos Panagiotidis
Keyword(s):  
Stream Flow ◽  
Goodness Of Fit ◽  
Probability Distributions ◽  
Specific Area ◽  
Low Flow ◽  
Urban Stream ◽  
Flow Data ◽  
Goodness Of Fit Tests ◽  
Chi Squared ◽  
Anderson Darling

Only a few scientific research studies, especially dealing with extremely low flow conditions, have been compiled so far, in Greece. The present study, aiming to contribute in this specific area of hydrologic investigation, generates synthetic low stream flow time series of an entire calendar year considering the stream flow data recorded during a center interval period of the year 2015. We examined the goodness of fit tests of eleven theoretical probability distributions to daily low stream flow data acquired at a certain location of the absolutely channelized urban stream which crosses the roads junction formed by Iokastis road an Chrisostomou Smirnis road, Agios Loukas residential area, Kavala city, NE Greece, using a 3-inches conventional portable Parshall flume and calculated the corresponding probability distributions parameters. The Kolmogorov-Smirnov, Anderson-Darling and Chi-Squared, GOF tests were employed to show how well the probability distributions fitted the recorded data and the results were demonstrated through interactive tables providing us the ability to effectively decide which model best fits the observed data. Finally, the observed against the calculated low flow data are plotted, compiling a log-log scale chart and calculate statistics featuring the comparison between the recorded and the forecasted low flow data.

scholarly journals A Probabilistic Approach to the Simulation of Non-Linear Stress-Strain Relationships for Oriented Strandboard Subject to In-Plane Tension

2011 ◽  
Vol 478 ◽  
pp. 54-63 ◽  
Author(s):  
Antony T. McTigue ◽  
Annette M. Harte
Keyword(s):  
Goodness Of Fit ◽  
Probability Distributions ◽  
Probabilistic Approach ◽  
Regression Equations ◽  
Goodness Of Fit Tests ◽  
Oriented Strandboard ◽  
Anderson Darling ◽  
Standardised Testing ◽  
Tension Loading ◽  
Tension Strength

This paper presents the results from an experimental test program conducted on commercially available oriented strandboard (OSB) panels and statistical analyses of the results. Standardised testing was used to determine the short-term behaviour of OSB/3 panels subjected to tension loading. A variety of thicknesses sourced from three different producers were used. Analysis of the results indicate that a quadratic expression in the form of  = a2 + b provides the best description of the relationship between stress (and strain ( up to the point of failure. It has also been shown that the coefficients a and b of the quadratic regression equations are negatively correlated to each other. Anderson-Darling goodness-of-fit tests were conducted on the results for tension strength and modulus of elasticity (MOE). The results indicate that the tension strength and MOE come from populations that follow either normal or lognormal probability distributions.

scholarly journals Testing for goodness rather than lack of fit of continuous probability distributions

PLoS ONE ◽  
2021 ◽  
Vol 16 (9) ◽  
pp. e0256499
Author(s):  
Stefan Wellek
Keyword(s):  
Goodness Of Fit ◽  
Alternative Hypothesis ◽  
Probability Distributions ◽  
Testing Procedure ◽  
Discrete Distributions ◽  
Testing Procedures ◽  
Continuous Type ◽  
Goodness Of Fit Tests ◽  
True Distribution

The vast majority of testing procedures presented in the literature as goodness-of-fit tests fail to accomplish what the term is promising. Actually, a significant result of such a test indicates that the true distribution underlying the data differs substantially from the assumed model, whereas the true objective is usually to establish that the model fits the data sufficiently well. Meeting that objective requires to carry out a testing procedure for a problem in which the statement that the deviations between model and true distribution are small, plays the role of the alternative hypothesis. Testing procedures of this kind, for which the term tests for equivalence has been coined in statistical usage, are available for establishing goodness-of-fit of discrete distributions. We show how this methodology can be extended to settings where interest is in establishing goodness-of-fit of distributions of the continuous type.

scholarly journals A best-fit probability distribution for the estimation of rainfall in northern regions of Pakistan

2016 ◽  
Vol 11 (1) ◽  
pp. 432-440 ◽  
Author(s):  
M. T. Amin ◽  
M. Rizwan ◽  
A. A. Alazba
Keyword(s):  
Probability Distribution ◽  
Goodness Of Fit ◽  
Probability Distributions ◽  
Future Research ◽  
Maximum Rainfall ◽  
Type Iii ◽  
Goodness Of Fit Tests ◽  
Pearson Type ◽  
Northern Regions ◽  
Best Fit

AbstractThis study was designed to find the best-fit probability distribution of annual maximum rainfall based on a twenty-four-hour sample in the northern regions of Pakistan using four probability distributions: normal, log-normal, log-Pearson type-III and Gumbel max. Based on the scores of goodness of fit tests, the normal distribution was found to be the best-fit probability distribution at the Mardan rainfall gauging station. The log-Pearson type-III distribution was found to be the best-fit probability distribution at the rest of the rainfall gauging stations. The maximum values of expected rainfall were calculated using the best-fit probability distributions and can be used by design engineers in future research.

scholarly journals A comparison of three parameter estimation methods of the gamma distribution of annual maximum low flow duration and deficit in the Upper Vistula catchment (Poland)

2018 ◽  
Vol 23 ◽  
pp. 00001
Author(s):  
Katarzyna Baran-Gurgul
Keyword(s):  
Gamma Distribution ◽  
Goodness Of Fit ◽  
Probability Distributions ◽  
Low Flow ◽  
Estimation Methods ◽  
Goodness Of Fit Test ◽  
Flow Duration ◽  
Distribution Parameters ◽  
Anderson Darling ◽  
Parameter Estimation Methods

Based on 30-year 24-hour flow sequences at 69 water gauging stations in the Upper Vistula catchment, it was determined that the probability distributions of the low flow duration and its maximum annual deficit can be described by the gamma distribution with the estimated parameters by the methods: MOM, the method of moments, LMOM, the method of linear moments, and MLE, the method of maximum likelihood. The stationarity of the time series was tested by the Mann-Kendall correlation using the Hamed and Rao variance correction. The low flows were defined by the SPA method, with the limit flow Q70%. The quality of the match was tested by the Anderson-Darling goodness of fit test. This test allowed accepting the gamma distribution in all analysed cases, regardless of the method used to estimate the distribution parameters, since the pv (p-values) values were greater than 5% (over 18% for Tmax and 7.5% for Vmax). The highest pv values for individual water gauging stations, as well as the highest 90% Tmax and Vmax quantiles were noted using LMOM to estimate the gamma distribution parameters. The highest 90% Tmax and Vmax quantiles were observed in the uppermost part of the studied area.

scholarly journals Estimation of flood frequency using statistical method: Mahanadi River basin, India

2020 ◽  
Vol 3 (1) ◽  
pp. 189-207
Author(s):  
Sandeep Samantaray ◽  
Abinash Sahoo
Keyword(s):  
Stream Flow ◽  
Goodness Of Fit ◽  
Central India ◽  
Extreme Value ◽  
Flood Frequency ◽  
Goodness Of Fit Test ◽  
Mahanadi River ◽  
Goodness Of Fit Tests ◽  
Study Results ◽  
Anderson Darling

Abstract Estimating stream flow has a substantial financial influence, because this can be of assistance in water resources management and provides safety from scarcity of water and conceivable flood destruction. Four common statistical methods, namely, Normal, Gumbel max, Log-Pearson III (LP III), and Gen. extreme value method are employed for 10, 20, 30, 35, 40, 50, 60, 70, 75, 100, 150 years to forecast stream flow. Monthly flow data from four stations on Mahanadi River, in Eastern Central India, namely, Rampur, Sundargarh, Jondhra, and Basantpur, are used in the study. Results show that Gumbel max gives better flow discharge value than the Normal, LP III, and Gen. extreme value methods for all four gauge stations. Estimated flood values for Rampur, Sundargarh, Jondhra, and Basantpur stations are 372.361 m3/sec, 530.415 m3/sec, 2,133.888 m3/sec, and 3,836.22 m3/sec, respectively, considering Gumbel max. Goodness-of-fit tests for four statistical distribution techniques applied in the present study are also evaluated using Kolmogorov–Smirov, Anderson–Darling, Chi-squared tests at critical value 0.05 for the four proposed gauge stations. Goodness-of-fit test results show that Gen. extreme value gives best results at Rampur, Sundergarh, and Jondhra gauge stations followed by LP III, whereas LP III is the best fit for Basantpur, followed by Gen. extreme value.

scholarly journals Best Fitted Distribution For Meteorological Data In Kuala Krai

2021 ◽  
Vol 3 (1) ◽  
pp. 16-25
Author(s):  
Siti Mariam Norrulashikin ◽  
Fadhilah Yusof ◽  
Siti Rohani Mohd Nor ◽  
Nur Arina Bazilah Kamisan
Keyword(s):  
Climate Change ◽  
Probability Distribution ◽  
Goodness Of Fit ◽  
Probability Distributions ◽  
Meteorological Data ◽  
Cumulative Distribution ◽  
Distribution Model ◽  
Data Series ◽  
Distribution Models ◽  
Goodness Of Fit Tests

Modeling meteorological variables is a vital aspect of climate change studies. Awareness of the frequency and magnitude of climate change is a critical concern for mitigating the risks associated with climate change. Probability distribution models are valuable tools for a frequency study of climate variables since it measures how the probability distribution able to fit well in the data series. Monthly meteorological data including average temperature, wind speed, and rainfall were analyzed in order to determine the most suited probability distribution model for Kuala Krai district. The probability distributions that were used in the analysis were Beta, Burr, Gamma, Lognormal, and Weibull distributions. To estimate the parameters for each distribution, the maximum likelihood estimate (MLE) was employed. Goodness-of-fit tests such as the Kolmogorov-Smirnov, and Anderson-Darling tests were conducted to assess the best suited model, and the test's reliability. Results from statistical studies indicate that Burr distributions better characterize the meteorological data of our research. The graph of probability density function, cumulative distribution function as well as Q-Q plot are presented.

scholarly journals Goodness of Fit Tests for Marshal-Olkin Extended Rayleigh Distribution

2019 ◽  
pp. 587-598
Author(s):  
Naz Saud ◽  
Sohail Chand
Keyword(s):  
Goodness Of Fit ◽  
Empirical Distribution ◽  
Rayleigh Distribution ◽  
Test Statistic ◽  
Goodness Of Fit Test ◽  
Goodness Of Fit Tests ◽  
Estimated Parameters ◽  
Percentage Points ◽  
Anderson Darling ◽  
Von Mises

A class of goodness of fit tests for Marshal-Olkin Extended Rayleigh distribution with estimated parameters is proposed. The tests are based on the empirical distribution function. For determination of asymptotic percentage points, Kolomogorov-Sminrov, Cramer-von-Mises, Anderson-Darling,Watson, and Liao-Shimokawa test statistic are used. This article uses Monte Carlo simulations to obtain asymptotic percentage points for Marshal-Olkin extended Rayleigh distribution. Moreover, power of the goodness of fit test statistics is investigated for this lifetime model against several alternatives.

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