On the use of the Weibull distribution in modeling and describing diameter distributions of clonal eucalypt stands

2020 ◽  
Vol 50 (10) ◽  
pp. 1050-1063 ◽  
Author(s):  
Luciane Naimeke Schmidt ◽  
Mateus Niroh Inoue Sanquetta ◽  
John Paul McTague ◽  
Gilson Fernandes da Silva ◽  
Clayton Vieira Fraga Filho ◽  
...  
Keyword(s):  
Weibull Distribution ◽  
Goodness Of Fit ◽  
Cumulative Distribution ◽  
Growth And Yield ◽  
Likelihood Method ◽  
Weibull Distributions ◽  
Recovery Method ◽  
Stand Attributes ◽  
Parameter Prediction ◽  
Diameter Distributions

Weibull distributions have been widely used to describe tree stem diameter distributions. However, there is a scarcity of studies that suggest the best Weibull formulation. The parameters of the Weibull distribution are usually predicted by either the parameter prediction method (PPM) or the parameter recovery method (PRM), although other methods have been proposed. Thus, this study aimed to evaluate the performance of eight Weibull formulations and compare methods of parameter prediction to describe diameter distributions of clonal eucalypt stands in Brazil. Data originated from remeasurements of 56 plots at ages 3, 5, and 6 years. Weibull distributions were fitted using the maximum likelihood method and evaluated in a goodness-of-fit indicators ranking. The right-truncated two-parameter formulation showed the best results and was used to evaluate the methods of parameter prediction. Stand attributes showed a strong relationship with shape and scale parameters. Regression models were developed and resulted in accurate estimates using PPM. PRM used a growth and yield system to estimate the stand attributes, followed by the moment-based method. The modified cumulative distribution function regression (CDFR) approach was also evaluated, and it presented the poorest results. Although the PPM showed excellent results, PRM is recommended in older stands with inventory because it implicitly promotes compatibility among stand attributes.

scholarly journals Predicting Diameter Distributions for Young Longleaf Pine Plantations in Southwest Georgia

2009 ◽  
Vol 33 (1) ◽  
pp. 25-28 ◽  
Author(s):  
Lichun Jiang ◽  
John R. Brooks
Keyword(s):  
Distribution Function ◽  
Goodness Of Fit ◽  
Longleaf Pine ◽  
Pinus Palustris ◽  
Cumulative Distribution ◽  
Diameter Distribution ◽  
Weibull Function ◽  
Recovery Method ◽  
Parameter Recovery ◽  
Diameter Distributions

Abstract Parameter prediction equations for the Weibull distribution function were developed based on four percentile functions and a parameter recovery method for longleaf pine (Pinus palustris Mill.) in Southwest Georgia. Four percentiles were expressed as functions of stand-level characteristics based on stepwise regression and seemingly unrelated regression. Using a percentile-based parameter recovery method (PCT), estimated diameter distributions were obtained from available stand-level variables. The PCT method was also compared with a cumulative distribution function (CDF) regression method. The PCT method produced consistently better goodness-of-fit statistics than the CDF method. The results indicate that diameter distribution in longleaf pine stands can be successfully characterized with the Weibull function.

scholarly journals Results of the Verification of the Statistical Distribution Model of Microseismicity Emission Characteristics

2016 ◽  
Vol 61 (3) ◽  
pp. 489-496
Author(s):  
Aleksander Cianciara
Keyword(s):  
Distribution Function ◽  
Weibull Distribution ◽  
Goodness Of Fit ◽  
Cumulative Distribution Function ◽  
Statistical Distribution ◽  
Cumulative Distribution ◽  
Distribution Model ◽  
Emission Characteristics ◽  
Goodness Of Fit Test ◽  
Empirical Cumulative Distribution Function

Abstract The paper presents the results of research aimed at verifying the hypothesis that the Weibull distribution is an appropriate statistical distribution model of microseismicity emission characteristics, namely: energy of phenomena and inter-event time. It is understood that the emission under consideration is induced by the natural rock mass fracturing. Because the recorded emission contain noise, therefore, it is subjected to an appropriate filtering. The study has been conducted using the method of statistical verification of null hypothesis that the Weibull distribution fits the empirical cumulative distribution function. As the model describing the cumulative distribution function is given in an analytical form, its verification may be performed using the Kolmogorov-Smirnov goodness-of-fit test. Interpretations by means of probabilistic methods require specifying the correct model describing the statistical distribution of data. Because in these methods measurement data are not used directly, but their statistical distributions, e.g., in the method based on the hazard analysis, or in that that uses maximum value statistics.

scholarly journals Goodness of Fit Tests in Modeling the Distribution of the Daily Rate of Return of the WIG20 Companies

2012 ◽  
Vol 10 (2) ◽  
pp. 103-113
Author(s):  
Kamila Bednarz
Keyword(s):  
Goodness Of Fit ◽  
Laplace Distribution ◽  
Cumulative Distribution ◽  
Rate Of Return ◽  
Likelihood Method ◽  
Return Distribution ◽  
Chi Square ◽  
Goodness Of Fit Tests ◽  
Daily Rate ◽  
Chi Square Test

Goodness of Fit Tests in Modeling the Distribution of the Daily Rate of Return of the WIG20 Companies In this paper a classic rate of return was examined. Due to a limited quantitative range, the study included only the modeling of the rate of return distribution of the WIG20 index and its companies by means of the Laplace distribution and the Gaussian distribution. Additionally, the goodness of fit tests and methods of estimating the aforementioned distributions parameters were thoroughly covered. When applying the Laplace distribution to modeling the rate of return distribution the parameters were determined by means of two methods: the method of moments and the maximum likelihood method. The maximum period was determined, for which usefulness of the distribution in modeling the rates of return distribution was observed, as well as the results of the chi-square test for class intervals with varying length ensuring equal probability, and for intervals with identical length considering two methods of determining the theoretical size: in accordance with the cumulative distribution function as well as on the basis of the probability density function.

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>

scholarly journals A Probabilistic Model of the Unidirectional Tensile Strength of Fiber-Reinforced Polymers for Structural Design

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Jianqing Zhang ◽  
Ruikun Zhang ◽  
Yihua Zeng
Keyword(s):  
Tensile Strength ◽  
Weibull Distribution ◽  
Probabilistic Model ◽  
Goodness Of Fit ◽  
Safety Margin ◽  
Reduction Factor ◽  
Fiber Reinforced Polymers ◽  
Statistical Uncertainty ◽  
Weibull Distributions ◽  
Frp Composites

In this paper, a statistical analysis of the tensile strength of FRP composites is conducted. A relatively large experimental database including 58 datasets is first constructed, and the Normal, Lognormal, and Weibull distributions are fitted to the data using a tail-sensitive Anderson–Darling statistic as the measure of goodness of fit. Fitting results show that the Normal, Lognormal, and Weibull distributions can be used to model the tensile strength of FRP composites. Then, the characteristic value for the tensile strength of FRP composites at a fixed percentile is analyzed. It is found that the Weibull distribution results in a higher safety margin in comparison to either the Normal or the Lognormal distribution. When the experimental justification, the theoretical justification, as well as the design conservativeness are taken into consideration, the Weibull distribution is the most recommended distribution to model the tensile strength of FRP composites. Furthermore, a probabilistic model considering the statistical uncertainty for the tensile strength for FRP composites is proposed. It is believed that the statistical uncertainty can be modeled as a reduction factor, and the recommended value of such factor for engineering design practices is provided based on regression analysis.

EXACT CONFIDENCE INTERVALS AND JOINT CONFIDENCE REGIONS FOR THE PARAMETERS OF THE WEIBULL DISTRIBUTIONS

2004 ◽  
Vol 11 (02) ◽  
pp. 133-140 ◽  
Author(s):  
ZHENMIN CHEN
Keyword(s):  
Weibull Distribution ◽  
Confidence Intervals ◽  
Reliability Function ◽  
Confidence Regions ◽  
Cumulative Distribution ◽  
Weibull Distributions ◽  
Confidence Bounds ◽  
Discussion Section ◽  
Mean Lifetime ◽  
Exact Confidence Intervals

The Weibull distribution is widely adopted as a lifetime distribution. One of the characteristics the Weibull distribution possesses is that its cumulative distribution function can be expressed by closed form. Parameter estimation for the Weibull distribution has been discussed by many authors. Various methods have been proposed for constructing confidence intervals and joint confidence regions for the parameters of the Weibull distribution based on censored data. This paper discusses those methods that deal with exact confidence intervals or exact joint confidence regions for the parameters. One of the applications of the joint confidence regions of the parameters is to find confidence bounds for the functions of the parameters. In this paper, confidence bounds for the mean lifetime and reliability function for the Weibull distributions are discussed. Some unresolved problems for the exact confidences and joint confidence regions are mentioned in the discussion section.

scholarly journals Bivariate exponentiated discrete Weibull distribution: statistical properties, estimation, simulation and applications

2019 ◽  
Vol 14 (1) ◽  
pp. 29-42 ◽  
Author(s):  
M. El- Morshedy ◽  
M. S. Eliwa ◽  
A. El-Gohary ◽  
A. A. Khalil
Keyword(s):  
Weibull Distribution ◽  
Joint Probability ◽  
Discrete Distribution ◽  
Real Data ◽  
Reliability Function ◽  
Statistical Properties ◽  
Cumulative Distribution ◽  
Likelihood Method ◽  
Model Parameters ◽  
Probability Mass Function

AbstractIn this paper, a new bivariate discrete distribution is defined and studied in-detail, in the so-called the bivariate exponentiated discrete Weibull distribution. Several of its statistical properties including the joint cumulative distribution function, joint probability mass function, joint hazard rate function, joint moment generating function, mathematical expectation and reliability function for stress–strength model are derived. Its marginals are exponentiated discrete Weibull distributions. Hence, these marginals can be used to analyze the hazard rates in the discrete cases. The model parameters are estimated using the maximum likelihood method. Simulation study is performed to discuss the bias and mean square error of the estimators. Finally, two real data sets are analyzed to illustrate the flexibility of the proposed model.

scholarly journals Diameter distribution model development of tropical hybrid Eucalyptus clonal plantations in Sumatera, Indonesia: A comparison of estimation methods

2022 ◽  
Vol 52 ◽  
Author(s):  
Joni Waldy ◽  
John A. Kershaw Jr ◽  
Aaron Weiskittel ◽  
Mark J. Ducey
Keyword(s):  
Growth And Yield ◽  
Estimation Methods ◽  
Hybrid Clone ◽  
Distribution Model ◽  
Diameter Distribution ◽  
Average Growth ◽  
Eucalyptus Hybrid ◽  
Stand Attributes ◽  
The Moment ◽  
Diameter Distributions

Background: Effective forest management and planning often requires information about the distribution of volume by size and product classes. Size-class models describe the diameter distribution and provide information by diameter class, such as the number of trees, basal area, and volume per unit of area. A successful diameter-distribution model requires high flexibility yet robust prediction of its parameters. To our knowledge, there are no studies regarding diameter distribution models for Eucalyptus hybrids in Indonesia. Therefore, the aim of this study was to compare different recovery methods for predicting parameters of the 3-parameter Weibull distribution for characterising diameter distributions of Eucalyptus hybrid clone plantations, on Sumatera Island of Indonesia. Methods: The parameter recovery approach was proposed to be compatible with stand-average growth and yield models developed based on the same data. Three approaches where compared: moment-based recovery, percentile-based prediction and hybrid methods. The ultimate goal was to recover Weibull parameters from future stand attributes, which were predicted from current stand attributes using regression models. Results: In this study, the moment method was found to give the overall lowest mean error-index and Kolmogorov– Smirnov (KS) statistic, followed by the hybrid and percentile methods. The moment-based method better fit long tails on both sides of the distribution and exhibited slightly greater flexibility in describing plots with larger variance than the other methods. Conclusions: The Weibull approach appeared relatively robust in determining diameter distributions of Eucalyptus hybrid clone plantation in Indonesia, yet some refinements may be necessary to characterize more complex distributions.

Bootstrap Analysis

2017 ◽  
Author(s):  
Russell Cheng
Keyword(s):  
Goodness Of Fit ◽  
Null Distribution ◽  
Real Data ◽  
Confidence Regions ◽  
Confidence Bands ◽  
Cumulative Distribution ◽  
Attractive Alternative ◽  
Bootstrap Analysis ◽  
Parametric Bootstrapping ◽  
Anderson Darling

Parametric bootstrapping (BS) provides an attractive alternative, both theoretically and numerically, to asymptotic theory for estimating sampling distributions. This chapter summarizes its use not only for calculating confidence intervals for estimated parameters and functions of parameters, but also to obtain log-likelihood-based confidence regions from which confidence bands for cumulative distribution and regression functions can be obtained. All such BS calculations are very easy to implement. Details are also given for calculating critical values of EDF statistics used in goodness-of-fit (GoF) tests, such as the Anderson-Darling A2 statistic whose null distribution is otherwise difficult to obtain, as it varies with different null hypotheses. A simple proof is given showing that the parametric BS is probabilistically exact for location-scale models. A formal regression lack-of-fit test employing parametric BS is given that can be used even when the regression data has no replications. Two real data examples are given.

Empirical comparison of skewed t-copula models for insurance and financial data

2020 ◽  
Vol 15 (4) ◽  
pp. 351-361
Author(s):  
Liwei Huang ◽  
Arkady Shemyakin
Keyword(s):  
Bayesian Estimation ◽  
Goodness Of Fit ◽  
Model Performance ◽  
Information Criteria ◽  
Stock Index ◽  
Likelihood Method ◽  
Special Importance ◽  
Copula Model ◽  
Technical Problems ◽  
T Copula

Skewed t-copulas recently became popular as a modeling tool of non-linear dependence in statistics. In this paper we consider three different versions of skewed t-copulas introduced by Demarta and McNeill; Smith, Gan and Kohn; and Azzalini and Capitanio. Each of these versions represents a generalization of the symmetric t-copula model, allowing for a different treatment of lower and upper tails. Each of them has certain advantages in mathematical construction, inferential tools and interpretability. Our objective is to apply models based on different types of skewed t-copulas to the same financial and insurance applications. We consider comovements of stock index returns and times-to-failure of related vehicle parts under the warranty period. In both cases the treatment of both lower and upper tails of the joint distributions is of a special importance. Skewed t-copula model performance is compared to the benchmark cases of Gaussian and symmetric Student t-copulas. Instruments of comparison include information criteria, goodness-of-fit and tail dependence. A special attention is paid to methods of estimation of copula parameters. Some technical problems with the implementation of maximum likelihood method and the method of moments suggest the use of Bayesian estimation. We discuss the accuracy and computational efficiency of Bayesian estimation versus MLE. Metropolis-Hastings algorithm with block updates was suggested to deal with the problem of intractability of conditionals.

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