Application of Extreme Value Statistics to Annual Maximum Magnitudes in Jordan Employing a Mixture Distribution

1999 ◽  
Vol 15 (4) ◽  
pp. 711-724 ◽  
Author(s):  
Abdallah I. Husein Malkawi ◽  
Fayez A. Abdulla ◽  
Samer A. Barakat ◽  
Mohanned S. Al-Sheriadeh
Keyword(s):  
Maximum Likelihood ◽  
Simplex Method ◽  
Mixture Distribution ◽  
Maximum Likelihood Estimators ◽  
Extreme Value ◽  
Annual Maximum ◽  
Maximum Earthquake Magnitude ◽  
Newton Raphson ◽  
Maximum Earthquake ◽  
Raphson Method

The concept of extreme value mixture distribution (EVmix) has been implemented in this study to estimate maximum earthquake magnitude occurrence. The EVmix model is applied to annual maximum earthquake magnitude occurrence in Jordan and conterminous regions spanning over the period 1918 to 1997. The maximum likelihood method, in conjunction with the two optimization methods, was employed for determining the statistical parameters of the Gumbel's asymptotic distribution, i.e., GI, and the extreme value distributions EVIII and EVmix. The Simplex method of Nelder and Mead (1965) was found to be more successful in obtaining the maximum likelihood estimators of the three given distributions than the Newton-Raphson method. The difficulties inherent to the Newton-Raphson method were overcome by the Simplex method. It is shown in this study that the EVmix model fits the observed annual maxima far better than GI and EVIII models. In addition, the maximum likelihood estimators obtained using the Simplex method were used to calculate the earthquake risk for a given return period and a design lifetime of structures.

Using Approximation Non-Bayesian Computation with Fuzzy Data to Estimation Inverse Weibull Parameters and Reliability Function

2018 ◽  
pp. 397
Author(s):  
Nadia Hashim Al-Noor ◽  
Shurooq A.K. Al-Sultany
Keyword(s):  
Maximum Likelihood ◽  
Expectation Maximization ◽  
Mean Squared Error ◽  
Maximum Likelihood Estimators ◽  
Reliability Function ◽  
Fuzzy Data ◽  
Monte Carlo Simulation Study ◽  
Weibull Parameters ◽  
Squared Error ◽  
Newton Raphson

        In real situations all observations and measurements are not exact numbers but more or less non-exact, also called fuzzy. So, in this paper, we use approximate non-Bayesian computational methods to estimate inverse Weibull parameters and reliability function with fuzzy data. The maximum likelihood and moment estimations are obtained as non-Bayesian estimation. The maximum likelihood estimators have been derived numerically based on two iterative techniques namely “Newton-Raphson” and the “Expectation-Maximization” techniques. In addition, we provide compared numerically through Monte-Carlo simulation study to obtained estimates of the parameters and reliability function in terms of their mean squared error values and integrated mean squared error values respectively.

The Calibration of Gravity, Entropy, and Related Models of Spatial Interaction

1972 ◽  
Vol 4 (2) ◽  
pp. 205-233 ◽  
Author(s):  
M Batty ◽  
S Mackie
Keyword(s):  
Maximum Likelihood ◽  
Spatial Interaction ◽  
Interaction Models ◽  
Spatial Interaction Models ◽  
Calibration Methods ◽  
The Family ◽  
Newton Raphson ◽  
Best Parameter ◽  
Parameter Values ◽  
Raphson Method

This paper presents a methodology for deriving best statistics for the calibration of spatial interaction models, and several procedures for finding best parameter values are described. The family of spatial interaction models due to Wilson is first outlined, and then some existing calibration methods are briefly reviewed. A procedure for deriving best statistics based on the principle of maximum-likelihood is then developed from the work of Hyman and Evans, and the methodology is illustrated using the example of a retail gravity model. Five methods for solving the maximum-likelihood equations are outlined: procedures based on a simple first-order iterative process, the Newton—Raphson method for several variables, multivariate Fibonacci search, search using the Simplex method, and search based on quadratic convergence, are all tested and compared. It appears that the Newton—Raphson method is the most efficient, and this is further tested in the calibration of disaggregated residential location models.

scholarly journals Fit of the 2011 Indonesian Mortality Table to Gompertz Law and Makeham Law using Maximum Likelihood Estimation

2019 ◽  
Vol 1 (2) ◽  
Author(s):  
Dino Agustin Putra ◽  
Nina Fitriyati ◽  
Mahmudi Mahmudi
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Lagrange Multiplier ◽  
Relative Error ◽  
Likelihood Estimation ◽  
Multiplier Method ◽  
Average Relative Error ◽  
Newton Raphson ◽  
Mortality Table ◽  
Raphson Method

AbstractThis research discusses the estimation of the parameters for Gompertz’s law and Makeham’s law using the Maximum Likelihood Estimation method. A numerical approach to estimate the parameters of Gompertz’s law is the Newton-Raphson method. In the Makeham’s law, we use the Lagrange multiplier method to solve constraints of 0.001<A<0.003, 10^(-6)<B<10^3 and 1.075<C<1.115, and Broyden as a method to estimate the parameter numerically. The estimation result shows that parameter B converges to 0.005749 and parameter C converges to 1.024738 in the Gompertz’s law. In the Makeham’s law, the estimated parameters that satisfied the constraints are A converges to 0.00300344,  B converges to 0.0002716465, and C converges to 1.113395. Based on the Average Relative Error (ARE) that calculated from the estimated for px, the 2011 Indonesian Mortality Table (the 2011 TMI) for men and for women are more accurate when approached using the Gompertz’s law than the Makeham’s law. The estimated for px uses the Gompertz’s law are very close to the px at the 2011 TMI (with Absolute Percentage Errors of less than 1%) at age intervals, for men: 0 – 10 years, 10 – 20 years, 20 – 30 years, and 60 – 70 years, and for women: 0 – 10 years, 10 – 20 years, and 70 – 80 years.Keywords: parameter estimation; Newton-Raphson method; Broyden method; Lagrange Multiplier method. AbstrakPenelitian ini membahas mengenai estimasi parameter hukum mortalitas Gompertz’s dan hukum mortalitas Makeham’s menggunakan metode Maximum Likelihood Estimation. Pendekatan numerik untuk estimasi parameter hukum mortalitas Gompertz dilakukan menggunakan metode Newton-Raphson. Untuk mengatasi syarat batas 0.001<A<0.003, 10^(-6)<B<10^3 dan 1.075<C<1.115, pada estimasi parameter hukum mortalita Makeham digunakan metode pengali Lagrange dan pendekatan numerik metode Broyden. Hasil estimasi menunjukkan bahwa parameter B konvergen ke 0,005749 dan parameter C konvergen ke 1,024738 pada hukum mortalitas Gompertz. Pada hukum mortalitas Makeham’s, hasil estimasi parameter yang memenuhi syarat batas adalah nilai A konvergen ke 0,00300344, B konvergen ke 0,0002716465, dan C konvergen ke 1,113395. Berdasarkan nilai Average Relative Error (ARE) yang dihitung untuk estimasi , Tabel Mortalita Indonesia (TMI 2011) untuk pria dan untuk wanita lebih sesuai jika didekati menggunakan hukum Gompertz daripada hukum Makeham. Estimasi  menggunakan pendekatan hukum Gompertz berada sangat dekat dengan nilai  pada TMI 2011 (dengan Mean Absolute Percentage Error kurang dari 1%) pada interval usia, untuk pria: 0 – 10 tahun, 10 – 20 tahun, 20 – 30 tahun, dan 60 – 70 tahun, dan untuk wanita: 0 – 10 tahun, 10 – 20 tahun, dan 70 – 80 tahun.Kata kunci: estimasi parameter; metode Newton-Raphson; metode Broyden; metode Pengali Lagrange.

scholarly journals The Jolly-Seber-Tag-Loss model with group heterogeneity

1969 ◽  
pp. 30-43
Author(s):  
Selina Beatriz Gonzalez ◽  
Dr. Laura Cowen
Keyword(s):  
Maximum Likelihood Estimators ◽  
Parameter Estimates ◽  
Population Parameters ◽  
R Software ◽  
Loss Model ◽  
Tag Loss ◽  
Subsequent Time ◽  
Group Heterogeneity ◽  
Newton Raphson ◽  
Raphson Method

Mark-recapture experiments are performed to estimate population parameters such as survival probabilities. Animals are captured, tagged, released, and recaptured at subsequent time periods in order to obtain parameter information. The Jolly-Seber-Tag-Loss (JSTL) model (Cowen & Schwarz, 2006) requires some individuals to be double tagged in order to account for the possibility of animals losing their tags. The Jolly-Seber-Tag-Loss model does not, however, consider the possibility of parameters being different among different groups of individuals, that is, group heterogeneity (for example, males may have higher capture probabilities than females). Our research extends the Jolly-Seber-Tag-Loss model to account for this possibility of group heterogeneity among parameters. We use a Newton-Raphson method to obtain maximum likelihood estimators and R software to create a program that estimates population parameters from tag histories. Our simulation study concludes that when group heterogeneity exists, accounting for this group heterogeneity results in more accurate parameter estimates than the original JSTL model. We present the group heterogeneous JSTL (g-hJSTL) for this purpose.

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