scholarly journals Estimation in the Pareto Distribution

1990 ◽  
Vol 20 (2) ◽  
pp. 201-216 ◽  
Author(s):  
Mette Rytgaard
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimator ◽  
Risk Premium ◽  
Unbiased Estimator ◽  
Minimum Variance ◽  
Credibility Theory ◽  
Likelihood Estimator ◽  
Minimum Variance Unbiased Estimator ◽  
The Moment ◽  
Alternative Estimator

AbstractIn the present paper, different estimators of the Pareto parameter α will be proposed and compared to each others.First traditional estimators of α as the maximum likelihood estimator and the moment estimator will be deduced and their statistical properties will be analyzed. It is shown that the maximum likelihood estimator is biased but it can easily be modified to an minimum-variance unbiased estimator of a. But still the coefficient of variance of this estimator is very large.For similar portfolios containing same types of risks we will expect the estimated α-values to be at the same level. Therefore, credibility theory is used to obtain an alternative estimator of α which will be more stable and less sensitive to random fluctuations in the observed losses.Finally, an estimator of the risk premium for an unlimited excess of loss cover will be proposed. It is shown that this estimator is a minimum-variance unbiased estimator of the risk premium. This estimator of the risk premium will be compared to the more traditional methods of calculating the risk premium.

scholarly journals A note on Hammersley's inequality for estimating the normal integer mean

2003 ◽  
Vol 2003 (34) ◽  
pp. 2147-2156 ◽  
Author(s):  
Rasul A. Khan
Keyword(s):  
Maximum Likelihood ◽  
Lower Bound ◽  
Maximum Likelihood Estimator ◽  
Random Sample ◽  
Unbiased Estimator ◽  
Minimum Variance ◽  
Likelihood Estimator ◽  
Sample Mean ◽  
Bhattacharyya Bounds ◽  
Limiting Property

LetX1,X2,…,Xnbe a random sample from a normalN(θ,σ2)distribution with an unknown meanθ=0,±1,±2,…. Hammersley (1950) proposed the maximum likelihood estimator (MLE)d=[X¯n], nearest integer to the sample mean, as an unbiased estimator ofθand extended the Cramér-Rao inequality. The Hammersley lower bound for the variance of any unbiased estimator ofθis significantly improved, and the asymptotic (asn→∞) limit of Fraser-Guttman-Bhattacharyya bounds is also determined. A limiting property of a suitable distance is used to give some plausible explanations why such bounds cannot be attained. An almost uniformly minimum variance unbiased (UMVU) like property ofdis exhibited.

scholarly journals Estimation a Stress-Strength Model for P (Yr:n1 < Xk:n2 ) Using the Lindley Distribution

2017 ◽  
Vol 40 (1) ◽  
pp. 105-121 ◽  
Author(s):  
Marwa Khalil
Keyword(s):  
Maximum Likelihood ◽  
Simulation Study ◽  
Real Data ◽  
Minimum Variance ◽  
Strength Model ◽  
Practical Application ◽  
Likelihood Estimator ◽  
Lindley Distribution ◽  
Minimum Variance Unbiased Estimator ◽  
Proposed Model

The problem of estimation reliability in a multicomponent stress-strength model, when the system consists of k components have strength each compo- nent experiencing a random stress, is considered in this paper. The reliability of such a system is obtained when strength and stress variables are given by Lindley distribution. The system is regarded as alive only if at least r out of k (r < k) strength exceeds the stress. The multicomponent reliability of the system is given by Rr,k . The maximum likelihood estimator (M LE), uniformly minimum variance unbiased estimator (UMVUE) and Bayes esti- mator of Rr,k are obtained. A simulation study is performed to compare the different estimators of Rr,k . Real data is used as a practical application of the proposed model.

scholarly journals Estimation of stress-strength reliability using record ranked set sampling scheme from the exponential distribution

Filomat ◽  
2015 ◽  
Vol 29 (5) ◽  
pp. 1149-1162 ◽  
Author(s):  
Mahdi Salehi ◽  
Jafar Ahmadi
Keyword(s):  
Maximum Likelihood ◽  
Exponential Distribution ◽  
Interval Estimation ◽  
Minimum Variance ◽  
Ranked Set Sampling ◽  
Strength Parameter ◽  
Likelihood Estimator ◽  
Data Set ◽  
Minimum Variance Unbiased Estimator ◽  
Upper Record

In this paper, point and interval estimation of stress-strength reliability based on upper record ranked set sampling (RRSS) from one-parameter exponential distribution are considered. Maximum likelihood estimator (MLE) as well as the uniformly minimum variance unbiased estimator (UMVUE) of stress-strength parameter are derived and their performance are studied. Also, some confidence intervals for stress-strength parameter based on upper RRSS are constructed and then compared on the basis of a simulation study. Finally, a data set has been analyzed for illustrative purposes.

Asymptotic Effect of Misspecification in the Random Part of the Multilevel Model

2004 ◽  
Vol 29 (2) ◽  
pp. 201-218 ◽  
Author(s):  
Johannes Berkhof ◽  
Jarl Kennard Kampen
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimator ◽  
Variance Components ◽  
Fixed Effects ◽  
Multilevel Model ◽  
Closed Form Expression ◽  
Model Parameters ◽  
Likelihood Estimator ◽  
The Moment ◽  
Random Part

The authors examine the asymptotic effect of omitting a random coefficient in the multilevel model and derive expressions for the change in (a) the variance components estimator and (b) the estimated variance of the fixed effects estimator. They apply the method of moments, which yields a closed form expression for the omission effect. In practice, the model parameters are estimated by maximum likelihood; however, since the moment estimator and the maximum likelihood estimator are both consistent, the presented expression for the change in the variance components estimator asymptotically holds for the maximum likelihood estimator as well. The results are illustrated with an analysis of mathematics performance data.

scholarly journals Uniform Minimum Variance Unbiased Estimator of Fractal Dimension

2021 ◽  
Vol 9 (1) ◽  
pp. 63-68
Author(s):  
Zeny Maureal ◽  
◽  
Elmer Castillano ◽  
Roberto Padua ◽  
◽  
...  
Keyword(s):  
Unbiased Estimator ◽  
Random Variable ◽  
Minimum Variance ◽  
Statistical Characteristics ◽  
Fractional Dimension ◽  
Power Law Distribution ◽  
Likelihood Estimator ◽  
Minimum Variance Unbiased Estimator ◽  
Point Estimator ◽  
The Relationship

The paper introduced the concept of a fractal distribution using a power-law distribution. It proceeds to determining the relationship between fractal and exponential distribution using a logarithmic transformation of a fractal random variable which turns out to be exponentially distributed. It also considered finding the point estimator of fractional dimension and its statistical characteristics. It was shown that the maximum likelihood estimator of the fractional dimension λ is biased. Another estimator was found and shown to be a uniformly minimum variance unbiased estimator (UMVUE) by Lehmann-Scheffe’s theorem.

The Comparison Between the Bayes Estimator and the Maximum Likelihood Estimator of the Reliability Function for Negative Exponential Distribution

2018 ◽  
pp. 439
Author(s):  
Hazim Mansour Gorgees ◽  
Bushra Abdualrasool Ali ◽  
Raghad Ibrahim Kathum
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimator ◽  
Exponential Distribution ◽  
Reliability Function ◽  
Bayes Estimator ◽  
Likelihood Estimator ◽  
Monte Carlo Simulation Technique ◽  
Negative Exponential Distribution ◽  
Negative Exponential ◽  
Better Than

     In this paper, the maximum likelihood estimator and the Bayes estimator of the reliability function for negative exponential distribution has been derived, then a Monte –Carlo simulation technique was employed to compare the performance of such estimators. The integral mean square error (IMSE) was used as a criterion for this comparison. The simulation results displayed that the Bayes estimator performed better than the maximum likelihood estimator for different samples sizes.

scholarly journals Transforming variables to central normality

2021 ◽  
Author(s):  
Jakob Raymaekers ◽  
Peter J. Rousseeuw
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimator ◽  
Simulation Study ◽  
Real Data ◽  
Data Sets ◽  
Transformation Parameter ◽  
Likelihood Estimator ◽  
Extensive Simulation ◽  
Highly Sensitive

AbstractMany real data sets contain numerical features (variables) whose distribution is far from normal (Gaussian). Instead, their distribution is often skewed. In order to handle such data it is customary to preprocess the variables to make them more normal. The Box–Cox and Yeo–Johnson transformations are well-known tools for this. However, the standard maximum likelihood estimator of their transformation parameter is highly sensitive to outliers, and will often try to move outliers inward at the expense of the normality of the central part of the data. We propose a modification of these transformations as well as an estimator of the transformation parameter that is robust to outliers, so the transformed data can be approximately normal in the center and a few outliers may deviate from it. It compares favorably to existing techniques in an extensive simulation study and on real data.

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