scholarly journals Bias Reduction for the Marshall-Olkin Extended Family of Distributions with Application to an Airplane’s Air Conditioning System and Precipitation Data

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 851
Author(s):  
Tiago M. Magalhães ◽  
Yolanda M. Gómez ◽  
Diego I. Gallardo ◽  
Osvaldo Venegas
Keyword(s):  
Maximum Likelihood ◽  
Air Conditioning ◽  
Extended Family ◽  
Score Function ◽  
Bias Reduction ◽  
Small Sample ◽  
Air Conditioning System ◽  
Finite Samples ◽  
Small Sample Sizes ◽  
Family Of Distributions

The Marshall-Olkin extended family of distributions is an alternative for modeling lifetimes, and considers more or less asymmetry than its parent model, achieved by incorporating just one extra parameter. We investigate the bias of maximum likelihood estimators and use it to develop an estimator with less bias than traditional estimators, by a modification of the score function. Unlike other proposals, in this paper, we consider a bias reduction methodology that can be applied to any member of the family and not necessarily to any particular distribution. We conduct a Monte Carlo simulation in order to study the performance of the corrected estimators in finite samples. This simulation shows that the maximum likelihood estimator is quite biased and the proposed estimator is much less biased; in small sample sizes, the bias is reduced by around 50 percent. Two applications, related to the air conditioning system of an airplane and precipitations, are presented to illustrate the results. In those applications, the bias reduction for the shape parameters is close to 25% and the bias reduction also reduces, among others things, the width of the 95% confidence intervals for quantiles lower than 0.594.

The Pricing Problem

1987 ◽  
Vol 1 (3) ◽  
pp. 349-366
Author(s):  
Jaxk H. Reeves ◽  
Ashim Mallik ◽  
William P. McCormick
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
High Efficiency ◽  
Asymptotic Properties ◽  
Likelihood Estimation ◽  
Small Sample ◽  
Sequential Procedure ◽  
Pricing Scheme ◽  
Simulation Results ◽  
Small Sample Sizes

A sequential procedure to select optimal prices based on maximum likelihood estimation is considered. Asymptotic properties of the pricing scheme and the concommitant estimation problem are examined. For small sample sizes, simulation results show that the proposed procedure has high efficiency relative to the best procedure when the parameter is known.

Maximum Likelihood Estimation for MA(1) Processes with a Root on or near the Unit Circle

1996 ◽  
Vol 12 (1) ◽  
pp. 1-29 ◽  
Author(s):  
Richard A. Davis ◽  
William T.M. Dunsmuir
Keyword(s):  
Maximum Likelihood ◽  
Maximum Likelihood Estimation ◽  
Unit Circle ◽  
Asymptotic Distribution ◽  
Moving Average ◽  
Likelihood Estimation ◽  
Small Sample ◽  
Practical Significance ◽  
True Parameter ◽  
Small Sample Sizes

This paper considers maximum likelihood estimation for the moving average parameter θ in an MA(1) model when θ is equal to or close to 1. A derivation of the limit distribution of the estimate θLM, defined as the largest of the local maximizers of the likelihood, is given here for the first time. The theory presented covers, in a unified way, cases where the true parameter is strictly inside the unit circle as well as the noninvertible case where it is on the unit circle. The asymptotic distribution of the maximum likelihood estimator subMLE is also described and shown to differ, but only slightly, from that of θLM. Of practical significance is the fact that the asymptotic distribution for either estimate is surprisingly accurate even for small sample sizes and for values of the moving average parameter considerably far from the unit circle.

scholarly journals Bias-Corrected Maximum Likelihood Estimators of the Parameters of the Unit-Weibull Distribution

2021 ◽  
Vol 50 (3) ◽  
pp. 41-53
Author(s):  
Andre Menezes ◽  
Josmar Mazucheli ◽  
F. Alqallaf ◽  
M. E. Ghitany
Keyword(s):  
Maximum Likelihood ◽  
Weibull Distribution ◽  
Parametric Bootstrap ◽  
Real Data ◽  
Bias Reduction ◽  
Maximum Likelihood Estimates ◽  
Data Sets ◽  
Analytical Methodology ◽  
Finite Samples ◽  
Modeling Data

It is well known that the maximum likelihood estimates (MLEs) have appealing statistical properties. Under fairly mild conditions their asymptotic distribution is normal, and no other estimator has a smaller asymptotic variance.However, in finite samples the maximum likelihood estimates are often biased estimates and the bias disappears as the sample size grows.Mazucheli, Menezes, and Ghitany (2018b) introduced a two-parameter unit-Weibull distribution which is useful for modeling data on the unit interval, however its MLEs are biased in finite samples.In this paper, we adopt three approaches for bias reduction of the MLEs of the parameters of unit-Weibull distribution.The first approach is the analytical methodology suggested by Cox and Snell (1968), the second is based on parametric bootstrap resampling method, and the third is the preventive approach introduced by Firth (1993).The results from Monte Carlo simulations revealed that the biases of the estimates should not be ignored and the bias reduction approaches are equally efficient. However, the first approach is easier to implement.Finally, applications to two real data sets are presented for illustrative purposes.

Estimation of number of degrees of freedom of nuclear reaction widths distributions

1969 ◽  
Vol 47 (6) ◽  
pp. 665-686 ◽  
Author(s):  
H. Lycklama ◽  
T. J. Kennett ◽  
L. B. Hughes
Keyword(s):  
Maximum Likelihood ◽  
Nuclear Reaction ◽  
Degrees Of Freedom ◽  
Minimum Variance ◽  
Small Sample ◽  
Finite Sample ◽  
Modified Maximum Likelihood ◽  
Method Of Maximum Likelihood ◽  
Chi Squared ◽  
Small Sample Sizes

The effects of small sample sizes have been studied in estimating the number of degrees of freedom of nuclear reaction widths distributions using the method of maximum likelihood, the method of moments, and the method of minimization of variance. It is found that the estimates are biased as a function of the sample size and the number of degrees of freedom of the widths distribution. Bias is reduced somewhat by applying the estimation techniques to the finite sample transformation of the chi-squared distribution, the beta distribution. The efficiency of each estimation technique is indicated by comparison of the variances of the estimates to the minimum variance obtainable. A modified maximum likelihood estimator is found to be unbiased and efficient.

Consistency and Efficiency of Ordinary Least Squares, Maximum Likelihood, and Three Type II Linear Regression Models

Methodology ◽  
2007 ◽  
Vol 3 (2) ◽  
pp. 81-88 ◽  
Author(s):  
João Maroco
Keyword(s):  
Maximum Likelihood ◽  
Measurement Errors ◽  
Major Axis ◽  
Ordinary Least Squares ◽  
Small Sample ◽  
Type Ii ◽  
Linear Regression Models ◽  
Sample Sizes ◽  
Reduced Major Axis ◽  
Small Sample Sizes

Abstract. Type I linear regression models, which allow for measurement errors only in the criterion variable, are frequently used in modeling research in psychology and the social sciences. Although there are frequently measurement errors and large natural variation both in the criterion and predictor variables, type II regression methods that account for these errors are seldom used in these fields of study. The consistency and efficiency of three type II regression methods (reduced major axis, Kendall's robust line-fit and Bartlett's three-group) were evaluated in comparison to ordinary least squares (OLS) and the maximum likelihood with known variance ratio used frequently in biometrics and econometrics. When predictors are measured with error, OLS slope estimates are biased toward zero, and the same bias was observed with both Kendall's and Bartlett's methods. Reduced major axis produced consistent estimates even for small sample sizes, whenever the measurement errors in X are similar in magnitude to measurement errors in Y, but there was a consistent bias when the measurement error in X was smaller/greater than in Y. Maximum likelihood estimates behaved erroneously for small sample sizes, but for larger sample sizes they converged to the expected values.

scholarly journals Exponentiated Nadarajah Haghighi Poisson Distribution

2019 ◽  
Vol 8 (5) ◽  
pp. 34
Author(s):  
Diouma Sira KA ◽  
George Otieno Orwa ◽  
Oscar Ngesa
Keyword(s):  
Maximum Likelihood ◽  
Poisson Distribution ◽  
Generating Functions ◽  
Air Conditioning ◽  
Estimation Method ◽  
Likelihood Estimation ◽  
Real Data ◽  
Data Set ◽  
Air Conditioning System ◽  
Maximum Likelihood Estimation Method

This paper discusses the Exponentiated Nadarajah-Haghighi Poisson distribution focusing on statistical properties such as the Quantile, Moments, Moment Generating Functions, Order statistics and Entropy. To estimate the parameters of the model, the Maximum Likelihood Estimation method is used. To demonstrate the performance of the estimators, a simulation study is carried out. A real data set from Air conditioning system is used to highlight the potential application of the distribution.

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