en>Matthiaspaul: Created disambiguation page with various meaning of Aperture value

2012-10-04T01:14:22Z

Created disambiguation page with various meaning of Aperture value

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'''Empirical likelihood''' (EL) is an estimation method in [[statistics]]. Empirical likelihood estimates require few assumptions about the error distribution compared to similar methods like [[maximum likelihood]]. EL can handle data well as long as it is [[independent and identically distributed]] (iid). EL performs well even when the distribution is asymmetric or censored. EL methods are also useful since they can easily incorporate constraints and prior information. [[Art Owen]] pioneered work in this area with his 1988 paper.

==Estimation procedure==
EL estimates are calculated by maximizing the empirical [[likelihood function]] subject to constraints based on the [[estimating function]] and the trivial assumption that the probability weights of the likelihood function sum to 1.<ref>Mittelhammer, Judge, and Miller (2000), 292.</ref> This procedure is represented:
: <math>
\max_{\pi_{i}, \theta} \sum_{i=1}^n \ln \pi_{i}
</math>
Subject to the constraints
:<math>
s.t. \sum_{i=1}^n \pi_{i} = 1, \sum_{i=1}^n \pi_{i} h(y_{i};\theta) = 0
</math><ref>Bera, Y. Bilias (2002), 77.</ref>

The value of the theta parameter can be found by solving the [[Lagrangian]]:
:<math>
\mathcal{L} = \sum_{i=1}^n \pi_{i} + \mu (1- \sum_{i=1}^n \pi_{i})-n\tau' \sum_{i=1}^n \pi_{i} h(y_{i};\theta)
</math><ref>Bera, Y. Bilias (2002), 77.</ref>

==See also==
* [[Bootstrapping (statistics)]]
* [[Jackknife (statistics)]]

==Notes==
{{reflist}}

==References==
* {{cite journal |last1=Bera |first1=Anil K. |last2=Bilias |first2=Yannis |year=2002 |title=The MM, ME, ML, EL, EF and GMM approaches to estimation: a synthesis |journal=Journal of Econometrics|issue=1-2,number 107 |pages=51–86}}
* {{cite book |last1=Mittelhammer |first1=Ron C. |last2=Judge |first2=George G. |last3=Miller |first3=Douglas J. |title=Econometric Foundations |publisher=Cambridge University Press |location=New York |year=2000 |isbn=0521623944}}
* Owen, Art B. "Empirical likelihood ratio confidence intervals for a single functional." Biometrika 75.2 (1988): 237-249. jstor

[[Category:Estimation theory]]
[[Category:Fitting probability distributions]]

Aperture value - Revision history

en>Matthiaspaul: Created disambiguation page with various meaning of Aperture value