Optimal Estimator with Respect to Expected Log-Likelihood

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Year:
2014
Type of Publication:
Article
Keywords:
Expected Log-Likelihood, Exponential Distribution, Log-Normal Distribution, Maximum Likelihood Estimator, Normal Distribution, Parametric Bootstrap Method, Simple Regression
Authors:
Dr. Kunio Takezawa
Journal:
IJISM
Volume:
2
Number:
6
Pages:
494-508
Month:
November
Abstract:
The maximum likelihood estimator does not always give optimal results in terms of the expected log-likelihood. The main purpose of statistical estimation is to accurately predict future phenomena. Therefore, we should investigate the estimator that gives the largest expected log-likelihood because it makes the best use of the information contained in data. Shrinkage estimators such as the ridge regression method were developed with this in mind. However, such shrinkage estimators produce differences to the maximum likelihood estimators in a specific direction. Differences in other directions may give better results. Therefore, we suggest a method that produces differences to a maximum likelihood estimator in a more flexible way. In our algorithm, pseudo-data are sampled from the generated population using a parametric bootstrap method. The pseudo-data are then used to optimize the constants in the estimator. The pseudo-data are divided into available and future pseudo-data. Then, the estimates are calculated using the available pseudo-data and the flexible estimator. We derive the expected log-likelihoods of the resulting estimates using future pseudo-data, and then optimize the constants by maximizing the expected log-likelihood. This method works well in several simulations, suggesting that the proposed method is beneficial when the aim is to maximize the expected log-likelihood.

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