Abstract
In many practical situations, we need to estimate different statistical characteristics based on a sample. In some cases, we know that the corresponding probability distribution belongs to a known finite-parametric family of distributions. In such cases, a reasonable idea is to use the Maximum Likelihood method to estimate the corresponding parameters, and then to compute the value of the desired statistical characteristic for the distribution with these parameters.
In some practical situations, we do not know any family containing the unknown distribution. We show that in such nonparametric cases, the Maximum Likelihood approach leads to the use of sample mean, sample variance, etc.
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References
Jaynes, E.T., Bretthorst, G.L.: Probability Theory: The Logic of Science. Cambridge University Press, Cambridge, UK (2003)
Sheskin, D.J.: Handbook of Parametric and Nonparametric Statistical Procedures. Chapman and Hall/CRC, Boca Raton, Florida (2011)
Acknowledgments
This work was supported by Chiang Mai University, Thailand. This work was also supported in part by the National Science Foundation grants HRD-0734825 and HRD-1242122 (Cyber-ShARE Center of Excellence) and DUE-0926721, and by an award “UTEP and Prudential Actuarial Science Academy and Pipeline Initiative” from Prudential Foundation.
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Kreinovich, V., Dumrongpokaphan, T. (2018). How to Estimate Statistical Characteristics Based on a Sample: Nonparametric Maximum Likelihood Approach Leads to Sample Mean, Sample Variance, etc.. In: Kreinovich, V., Sriboonchitta, S., Chakpitak, N. (eds) Predictive Econometrics and Big Data. TES 2018. Studies in Computational Intelligence, vol 753. Springer, Cham. https://doi.org/10.1007/978-3-319-70942-0_11
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DOI: https://doi.org/10.1007/978-3-319-70942-0_11
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