Abstract
Given a parameter of interest, such as a population mean μ or population proportion p, the objective of point estimation is to use a sample to compute a number that represents in some sense a good guess for the true value of the parameter. The resulting number is called a point estimate. In Section 7.1, we present some general concepts of point estimation. In Section 7.2, we describe and illustrate two important methods for obtaining point estimates: the method of moments and the method of maximum likelihood.
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Notes
- 1.
Following earlier notation, we could use \( \hat{\Theta } \) (an uppercase theta) for the estimator, but this is cumbersome to write.
- 2.
Since ln[g(x)] is a monotonic function of g(x), finding x to maximize ln[g(x)] is equivalent to maximizing g(x) itself. In statistics, taking the logarithm frequently changes a product to a sum, which is easier to work with.
- 3.
This conclusion requires checking the second derivative, but the details are omitted.
Bibliography
DeGroot, Morris, and Mark Schervish, Probability and Statistics (3rd ed.), Addison-Wesley, Boston, MA, 2002. Includes an excellent discussion of both general properties and methods of point estimation; of particular interest are examples showing how general principles and methods can yield unsatisfactory estimators in particular situations.
Efron, Bradley, and Robert Tibshirani, An Introduction to the Bootstrap, Chapman and Hall, New York, 1993. The bible of the bootstrap.
Hoaglin, David, Frederick Mosteller, and John Tukey, Understanding Robust and Exploratory Data Analysis, Wiley, New York, 1983. Contains several good chapters on robust point estimation, including one on M-estimation.
Hogg, Robert, Allen Craig, and Joseph McKean, Introduction to Mathematical Statistics (6th ed.), Prentice Hall, Englewood Cliffs, NJ, 2005. A good discussion of unbiasedness.
Larsen, Richard, and Morris Marx, Introduction to Mathematical Statistics (4th ed.), Prentice Hall, Englewood Cliffs, NJ, 2005. A very good discussion of point estimation from a slightly more mathematical perspective than the present text.
Rice, John, Mathematical Statistics and Data Analysis (3rd ed.), Duxbury Press, Belmont, CA, 2007. A nice blending of statistical theory and data.
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Devore, J.L., Berk, K.N. (2012). Point Estimation. In: Modern Mathematical Statistics with Applications. Springer Texts in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0391-3_7
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