Abstract
This chapter provides an overview of point estimation. The intersection of a nonempty collection of sigma-fields is a sigma-field. In mathematical statistics, there are two synonyms used for the word random variable—estimate and statistic. The word estimate is used in the problems of estimation, and the word statistic is used in the problems of hypothesis testing. The chapter discusses two properties of estimates—unbiasedness and consistency. The random variable U is called an unbiased estimate of θ, if the expectation of U exists and if EU = θ, whatever be the value of θ. Unbiasedness follows from a simple application of the linear properties of expectation. Unbiasedness forces the distribution of the estimator to be centered at the true parameter value. An unbiased estimate, whose variance is the minimum variance in the Cramér–Rao inequality, is a maximum likelihood estimate.
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