If ^(x) is a maximum likelihood estimate for , then g( ^(x)) is a maximum likelihood estimate for g( ). An estimator is a function of the data. • Obtaining a point estimate of a population parameter • Desirable properties of a point estimator: • Unbiasedness • Efficiency • Obtaining a confidence interval for a mean when population standard deviation is known • Obtaining a confidence interval for a mean when population standard deviation is … Small-Sample Estimator Properties Nature of Small-Sample Properties The small-sample, or finite-sample, distribution of the estimator βˆ j for any finite sample size N < ∞ has 1. a mean, or expectation, denoted as E(βˆ j), and 2. a variance denoted as Var(βˆ j). Only once we’ve analyzed the sample minimum can we say for certain if it is a good estimator or not, but it is certainly a natural first choice. Indeed, any statistic is an estimator. The small-sample properties of the estimator βˆ j are defined in terms of the mean ( ) When we want to study the properties of the obtained estimators, it is convenient to distinguish between two categories of properties: i) the small (or finite) sample properties, which are valid whatever the sample size, and ii) the asymptotic properties, which are associated with large samples, i.e., when tends to . • In statistics, an estimator is a rule for calculating an estimate of a given quantity based on observed data • Example- i. X follows a normal distribution, but we do not know the parameters of our distribution, namely mean (μ) and variance (σ2 ) ii. Some of the properties are defined relative to a class of candidate estimators, a set of possible T(") that we will denote by T. The density of an estimator T(") will be denoted (t, o), or when it is necessary to index the estimator, T(t, o). ˆ. Properties of estimators. Properties of Good Estimators ¥In the Frequentist world view parameters are Þxed, statistics are rv and vary from sample to sample (i.e., have an associated sampling distribution) ¥In theory, there are many potential estimators for a population parameter ¥What are characteristics of good estimators? But the sample mean Y is also an estimator of the popu-lation minimum. WHAT IS AN ESTIMATOR? Large-sample properties of estimators I asymptotically unbiased: means that a biased estimator has a bias that tends to zero as sample size approaches in nity. is unbiased for . Then relative e ciency of ^ 1 relative to ^ 2, Relative e ciency: If ^ 1 and ^ 2 are both unbiased estimators of a parameter we say that ^ 1 is relatively more e cient if var(^ 1)

properties of a good estimator pdf

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