This class of estimators has an important property. We would like to have an estimator with smaller bias and smaller variance : if one can nd several unbiased estimators, we want to use an estimator with smaller vari-ance. θ. 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. We say that . is unbiased for . PROPERTIES OF ESTIMATORS (BLUE) KSHITIZ GUPTA 2. WHAT IS AN ESTIMATOR? 2.4.1 Finite Sample Properties of the OLS and ML Estimates of An estimator is a function of the data. ECONOMICS 351* -- NOTE 3 M.G. 1.1 Unbiasness. 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). This flexibility in Relative e ciency (Def 9.1) Suppose ^ 1 and ^ 2 are two unbi-ased estimators for , with variances, V( ^ 1) and V(^ 2), respectively. T. is some function. 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? 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 . Example: Suppose X 1;X 2; ;X n is an i.i.d. Properties of estimators. Our first choice of estimator for this parameter should prob-ably be the sample minimum. Then relative e ciency of ^ 1 relative to ^ 2, For example, if is a parameter for the variance and ^ is the maximum likelihood estimator, then p ^ is the maximum likelihood estimator for the standard deviation. Point estimators. ˆ= T (X) be an estimator where . properties at the same time, and sometimes they can even be incompatible. θ. • 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 … But the sample mean Y is also an estimator of the popu-lation minimum. ˆ. I When no estimator with desireable small-scale properties can be found, we often must choose between di erent estimators on the basis of asymptotic properties X. be our data. If ^(x) is a maximum likelihood estimate for , then g( ^(x)) is a maximum likelihood estimate for g( ). 1 Estimators. • 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. If we have a parametric family with parameter θ, then an estimator of θ is usually denoted by θˆ. random sample from a Poisson distribution with parameter . Let . The small-sample properties of the estimator βˆ j are defined in terms of the mean ( ) 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). Abbott 2. Indeed, any statistic is an estimator. 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. Let . 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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