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estimation - Stat 411 Lecture Notes Point estimation Ryan...

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Stat 411 – Lecture Notes Point estimation *†‡§ Ryan Martin Spring 2012 1 Introduction The statistical inference problem starts with the identification of a population of interest, about which something is unknown. For example, before introducing a law that homes be equipped with radon detectors, government officials should first ascertain whether radon levels in local homes are, indeed, too high. The most efficient (and, surprisingly, often the most accurate) means to gather this information is to take a sample of local homes and record the radon levels in each. 1 Now that the sample is obtained, how should this information be used to answer the question of interest? Suppose that officials are interested in the mean radon level for all homes in their community—this quantity is unknown, otherwise, there’d be no reason to take the sample in the first place. After some careful exploratory data analysis, the statistician working the project determines a statistical model, i.e., the functional form of the PDF that characterizes radon levels in homes in the community. Now the statistician has a model, which depends on the unknown mean radon level (and possibly other unknown population characteristics), and a sample from that distribution. His/her charge is to use these two pieces of information to make inference about the unknown mean radon level. The simplest of such inferences is simply to estimate this mean. In this section we will discuss some of the basic principles of statistical estimation. This will be an important theme throughout the course. * Version: January 23, 2012 Please do not distribute these notes without the author’s consent ( [email protected] ) These notes are meant solely to supplement in-class lectures. The author makes no guarantees that these notes are free of typos or other, more serious errors. § Parts are taken from A. DasGupta’s Stat 528 lecture notes at Purdue in Fall 2005. 1 In Stat 411 we will take the sample as given, that is, we will not consider the question of how the sample is obtained. In general it is not an easy task to obtain a bona fide completely random sample; carefully planning of experimental or survey designs is necessary. For example, despite the convenience, government officials should not simply choose to study the radon levels just in the homes in their own neighborhood—why? 1
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2 Notation and terminology The starting point is a statement of the model. Let X 1 , . . . , X n be a sample from a distribution with CDF F θ , depending on a parameter θ which is unknown. In some cases, it will be important to also know the parameter space Θ, the set of possible values of θ . 2 Point estimation is the problem of find a function of the data that provides a “good” estimate of the unknown parameter θ .
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