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sta 2006 0708_chapter_2

sta 2006 0708_chapter_2 - STA2006 2007-2008 Term 2 Chapter...

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STA2006 2007-2008 Term 2 Chapter 2: Point Estimation (Section 6.2) This is the first topic in statistical inference: Estimate the unknown but fixed parameter of the model (or population) by using the data. The model is usually chosen by 1. using Exploratory Data Analysis 2. using the specific domain knowledge such as financial and economic theories, engineering practice, medical \ biological background, and so on. 3. mathematical convenience. (Poor!) The model is in the form of p.d.f. or p.m.f. paramertized by θ which could be a scalar or a vector. The set of all possible θ is denoted by Ω and is called the parameter space. The model then is represented as { f ( x ; θ ) : θ Ω } . Example 1: Suppose Sam is interested in how much time he spent in waiting for school bus. According to the queueing theory, the time needed to wait for a bus to arrive is exponentially distributed. Let X be such waiting time. Then, under the model, X has p.d.f. f ( x ; θ ) = exp {- x/θ } for x > 0 and Ω = { θ : θ > 0 } is the positive real line. Suppose the data X 1 , . . . , X n are random sampled from the model whose parameter θ is unknown but fixed. The primary issue of estimation problem is to find a random variable as a function of the data X 1 , . . . , X n (denoted as u ( X 1 , . . . , X n ) in the textbook) such that u can be used to estimate θ . u ( X 1 , . . . , X n ) is called the estimator of θ and its realization u ( x 1 , . . . , x n ) is called the estimate of θ . 1 Method of Maximum Likelihood This is a systematic way of finding ”good” estimate called the maximum likelihood estimate (MLE) for a general class of models. It is due to late Sir Ronald A. Fisher (1890-1962) (http://www-groups.dcs.st-and.ac.uk/ history/ 1
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Mathematicians/Fisher.html) who is one of the pioneers in modern mathe- matical statistics. Define the likelihood function of the model as the joint p.d.f. of the data. Denote that by L ( θ ). That is, L ( θ ) = f ( x 1 , . . . , x n ; θ ). Note that the interpretations of the left hand side and the right hand side are different. The right hand side is the p.d.f. which treats x 1 , . . . , x n as the input of the function and θ as the parameter. However, left hand side treats
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