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Chapter 6: Point Estimation 6.1.Concepts of Point Estimation Point Estimator A point estimator of a parameter is a single number that can be regarded as an approximation of . A point estimator can be obtained by selecting a suitable statistic and computing its value from the given sample data. Point Estimator A point estimator of a parameter is a single number that can be regarded as an approximation of . A point estimator can be obtained by selecting a suitable statistic and computing its value from the given sample data. Point Estimator A point estimator of a parameter is a single number that can be regarded as an approximation of . A point estimator can be obtained by selecting a suitable statistic and computing its value from the given sample data. Unbiased Estimator A point estimator ˆ is said to be an ˆ E ( ) unbiased estimator of if for every possible value of . If ˆ is not biased, the difference E (ˆ) is called the bias of ˆ . The pdf’s of a biased estimator ˆ1 and an unbiased estimator ˆ2 for a parameter . pdf of ˆ2 pdf of ˆ1 Bias of 1 The pdf’s of a biased estimator ˆ1 and an unbiased estimator ˆ2 for a parameter . pdf of ˆ2 pdf of ˆ1 Bias of 1 Unbiased Estimator When X is a binomial rv with parameters n and p, the sample proportion pˆ X / n is an unbiased estimator of p. Principle of Unbiased Estimation When choosing among several different estimators of , select one that is unbiased. Unbiased Estimator Let X1, X2,…,Xn be a random sample from a distribution with mean and 2 variance . Then the estimator ˆ S 2 2 X i X n 1 is an unbiased estimator. 2 Unbiased Estimator If X1, X2,…,Xn is a random sample from a distribution with mean , then X is an unbiased estimator of . If in addition the distribution is continuous and symmetric, then X and any trimmed mean are also unbiased estimators of . Principle of Minimum Variance Unbiased Estimation Among all estimators of that are unbiased, choose the one that has the minimum variance. The resulting ˆ is called the minimum variance unbiased estimator (MVUE) of . Graphs of the pdf’s of two different unbiased estimators pdf of ˆ1 pdf of ˆ2 MVUE for a Normal Distribution Let X1, X2,…,Xn be a random sample from a normal distribution with parameters and . Then the estimator ˆ X is the MVUE for . A biased estimator that is preferable to the MVUE pdf of ˆ1 (biased) pdf of ˆ2 (the MVUE) The Estimator for X, X, X e , X tr (10) 1. If the random sample comes from a normal distribution, then is the best estimator since it has minimum variance among all unbiased estimators. The Estimator for X, X, X e , X tr (10) 2. If the random sample comes from a Cauchy distribution, then X is good (the MVUE is not known). X and X e are quite bad. The Estimator for X, X, X e , X tr (10) 3. If the underlying distribution is uniform, the best estimator is X e this estimator is influenced by outlying observations, but the lack of tails makes this impossible. The Estimator for X, X, X e , X tr (10) 4. The trimmed mean X tr (10) works reasonably well in all three situations but is not the best for any of them. Standard Error The standard error of an estimator ˆ is ˆ = V ( ) . If its standard deviation ˆ the standard error itself involves unknown parameters whose values can be estimated, substitution into ˆ yields the estimated standard error of the estimator, denoted ˆˆ or sˆ . 6.2. Methods of Point Estimation Moments:Let X1, X2,…,Xn be a random sample from a pmf or pdf f (x). For k = 1, 2,… the kth population moment, or kth moment of the distribution f (x) is estimated be the kth sample moment 1 n k i 1 X i . n Moment Estimators Let X1, X2,…,Xn be a random sample from a distribution with pmf or pdf f ( x;1 ,..., m ), where 1 ,..., m are parameters whose values are unknown. Then the moment estimators 1 ,..., m are obtained by equating the first m sample moments to the corresponding first m population moments and solving for 1 ,..., m . Likelihood Function Let X1, X2,…,Xn have joint pmf or pdf f ( x1 ,..., xn ;1 ,..., m ) where parameters 1 ,..., m have unknown values. When x1,…,xn are the observed sample values and f is regarded as a function of 1 ,..., m , it is called the likelihood function. Maximum Likelihood Estimators The maximum likelihood estimates (mle’s) ˆ1 ,...,ˆm are those values of the i 's that maximize the likelihood function so that f ( x1 ,..., xn ;ˆ1 ,...,ˆm ) f ( x1,..., xn ;1,...,m ) for all 1 ,..., m When the Xi’s are substituted in the place of the xi’s, the maximum likelihood estimators result. The Invariance Principle Let ˆ1 ,...,ˆm be the mle’s of the parameters 1 ,..., m Then the mle of any function h(1 ,..., m ) of these parameters is the function h(ˆ1 ,...,ˆm ) of the mle’s. Desirable Property of the Maximum Likelihood Estimate Under very general conditions on the joint distribution of the sample, when the sample size n is large, the maximum likelihood estimator of any parameter ˆ [ E ( ) ] and has is approx. unbiased variance that is nearly as small as can be achieved by any estimator. mleˆ MVUE of

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