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Parameter Estimation Framework
- 1. Population Y
- 2. E[Y] = @
- 3. Var(Y) = sig2
- 4. Assume Y1, Y2, . . ., Yn is a random sample from population
- 5. Assume Yi are independtly and identically distributed (iid)
- 6. ^ indicates estimate
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Estimation of @ Method 1: Method of Moments
- 1. E[Yi] = @ "population moment"
- 2. `Y = ^@ = sum1-n(Yi/n) "sample moment"
- "sum..." is the Estimator, which is an expression
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Estimation of @ Method 2: Minimize
- 1. s = sum1-n(yi - @)2 given yi(data) = s(@)
- 2. Goal is to find @ that minimizes expression
- 3. @ in 1 known as "Least Squares Estimator of @"
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Best Linear Unbiased Estimator(BLUE)
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Matrix Notation for Estimation
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Ybar =
^mu = (1/n)sum(Yi), where Yi~N(mu,sig2)
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s2 =
^sig = [sum(Yi - Ybar)2]/(n - 1), where Yi~N(mu,sig2)
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Show E(s2) = sig2, that is show that s2 is an unbiased estimator of population variance.
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T-distribution
- t = (Ybar - mu)/[s/sqrt(n)]
- = (Ybar - mu)/se(Ybar)~tn-1
- Derivation:
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Properties of t distribution
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Ingredients of Hypothesis Test
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Var(Ybar) =
^sig2/n = s2/n
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se(Ybar) =
^sig/sqrt(n) = s/sqrt(n)
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Elements of T distribution
- Let Y1...Yn be a random sample from a population Yi~N(mu,sig2)
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Probability of rejecting Ho when it is true
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Rejection Rules
Ho: mu = muo
H1: mu > muo
- Reject Ho and accept H1:
- If p <= alpha and t >= tc
- Fail to reject Ho:
- If p > alpha and t < tc
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Rejection Rules
Ho: mu = muo
H1: mu < muo
- Reject Ho and accept H1:
- If p <= alpha and t <= -tc
- Fail to reject Ho:
- If p > alpha and t > -tc
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Rejection Rules
Ho: mu = muo
H1: mu not equal to muo
"Two-tailed test"
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