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Parameter Estimation Framework
 1. Population Y
 2. E[Y] = @
 3. Var(Y) = sig^{2}
 4. Assume Y_{1}, Y_{2}, . . ., Y_{n} is a random sample from population
 5. Assume Y_{i} are independtly and identically distributed (iid)
 6. ^ indicates estimate

Estimation of @ Method 1: Method of Moments
 1. E[Y_{i}] = @ "population moment"
 2. `Y = ^@ = sum_{1n}(Y_{i}/n) "sample moment"
 "sum..." is the Estimator, which is an expression

Estimation of @ Method 2: Minimize
 1. s = sum_{1n}(y_{i}  @)^{2} given y_{i}(data) = s(@)
 2. Goal is to find @ that minimizes expression
 3. @ in 1 known as "Least Squares Estimator of @"


Best Linear Unbiased Estimator(BLUE)


If Y~N(@, sig^{2}) => `Y~

Matrix Notation for Estimation




Matrix ChiSquare: X^{2}_{(m)}

Ybar =
^mu = (1/n)sum(Y_{i}), where Y_{i}~N(mu,sig^{2})

s^{2} =
^sig = [sum(Y_{i}  Ybar)^{2}]/(n  1), where Y_{i}~N(mu,sig^{2})

Show E(s^{2}) = sig^{2}, that is show that s^{2} is an unbiased estimator of population variance.

Tdistribution
 t = (Ybar  mu)/[s/sqrt(n)]
 = (Ybar  mu)/se(Ybar)~t_{n1}
 Derivation:

Properties of t distribution


Ingredients of Hypothesis Test

Var(Ybar) =
^sig^{2}/n = s^{2}/n

se(Ybar) =
^sig/sqrt(n) = s/sqrt(n)

Elements of T distribution
 Let Y_{1}...Y_{n} be a random sample from a population Yi~N(mu,sig^{2})


Probability of rejecting H_{o} when it is true

Rejection Rules
H_{o}: mu = mu_{o}
H_{1}: mu > mu_{o}
 Reject H_{o} and accept H_{1}:
 If p <= alpha and t >= t_{c}
 Fail to reject H_{o}:
 If p > alpha and t < t_{c}


Rejection Rules
H_{o}: mu = mu_{o}
H_{1}: mu < mu_{o}
 Reject H_{o} and accept H_{1}:
 If p <= alpha and t <= t_{c}
 Fail to reject H_{o}:
 If p > alpha and t > t_{c}


Rejection Rules
H_{o}: mu = mu_{o}
H_{1}: mu not equal to mu_{o
}"Twotailed test"

