Stat 400 In Class Worksheet 12
T.A. Emily King
April 30, 2008
Goal: Find a point estimator ˆ
θfor a quantity θ, given a random sample
X1, X2, . . . Xnfrom a pdf.
•Method 1: Method of Moments
–Step 1: Find a formula for θin terms of E(X), E(X2), E(X3),etc.
–Note: Recall that V(X) = E(X2)−[E(X)]2.
–Step 2: Replace each E(Xk) in the formula that you found in Step 1
with 1
nPn
i=1 Xk
i. When k= 1, this is simply Xthis is your formula
for the point estimator ˆ
θ.
–Step 3: Given values from a random sample, x1, x2, . . . , xn, plug in
the values to find the point estimate.
•Method 2: Maximum Likelihood Estimator (MLE)
The pdf is of the form f(x;θ).
–Step 1: Since X1, X2, . . . Xnare independent readings from the same
pdf, their joint pdf is Qn
i=1 f(xi;θ). This is the likelihood function.
–Step 2: We want to maximize the likelihood function with respect
to θ. This is usually accomplished by maximized the loglikelihood
function (because the algebra and calculus necessary to solve the
problem are easier).
–Step 3: The value of θwhich maximizes the (log)likelihood function
is the MLE ˆ
θ.
–Step 4: Given values from a random sample, x1, x2, . . . , xn, plug in
the values to find the point estimate.
