STAT 330 Lecture, July 10th, 2012
Chapter 6: Point estimation
Section 6.1: Introduction
Background: Suppose X ,...,X are iid random variables from pf (discrete case) or pdf
(continuous case), f(x;θ). Here θ is unknown; it can be scalar or vector θ = (θ1,...,θk) .
Purpose: We would like to estimate θ based on X ,..1,X . n
For example, if X 1...,X ∼nN(µ,1), then θ =
For example, if X 1...,X ∼nN(µ,σ ), then θ =
• Θ: parameter space; all possible values of θ.
For example, if X1,...,X n N(µ,1), then Θ =
For example, if X ,...,X ∼ N(µ,σ ), then Θ =
• Data: (X ,...,X ).
• Observed data: observed value of (X ,...,X ); denoted by (x ,...,x ).
1 n 1 n
• Statistic: function of data & does not depend on θ; denoted by T = T(X ,...,X ). For
example, if X 1...,X ∼nN(µ,1), then X = ¯n 1 n Xiis a statistic; n(X n µ)/σ
is NOT a statistic.
Estimator & Estimate
(1) If the statistic T = T(X ,1..,X ) ns used to estimate θ, then T = T(X ,...,1 ) is n
called the estimator of θ.
(2) The observed value of T, t = t(x1,...,x n is called the estimate of θ.
1 Example Suppose X ,1..,X ∼nN(µ,1) and the observed value of (1 ,...nX ) i1 (x ,.n.,x ).
X n X i is
¯n= xi is
2 Section 6.2: Method of moments
Problem of interest: Suppose X ,...,1 are iin random variables from pf (discrete case)
or pdf (continuous case), f(x;θ). Our purpose is to estimate θ with θ = (θ ,.1.,θ ) .k
Let µ j E(X ), 1 = 1,...,k: the ﬁrst k population moments. Sometimes, µ is a funjtion
of θ, and therefore we also write it as µj(θ1,...,θk).
Let M = 1 ▯ n X , j = 1,...,k: the ﬁrst k sample moments. Note that E(M ) = µ .
j n i=1 i j j
Method of moments (MM):
The idea of method of moments is to choose the estimators θ ,...1θ such khat the ﬁrst k
population moments are equal to the ﬁrst k sample moments. That is,
µ (θ ,...,θ ) = M = 1 X , j = 1,...,k.
j 1 k j n i
ˆ ˆ ˆ T
The estimator θ = (θ ,1..,θ )kis called the Method of moment (