University of California, Los Angeles Department of Statistics
Statistics 100B Instructor: Nicolas Christou Method of moments – Examples
Very simple!
The method of moments is based on the assumption that the sample moments are good estimates of the corresponding population moments.
Definition:
Population moments
EX = μ is the first population moment EX2 is the second population moment .
.
EXk is the kth population moment
n i=1 i
Therefore, X ̄ = 1 n X is a good estimator of EX = μ. Similarly, 1 n X2 is a good
n i=1 i n i=1 i estimator of EX2, etc.
Example 1:
Suppose X1, X2, · · · , Xn is a random sample from a Poisson distribution with mean λ. Find the moment estimator of λ.
Sample moments
X ̄ = 1 n X is the first sample moment.
n i=1 i
1 n X2 is the second sample moment.
n i=1 i .
.
1 n Xk is the kth sample moment.
Example 2:
Let X follow the uniform distribution on the interval (0, θ), and X1, X2, · · · , Xn denote i.i.d. random variables from this distribution. Find the method of moments estimator of θ.
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Example 3:
If X1, X2, · · · , Xn denotes a random sample from N(μ, σ), find the method of moments esti- mators of μ and σ2.
Example 4:
If X1, X2, · · · , Xn denotes a random sample from N(0, σ), find the method of moments esti- mators of σ2.
Example 5:
Let X1, · · · , Xn denote a random sample from the probability density function f(x; θ) = (θ+1)xθ, 0
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