Contents
How to equate sample moments to theoretical moments?
Equate the second sample moment about the mean M 2 ∗ = 1 n ∑ i = 1 n ( X i − X ¯) 2 to the second theoretical moment about the mean E [ ( X − μ) 2]. Continue equating sample moments about the mean M k ∗ with the corresponding theoretical moments about the mean E [ ( X − μ) k], k = 3, 4, … until you have as many equations as you have parameters.
Which is the method of moments estimator of μ?
We just need to put a hat (^) on the parameters to make it clear that they are estimators. Doing so, we get that the method of moments estimator of μ is: μ ^ M M = X ¯. (which we know, from our previous work, is unbiased). The method of moments estimator of σ 2 is: σ ^ M M 2 = 1 n ∑ i = 1 n ( X i − X ¯) 2.
How to equate the first sample moment to the second?
Equate the first sample moment about the origin M 1 = 1 n ∑ i = 1 n X i = X ¯ to the first theoretical moment E ( X). Equate the second sample moment about the mean M 2 ∗ = 1 n ∑ i = 1 n ( X i − X ¯) 2 to the second theoretical moment about the mean E [ ( X − μ) 2].
How to calculate the K T H sample moment?
M k ∗ = 1 n ∑ i = 1 n ( X i − X ¯) k is the k t h sample moment about the mean, for k = 1, 2, … Equate the first sample moment about the origin M 1 = 1 n ∑ i = 1 n X i = X ¯ to the first theoretical moment E ( X). Equate the second sample moment about the origin M 2 = 1 n ∑ i = 1 n X i 2 to the second theoretical moment E ( X 2).
How to calculate the method of moments for the mean?
Equating the first theoretical moment about the origin with the corresponding sample moment, we get: And, equating the second theoretical moment about the origin with the corresponding sample moment, we get: Now, the first equation tells us that the method of moments estimator for the mean μ is the sample mean:
How to calculate RST moment and theoretical second moment?
rst moment is E(X) = and the theoretical second moment is E(X2) = ( +1). Now we calculate the empirical counterparts: Second Step: Calculate the sample moments: sample rst moment = 200+300+350+450 4 = 325 sample second moment = 2002 +3002 +3502 +4502.