# method of moments with variance=$\sigma^2$

I am trying to estimate the value of a parameter by equating variance from a distribution to the sample variance... i.e. using method of moments estimation. Would it better to use the variance formula with $1/(n-1)$ in the denominator or $1/n$? Why is one better than the other?

thanks

I'll give an example: say I have been told Random variables from the 'geometric distribution' have a $\sum_{1}^n(X_i - \bar{X})^2$= 10. Then in order to estimate the parameter 'p' where $\frac{1-p}{p^2}$ is the variance of the geometric distribution.

do I in this circumstance use: $\frac{10}{n-1}=\frac{1-p}{p^2}$ and then given 'n' solve quadratic for p. or should I be using $\frac{10}{n}$ instead on the LHS.

• How do you define best? You need to specify that to get an answer. In a different context where you want to choose a method to estimate variance, if you make the criterion minimum variance unbiased you would use the estimate with n-1 in the denominator. If the criteria is minimum mean square error, dividing by n could be best. In the case of a normal distribution this would be the mle. – Michael R. Chernick Sep 29 '12 at 12:59
• I now specified a particular context. I've been reading up on variance in wikipedia and there is mention that unbiased variance may be appropriate in the situation i described...though i'm still unsure. – plantt Sep 29 '12 at 13:34

Regarding the method of moments and you question you can find the answer here in Wikipedia. When given a family of distributions where the distribution is determined by the value of one or more unknown parameters you can take the non central moments and given that they are a function of the unknown parameters solve k equations in k unknowns where the k equations equate the first k non central moments to the function of the parameters that they are equal to. Use the sample estimate for the moments and then solve for the parameters. If you have k parameters you have to have a case where the first k moments exist to solve. The sample moments are for a sample X$_1$, X$_2$,...,X$_n$ given by
jth moment: ∑X$_i$$^j$ /n where i runs from 1 to n and j can be 1, 2, ...,k.
• Say i do not know what the raw moments are... but I only know that $\sum_{1}^n(X_i - \bar{X})^2$= 10 (I know in practice raw moments are easier to calculate than the previous expression...but assuming we do not know the raw moments). In this circumstance I must choose between 1/n or 1/(n-1). Then I think, I ought to choose 1/(n-1) since it gives sample variance. – plantt Sep 30 '12 at 10:24