#Method of moments how to#

except probably for starting points for ML. Method of Moments Estimator Population moments: j E(Xj), the j-th moment of X.

Having a strong functional dependence between the variance and the shape parameter slows down convergence, as well.Īs a bottom line, I don't see much use for MM. Further suppose the probability distribution of X is completely determined by its moments, i.e. Secondly, we introduce a method of moments for estimating the unknown parameter, the description of the method is presented as follows. Suppose X is a random variable and that all of the moments. (By particular case, I mean that generally people speculate whether you have multiple local maxima or not, but in the case of $t$-distribution, existence of local maxima can be demonstrated analytically.) So you have to try a bunch of different starting values for it. In probability theory, the method of moments is a way of proving convergence in distribution by proving convergence of a sequence of moment sequences. Denote (X) our observed data and (m (X)) the vector of observed moments. ML does not have these theoretical problems, but in practice, it is found through numerical maximization, and in the particular case of $t$-distributions is prone to multiple local maxima. The method of simulated moments approach to estimating model parameters is to minimize a certain distance between observed moments and simulated moments with respect to the parameters that generate the simulated model.

#Method of moments how to#

High variability in the estimates, even in the situation when the estimator is defined and consistent, will also make MM less asymptotically efficient than ML. How to Use Method of Moments Like a Pro Obtain the moment generating function Derive the moments Obtain parameter estimators Calculate the estimates of the. is >10, then you can find yourself in a situation where the MM estimator is not even consistent. If you come to the data without knowing that your d.f. MoM is merely a proposal that to estimate a population moment or a function of population moments use the corresponding sample moments( or the function of. E.g., for the sum of the fourth moments to be asymptotically normal, you need to have finite 8th order moments for it to converge in the sense of the law of large numbers, you may need just another bit of a moment on top of the fourth, but at any rate, you have limitations on the degrees of freedom of your $t$ distribution. This methodology can be traced back to Pearson ( 1894) who used it to fit a simple mixture model. It works by finding values of the parameters that result in a match between the sample moments and the population moments (as implied by the model). For one thing, the ML estimate is some sort of robust mean, with tails downweighted, as opposed to MM-based brute force arithmetic mean.įirst of all, for the MM to work, you will need to have higher order moments to ensure that the sums necessary for the MM converge. The method of moments is a technique for estimating the parameters of a statistical model. Of course the parameter estimates will be different from ML and MM.