Editing Point estimation (section)

=== Method of moments (MOM) ===
The [[Method of moments (statistics)|method of moments]] was introduced by K. Pearson and P. Chebyshev in 1887, and it is one of the oldest methods of estimation. This method is based on [[law of large numbers]], which uses all the known facts about a population and apply those facts to a sample of the population by deriving equations that relate the population moments to the unknown parameters. We can then solve with the sample mean of the population moments.<ref>{{Cite book|title=The Concise Encyclopedia of Statistics|publisher=Dodge, Y.|year=2008|location=Springer}}</ref> However, due to the simplicity, this method is not always accurate and can be biased easily.

Let (X<sub>1</sub>, X<sub>2</sub>,…X<sub>n</sub>) be a random sample from a population having p.d.f. (or p.m.f) f(x,θ), θ = (θ<sub>1</sub>, θ<sub>2</sub>, …, θ<sub>k</sub>). The objective is to estimate the parameters θ<sub>1</sub>, θ<sub>2</sub>, ..., θ<sub>k</sub>. Further, let the first k population moments about zero exist as explicit function of θ, i.e. μ<sub>r</sub> = μ<sub>r</sub>(θ<sub>1</sub>, θ<sub>2</sub>,…, θ<sub>k</sub>), r = 1, 2, …, k. In the method of moments, we equate k sample moments with the corresponding population moments. Generally, the first k moments are taken because the errors due to sampling increase with the order of the moment. Thus, we get k equations μ<sub>r</sub>(θ<sub>1</sub>, θ<sub>2</sub>,…, θ<sub>k</sub>) = m<sub>r</sub>, r = 1, 2, …, k. Solving these equations we get the method of moment estimators (or estimates) as

m<sub>r</sub> = 1/n ΣX<sub>i</sub><sup>r</sup>.<ref name=":1" /> See also [[generalized method of moments]].