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they exist in closed form, are usually well known and published in various
locations.
Another, more crude technique for estimating parameters of a distribution
is called the method of moments . This method takes the moments of the
distribution E[X k ]asfunctions oftheunknown parameters toformasystem
of equations. So, if the distribution has two unknown parameters, the
method of moments uses the first two moments of the distribution to form a
systemoftwoequations.Theobservedvaluesofthesamplemoments,which
are easily calculated, are then substituted into the equations and the system
of equations is solved.
For example, the gamma distribution has its shape and scale parameters. To
use the method of moments to estimate these parameters requires the first
two moments: m 1 = k×p and m 2 = p 2 ×k×(k+1). Given a sample, assumed
to be drawn from a gamma distribution, the first two moments are simply
the mean of the data and the mean of the square of the data. Those are
substituted into m 1 and m 2 and then you solve for p and k.
But, where do moment functions come from? The hard way, as explained
in the definition of expectation and variance, is to manually perform the
integral for the appropriate moment. However, most well-known
distributions have a so-called moment generating function . The moment
generating function is defined as E[e sX ], and produces a moment function
when it is differentiated a number of times. If the second moment is
required, the function is differentiated twice; for the fifth moment, five
times, and so on. The function is then evaluated at s=0 to produce the
moment function. This is where the functions used in the earlier gamma
distribution came from. In general, these generating functions, along with
otherformsofgeneratingfunctionsusefulforcalculatingthingslikerandom
sums of random variables, are available in tables and other sources, such as
Wikipedia and Wolfram Alpha.
The Delta Method
The method of moments , essentially, provides a tool for computing the
expectation of the power of a random variable X. These powers of X are
then used to find estimates of parameters of the distribution, though better
methods may also be available. The delta
method is a tool used to
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