Examples are typical Gaussian mixture models as well as many heavy-tailed distributions that result from compounding (i. Technically, this is true because
is the cumulant generating function of the sufficient statistic. The resulting distribution is simply the same as the above distribution for a scalar-valued random variable with each occurrence of the scalar x replaced by the vector
The dimensions k of the random variable need not match the dimension d of the parameter vector, nor (in the case of a curved exponential function) the dimension s of the natural parameter
{\displaystyle {\boldsymbol {\eta }}}
and sufficient statistic T(x). , Tn) as sufficient statistic.
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We only want the if part of the if and only if, which is covered in Section 6. a product of two “allowed” factors. . If F is absolutely continuous with a density more tips here
f
(
x
)
{\displaystyle f(x)}
with respect to a reference measure
d
x
{\displaystyle \,{\rm {d\,}}x\,}
(typically Lebesgue measure), one can write
d
F
(
x
)
=
f
(
x
)
x
{\displaystyle \,{\rm {d\,}}F(x)=f(x)~{\rm {d\,}}x\,}
. .