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statistical model∗
CWoo†
2013-03-21 17:46:48
Let X = (X1 , . . . , Xn ) be a random vector with a given realization X(ω) =
(x1 , . . . , xn ), where ω is the outcome (of an observation or an experiment) in
the sample space Ω. A statistical model P based on X is a set of probability
distribution functions of X:
P = {FX }.
If it is known in advance that this family of distributions comes from a set of
continuous distributions, the statistical model P can be equivalently defined as
a set of probability density functions:
P = {fX }.
As an example, a coin is tossed n times and the results are observed. The
probability of landing a head during one toss is p. Assume that each toss is
independent of one another. If X = (X1 , . . . , Xn ) is defined to be the vector of
the n ordered outcomes, then a statistical model based on X can be a family of
Bernoulli distributions
P={
n
Y
pxi (1 − p)1−xi },
i=1
where Xi (ω) = xi and xi = 1 if ω is the outcome that the ith toss lands a head
and xi = 0 if ω is the outcome that the ith toss lands a tail.
Next, suppose X is the number of tosses where a head is observed, then a
statistical model based on X can be a family binomial distributions:
n x
P={
p (1 − p)n−x },
x
where X(ω) = x, where ω is the outcome that x heads (out of n tosses) are
observed.
∗ hStatisticalModeli created: h2013-03-21i by: hCWooi version: h36107i Privacy setting:
h1i hDefinitioni h62A01i
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A statistical model is usually parameterized by a function, called a parameterization
Θ → P given by θ 7→ Fθ so that P = {Fθ | θ ∈ Θ},
where Θ is called a parameter space. Θ is usually a subset of Rn . However, it
can also be a function space.
In the first part of the above example, the statistical model is parameterized
by
n
Y
p 7→
pxi (1 − p)1−xi .
i=1
If the parameterization is a one-to-one function, it is called an identifiable
parameterization and θ is called a parameter. The p in the above example is a
parameter.
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