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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 † This text is available under the Creative Commons Attribution/Share-Alike License 3.0. You can reuse this document or portions thereof only if you do so under terms that are compatible with the CC-BY-SA license. 1 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. 2