Let be a random vector with a given realization , where is the outcome (of an observation or an experiment) in the sample space . A statistical model based on X is a set of probability distribution functions of X:
As an example, a coin is tossed times and the results are observed. The probability of landing a head during one toss is . Assume that each toss is independent of one another. If is defined to be the vector of the ordered outcomes, then a statistical model based on X can be a family of Bernoulli distributions
where and if is the outcome that the th toss lands a head and if is the outcome that the th toss lands a tail.
Next, suppose is the number of tosses where a head is observed, then a statistical model based on can be a family binomial distributions:
where , where is the outcome that heads (out of tosses) are observed.
A statistical model is usually parameterized by a function, called a parameterization
where is called a parameter space. is usually a subset of . However, it can also be a function space.
In the first part of the above example, the statistical model is parameterized by
If the parameterization is a one-to-one function, it is called an identifiable parameterization and is called a parameter. The in the above example is a parameter.
|Date of creation||2013-03-22 14:33:18|
|Last modified on||2013-03-22 14:33:18|
|Last modified by||CWoo (3771)|