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In a parametric inference problem, the set of probability distributions that could have generated the data is put into correspondence with a set of real numbers (or real vectors). This set is called parameter space and its members are called parameters.


Suppose you observe a sample of independent draws from a normal distribution whose mean and variance are unknown and need to be estimated.

Consider the set of all normal distributions, that is, the set of all probability distribution that could have generated the sample. Each of these normal distributions can be put into correspondence with a $2$-dimensional vector[eq1]where mu is the mean of the distribution and sigma^2 is its variance. The set of all such vectors is called parameter space, while any one of these vectors is called a parameter.

Note that the variance cannot be negative, while there are no constraints on the mean. Therefore, the parameter space, denoted by $\Theta $, is[eq2]In other words, the parameter space $\Theta $ is the set of all couples [eq3] such that mu and sigma^2 are real numbers and sigma^2 is non-negative. Any element of $\Theta $ is a parameter.

More details

You can go to lecture entitled Statistical inference to learn more about parameters and parametric inference. You can also have a look at a related glossary entry: Parameter space.

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