Statistical parameter

A statistical parameter is a parameter that indexes a family of probability distributions. It can be regarded as a numerical characteristic of a population or a statistical model.^[1]

Definition

Among parameterized families of distributions are the normal distributions, the Poisson distributions, the binomial distributions, and the exponential family of distributions. The family of normal distributions has two parameters, the mean and the variance: if these are specified, the distribution is known exactly. The family of chi-squared distributions, on the other hand, has only one parameter, the number of degrees of freedom.

In statistical inference, parameters are sometimes taken to be unobservable, and in this case the statistician's task is to infer what they can about the parameter based on observations of random variables distributed according to the probability distribution in question, or, more concretely stated, based on a random sample taken from the population of interest. In other situations, parameters may be fixed by the nature of the sampling procedure used or the kind of statistical procedure being carried out (for example, the number of degrees of freedom in a Pearson's chi-squared test).

Even if a family of distributions is not specified, quantities such as the mean and variance can still be regarded as parameters of the distribution of the population from which a sample is drawn. Statistical procedures can still attempt to make inferences about such population parameters. Parameters of this type are given names appropriate to their roles, including:

• location parameter
• dispersion parameter or scale parameter
• shape parameter

Where a probability distribution has a domain over a set of objects that are themselves probability distributions, the term concentration parameter is used for quantities that index how variable the outcomes would be.

Quantities such as regression coefficients, are statistical parameters in the above sense, since they index the family of conditional probability distributions that describe how the dependent variables are related to the independent variables.

Examples

A parameter is to a population as a statistic is to a sample. At a particular time, there may be some parameter for the percentage of all voters in a whole country who prefer a particular electoral candidate. But it is impractical to ask every voter before an election occurs what their candidate preferences are, so a sample of voters will be polled, and a statistic, the percentage of the polled voters who preferred each candidate, will be counted. The statistic is then used to make inferences about the parameter, the preferences of all voters. Similarly, in some forms of testing of manufactured products, rather than destructively testing all products, only a sample of products are tested, to gather statistics supporting an inference that all the products meet product design parameters.

See also

• Precision (statistics), another parameter not specific to any one distribution
• Parametrization (i.e., coordinate system)
• Parsimony (with regards to the trade-off of many or few parameters in data fitting)

References