Noncentrality parameter
Encyclopedia
Noncentrality parameters are parameters of families of probability distribution
s which are related to other "central" families of distributions. If the noncentrality parameter of a distribution is zero, the distribution is identical to a distribution in the central family. For example, the Student's t-distribution is the central family of distributions for the Noncentral t-distribution
family.
Noncentrality parameters are used in the following distributions:
In general, noncentrality parameters occur in distributions that are transformations of a normal distribution. The "central" versions are derived from normal distributions that have a mean
of zero; the noncentral versions generalize to arbitrary means. For example, the standard (central) chi-squared distribution is the distribution of a sum of squared independent standard normal distributions, i.e. normal distributions with mean 0, variance
1. The noncentral chi-squared distribution generalizes this to normal distributions with arbitrary mean and variance.
Each of these distributions has a single noncentrality parameter. However, there are extended versions of these distributions which have two noncentrality parameters: the doubly noncentral beta distribution, the doubly noncentral F distribution and the doubly noncentral t distribution. These types of distributions occur for distributions that are defined as the quotient of two independent distributions. When both source distributions are central (either with a zero mean or a zero noncentrality parameter, depending on the type of distribution), the result is a central distribution. When one is noncentral, a (singly) noncentral distribution results, while if both are noncentral, the result is a doubly noncentral distribution. As an example, a t-distribution is defined (ignoring constant values) as the quotient of a normal distribution and the square root of an independent chi-squared distribution. Extending this definition to encompass a normal distribution with arbitrary mean produces a noncentral t-distribution
, while further extending it to allow a noncentral chi-squared distribution in the denominator while produces a doubly noncentral t-distribution.
Note also that there are some "noncentral distributions" that are not usually formulated in terms of a "noncentrality parameter": see noncentral hypergeometric distributions
, for example.
The noncentrality parameter of the t-distribution may be negative or positive while the noncentral parameters of the other three distributions must be greater than zero.
Probability distribution
In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
s which are related to other "central" families of distributions. If the noncentrality parameter of a distribution is zero, the distribution is identical to a distribution in the central family. For example, the Student's t-distribution is the central family of distributions for the Noncentral t-distribution
Noncentral t-distribution
In probability and statistics, the noncentral t-distribution generalizes Student's t-distribution using a noncentrality parameter. Like the central t-distribution, the noncentral t-distribution is primarily used in statistical inference, although it may also be used in robust modeling for data...
family.
Noncentrality parameters are used in the following distributions:
- Noncentral t-distributionNoncentral t-distributionIn probability and statistics, the noncentral t-distribution generalizes Student's t-distribution using a noncentrality parameter. Like the central t-distribution, the noncentral t-distribution is primarily used in statistical inference, although it may also be used in robust modeling for data...
- Noncentral chi-squared distribution
- Noncentral chi-distribution
- Noncentral F-distributionNoncentral F-distributionIn probability theory and statistics, the noncentral F-distribution is a continuous probability distribution that is a generalization of the F-distribution...
- Noncentral beta distributionNoncentral beta distributionIn probability theory and statistics, the noncentral beta distribution is a continuous probability distribution that is a generalization of the beta distribution.- Probability density function :...
In general, noncentrality parameters occur in distributions that are transformations of a normal distribution. The "central" versions are derived from normal distributions that have a mean
Mean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....
of zero; the noncentral versions generalize to arbitrary means. For example, the standard (central) chi-squared distribution is the distribution of a sum of squared independent standard normal distributions, i.e. normal distributions with mean 0, variance
Variance
In probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...
1. The noncentral chi-squared distribution generalizes this to normal distributions with arbitrary mean and variance.
Each of these distributions has a single noncentrality parameter. However, there are extended versions of these distributions which have two noncentrality parameters: the doubly noncentral beta distribution, the doubly noncentral F distribution and the doubly noncentral t distribution. These types of distributions occur for distributions that are defined as the quotient of two independent distributions. When both source distributions are central (either with a zero mean or a zero noncentrality parameter, depending on the type of distribution), the result is a central distribution. When one is noncentral, a (singly) noncentral distribution results, while if both are noncentral, the result is a doubly noncentral distribution. As an example, a t-distribution is defined (ignoring constant values) as the quotient of a normal distribution and the square root of an independent chi-squared distribution. Extending this definition to encompass a normal distribution with arbitrary mean produces a noncentral t-distribution
Noncentral t-distribution
In probability and statistics, the noncentral t-distribution generalizes Student's t-distribution using a noncentrality parameter. Like the central t-distribution, the noncentral t-distribution is primarily used in statistical inference, although it may also be used in robust modeling for data...
, while further extending it to allow a noncentral chi-squared distribution in the denominator while produces a doubly noncentral t-distribution.
Note also that there are some "noncentral distributions" that are not usually formulated in terms of a "noncentrality parameter": see noncentral hypergeometric distributions
Noncentral hypergeometric distributions
In statistics, the hypergeometric distribution is the discrete probability distribution generated by picking colored balls at random from an urn without replacement....
, for example.
The noncentrality parameter of the t-distribution may be negative or positive while the noncentral parameters of the other three distributions must be greater than zero.