Probability Distributions > Chi-bar-squared distribution
What is a chi-bar-squared distribution?
A chi bar squared distribution is two or more chi-square distributions, mixed over their degrees of freedom. You’ll often find chi-bar-squared distributions when testing a hypothesis with an inequality [1]. More specifically, when testing a hypothesis where the alternate hypothesis has linear inequality constraints on means of normal distributions with known variances. Linear equality constraints are restrictions that can be expressed as a system of linear equations. As a simple example, the constraint that the means of population 1 and population 2 are equal can be expressed as the equation: μ1 = μ2.
A classic base for a hypothesis test is on -2 log Λ, where Λ is a likelihood-ratio statistic. The likelihood-ratio statistic is a test statistic that is used to compare the likelihood of the data under the null hypothesis to the likelihood of the data under the alternate hypothesis; the test distribution will usually follow a chi-bar-squared distribution. A test on -2 log A will have an asymptotic normal distribution as the number of populations increases to infinity [2]. Wollan [3] showed that large-sample likelihood-ratio tests for hypothesis involving inequality constraints will result in chi-bar-squared distributions, given appropriate regularity conditions.
Properties
Expected value: 
and 
Variance 
Survival function:
If {Pn} is a sequence of probability distributions with the following conditions:
- The distributions have support in the nonnegative integers,
- The mean μ and variance σ2 are finite and nonzero,
Then the survival function is [4]
Where:
- Yn is a chi bar squared distributed random variable associated with Pn.
- pn = pn (j).
Practical use of chi-bar-squared distributions
While chi-bar-squared distributions often occur, the weights are often intractable and challenging to calculate. Despite attempts from many authors to solve this issue, it is generally accepted that intractability is a “pervasive problem.” However, if a normal approximation is justified, you don’t need to know every chi-bar-square coefficient to perform analysis; you only need to know the mean and variance for the chi-bar-squared distribution [1].
References
- Dykstra R 1991 Asymptotic normality for chi-bar-square distributions Can. J. Stat. 19 297–306
- Barlow, R. E., Bartholomew, D. J., Bremner, J. M., and Brunk, H. D. (1972). Statistical Inference under Order Restrictions, New York: Wiley.
- Wollan. P.C. (1985). Estimation and hypothesis testing under inequality constraints. Ph.D. Thesis. Universiy of Iowa.
- Johnson, Kotz, and Balakrishnan, (1994), Continuous Univariate Distributions, Volumes I and II, 2nd. Ed., John Wiley and Sons.