Chi Bar Squared Distribution

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What is a chi bar squared distribution?

A chi bar squared distribution is a mixture of 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 of Chi Bar Squared Distribution

Expected value:

and:

Consequently, the variance is:

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]

chi-bar squared survival function

Where:

  • Yn is a chi bar squared distributed random variable associated with Pn.
  • pn = pn (j).

Practical Use

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

[1] Dykstra R 1991 Asymptotic normality for chi-bar-square distributions Can. J. Stat. 19 297–306
[2] Barlow, R. E., Bartholomew, D. J., Bremner, J. M., and Brunk, H. D. (1972). Statistical Inference under Order Restrictions, New York: Wiley.
[3] Wollan. P.C. (1985). Estimation and hypothesis testing under inequality constraints. Ph.D. Thesis. Universiy of Iowa.
[4] Johnson, Kotz, and Balakrishnan, (1994), Continuous Univariate Distributions, Volumes I and II, 2nd. Ed., John Wiley and Sons.

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