Chi-Bar-Squared Distribution

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A chi-bar-squared distribution is a mixture of chi-square distributions, mixed over their degrees of freedom. You’ll often find them when testing a hypothesis with an inequality [1].

More specifically, if you are testing a hypothesis where the alternate hypothesis has linear inequality constraints on means of normal distributions with known variances, a classic base for a hypothesis test is on -2 log Λ, where Λ is a likelihood-ratio statistics; 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.

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]


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

Properties of Chi-Bar-Squared Distribution

Expected value:


Consequently, the variance is

Practical Use

While chi-bar-square distributions frequently 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].


[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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