 # 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 .

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 statistic; 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 . Wollan  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:

Where:

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

## 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 .

## 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.

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Stephanie Glen. "Chi-Bar-Squared Distribution" From StatisticsHowTo.com: Elementary Statistics for the rest of us! https://www.statisticshowto.com/chi-bar-squared-distribution/
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