Non central chi distribution

ABSTRACT

In probability theory and statistics, the noncentral chi distribution is a noncentral generalization of the chi distribution. It is also known as the generalized Rayleigh distribution.

If squares of k independent standard normal random variables are added, it gives rise to central Chi-squared distribution with ‘k’ degrees of freedom. Instead, if squares of k independent normal random variables with non-zero means are added, it gives rise to non-central Chi-squared distribution. Non-central Chi-square distribution is related to Ricean distribution, whereas the central-chi Squared Distribution is related to Rayleigh Distribution.

The non-central Chi-squared distribution is a generalization of Chi Square Distribution. A non-central Chi squared distribution is defined by two parameters: 1) degrees of freedom and 2) non-centrality parameter.

Introduction

While the chi-square distribution characterizes how the chi-square test statistic is distributed when the null hypothesis is assumed to be true, the noncentral chi-square distribution instead shows how the chi-square test statistic is distributed when the alternative hypothesis is assumed to be true (i.e. when the null hypothesis is assumed to be false). As such it is useful in calculating the power of various chi-square tests.

Properties

Formulas

Method of Generating non-central Chi-squared random variable:

Questions

Applications

Conclusion

References

Steier, J. F. and Fouladi, R. T. (1997) Noncentrality interval estimation and the evaluation of statistical models

http://www.statpower.net/Steiger%20Biblio/Steiger&Fouladi97.PDF

Krishnamoorthy, K. (2006) Handbook of statistical distributions with applications. Chapman and Hall

https://www.academia.edu/41846183/Handbook_of_Statistical_Distributions_with_Applications

Benton, D. and Krishnamoorthy, K. (2003) Computing discrete mixtures of continuous

distributions: noncentral chisquare, noncentral t and the distribution of the square of the sample multiple correlation coefficient. Computational Statistics & Data Analysis 43. 249 – 267

https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.560.9968&rep=rep1&type=pdf