Chi-Squared Distribution

A $\chi^2$ distribution is a Gamma Distribution with $\theta \equiv 2$ and $\alpha \equiv r/2$, where $r$ is the number of Degrees of Freedom. If $Y_i$ have Normal Independent distributions with Mean 0 and Variance 1, then

$$\chi^2\equiv \sum_{i=1}^r {Y_i}^2$$ (1)

is distributed as $\chi^2$ with $r$ Degrees of Freedom. If ${\chi_i}^2$ are independently distributed according to a $\chi^2$ distribution with $r_1$, $r_2$, ..., $r_k$ Degrees of Freedom, then

$$\sum_{j=1}^k {\chi_j}^2$$ (2)

is distributed according to $\chi^2$ with $r\equiv \sum_{j=1}^k r_j$ Degrees of Freedom.

$$P_r(x) = \cases{ {x^{r/2-1}e^{-x/2}\over\Gamma({\textstyle{... ...) 2^{r/2}} & for $0 \leq x < \infty$\cr 0 & for $x < 0$.\cr}$$ (3)

The cumulative distribution function is then

$$D_r(\chi^2) = \int_0^{\chi^2} {t^{r/2-1}e^{-t/2}\,dt\over \Gamma({\textstyle{1\over 2}}r) 2^{r/2}}$$

$$= {\gamma({\textstyle{1\over 2}}r, {\textstyle{1\over 2}}\chi^2)\ov... ...style{1\over 2}}r)} = P({\textstyle{1\over 2}}r, {\textstyle{1\over 2}}\chi^2),$$ (4)

where $P(a,z)$ is a Regularized Gamma Function. The Confidence Intervals can be found by finding the value of $x$ for which $D_r(x)$ equals a given value. The Moment-Generating Function of the $\chi^2$ distribution is

$$M(t) = (1-2t)^{-r/2}$$ (5)

$$R(t) \equiv \ln M(t) = - {\textstyle{1\over 2}}r \ln(1-2t)$$ (6)

$$R'(t) = {r\over 1-2t}$$ (7)

$$R''(t) = {2r\over (1-2t)^2},$$ (8)

so

$$\mu = R'(0) = r$$ (9)

$$\sigma^2 = R''(0) = 2r$$ (10)

$$\gamma_1 = 2\sqrt{2\over r}$$ (11)

$$\gamma_2 = {12\over r}.$$ (12)

The $n$th Moment about zero for a distribution with $r$ Degrees of Freedom is

$$m_n'=2^n{\Gamma(n+{\textstyle{1\over 2}}r)\over\Gamma({\textstyle{1\over 2}}r)} = r(r+2)\cdots(r+2n-2),$$ (13)

and the moments about the Mean are

$$\mu_2 = 2r$$ (14)

$$\mu_3 = 8r$$ (15)

$$\mu_4 = 12r(r+4).$$ (16)

The $n$th Cumulant is

$$\kappa_n=2^n\Gamma(n) ({\textstyle{1\over 2}}r)=2^{n-1}(n-1)! r.$$ (17)

The Moment-Generating Function is

$$M(t) = e^{rt/\sqrt{2r}}\left({1-{2t\over\sqrt{2r}}}\right)^{-r/2}$$

$$= \left[{e^{t\sqrt{2/r}}\left({1-\sqrt{2\over r}\,t}\right)}\right]^{-r/2}$$

$$= \left[{1-{t^2\over r}-{1\over 3}\left({2\over r}\right)^{3/2}t^3-\ldots}\right]^{-r/2}.$$ (18)

As $r\to\infty$,

$$\lim_{r\to\infty}M(t)=e^{t^2/2},$$ (19)

so for large $r$,

$$\sqrt{2\chi^2}=\sqrt{\sum_i {(x_i-\mu_i)^2\over{\sigma_i}^2}}$$ (20)

is approximately a Gaussian Distribution with Mean $\sqrt{2r}$ and Variance $\sigma^2 = 1$. Fisher showed that

$${\chi^2-r\over\sqrt{2r-1}}$$ (21)

is an improved estimate for moderate $r$. Wilson and Hilferty showed that

$$\left({\chi^2\over r}\right)^{1/3}$$ (22)

is a nearly Gaussian Distribution with Mean $\mu=1-2/(9r)$ and Variance $\sigma^2=2/(9r)$.

In a Gaussian Distribution,

$$P(x)\,dx={1\over\sigma\sqrt{2\pi}} e^{-(x-\mu)^2/2\sigma^2}\,dx,$$ (23)

let

$$z \equiv (x-\mu )^2/\sigma^2.$$ (24)

Then

$$dz = {2(x-\mu)\over \sigma^2} \,dx = {2\sqrt{z}\over \sigma} \,dx$$ (25)

so

$$dx={\sigma\over 2\sqrt{z}} dz.$$ (26)

But

$$P(z)\,dz=2 P(x)\,dx,$$ (27)

so

$$P(x)\,dx=2\,{1\over \sigma\sqrt{2\pi}} e^{-z/2}\,dz = {1\over \sigma\sqrt{\pi}} e^{-z/2}\,dz.$$ (28)

This is a $\chi^2$ distribution with $r = 1$, since

$$P(z)\,dz={z^{1/2-1}e^{-z/2}\over \Gamma({1\over 2})2^{1/2}}\,dz = {x^{-1/2}e^{-1/2}\over \sqrt{2\pi}}\,dz.$$ (29)

If $X_i$ are independent variates with a Normal Distribution having Means $\mu_i$ and Variances ${\sigma_i}^2$ for $i=1$, ..., $n$, then

$${\textstyle{1\over 2}}\chi^2\equiv \sum_{i=1}^n {(x_i-\mu_i)^2\over 2{\sigma_i}^2}$$ (30)

is a Gamma Distribution variate with $\alpha=n/2$,

$$P({\textstyle{1\over 2}}\chi^2)\,d({\textstyle{1\over 2}}\ch... ...{1\over 2}}\chi^2)^{(n/2)-1}\,d({\textstyle{1\over 2}}\chi^2).$$ (31)

The noncentral chi-squared distribution is given by

$$P(x)= 2^{-n/2}e^{-(\lambda+x)/2}x^{n/2-1}F({\textstyle{1\over 2}}n, {\textstyle{1\over 4}}\lambda x),$$ (32)

where

$$F(a,z)\equiv {{}_0F_1(; a; z)\over \Gamma(a)},$$ (33)

${}_0F_1$ is the Confluent Hypergeometric Limit Function and $\Gamma$ is the Gamma Function. The Mean, Variance, Skewness, and Kurtosis are

$$\mu = \lambda+n$$ (34)

$$\sigma^2 = 2(2\lambda+n)$$ (35)

$$\gamma_1 = {2\sqrt{2}\,(3\lambda+n)\over (2\lambda+n)^{3/2}}$$ (36)

$$\gamma_2 = {12(4\lambda+n)\over (2\lambda+n)^2}.$$ (37)

See also Chi Distribution, Snedecor's F-Distribution

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 940-943, 1972.

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 535, 1987.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Incomplete Gamma Function, Error Function, Chi-Square Probability Function, Cumulative Poisson Function.'' §6.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 209-214, 1992.

Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, pp. 115-116, 1992.