Non-Central t and F-distributions

Suppose $X\sim N(\mu,1)$ and $W \sim \chi^2_{n}(0)$ and that $X$ and $W$ are independent. Then the random variable $T'$ defined by

$$T' = \frac{X}{\sqrt{W/n}}$$

has a non-central t-distribution with $n$ df and non-centrality parameter $\mu$.

No attempt will be made to derive the pdf of the non-central $t$. Clearly, when $\mu = 0, \ T'$ reduces to the central $t$ distribution.

Let $W_1 \sim \chi^2_{n_1} (\lambda )$ and $W_2 \sim \chi^2_{n_2} (0)$ be independent random variables. Then the random variable $F'$ defined by

$$F' = \frac{W_1/n_1}{W_2/n_2}$$

has a non-central $F$-distribution with non-centrality parameter $\lambda$. We write $F' \sim F_{n_1,n_2}(\lambda )$.

The $F'$ statistic has a non-central $F$ distribution with probability density function

$$g(x) = \sum_{r=0}^\infty e^{\frac{- \lambda}{2}} \times {{(... ...2}n_1+r-1}} \over {[1 + (n_1/n_2)x]^{\frac{1}{2}(n_1+n_2)+r}}}$$

where $n_1, n_2$ are the degrees of freedom, $\lambda$ is the non-centrality parameter defined by

$$\lambda = {{\sum_{i=1}^p n_i(\tau_i - \hat \tau_i)^2} \over {\sum_i \tau_i^2}}$$ (6.9)

If all means are equal, $\lambda =0$ and $g(x)$ is the pdf of a central $F$ variable.

The terms of the form $B(a,b)$ are beta functions. See the appendix for S-Plus functions.