The Transformation $F(X)$

Theorem 5..2
(Probability Integral Transform) Let the random variable $X$ have cdf $F(x)$. If $F(x)$ is continuous, the random variable $Z$ produced by the transformation

$$Z = F(X)$$ (5.8)

has the uniform probability distribution over the interval $0\leq z \leq 1$.

Proof. See HC 4.1, p 161.

The above result is a useful ploy for inference using order statistics and the following diagram (Figure 5.5) illustrates the connections amongst the underlying distribution, the observed data and their order statistics, and the transform of these to a set of data ($Z_i$) whose distribution is known.

The top row is an expression for Theorem 5.2. The second row is the one-to-one mapping of samples from $F(x)$ to samples from the uniform distribution. The ordered variables $Y_1 \ldots Y_n$ are transformed to ordered $Z_i$ and properties about the original data, $X$, may be discerned from the $Z_{(1)} \ldots Z_{(n)}$.

Figure 5.4: Probability Integral Transform of order statistics.

Theorem 5..3
Consider $(Y_1, Y_2, \dots , Y_n)$, the vector of order statistics from a random sample of size $n$ from a population with a continuous cdf $F$. Then the joint pdf of the random variables

$$Z_i =F(Y_i), \quad\quad i=1,2,\dots ,n$$ (5.9)

is given by

$$\begin{array}{c} f_{\bf Z}(z_1,z_2,\dots ,z_n) = \end{arr... ...s < z_n < 1\\ 0 \ \ \ \mbox{elsewhere} \end{array} \right.$$ (5.10)

Proof. See HC 11.2, p 502.

Theorem 5..4
The marginal pdf of $Z_r=F(Y_r)$ is given by

$$f_{Z_r}(z_r)= \frac{n!}{(r-1)!(n-r)!}z_r^{r-1} (1-z_r)^{n-r} \ , 0 \, < \, z_r \, < \, 1.$$ (5.11)

It can be seen that $Z_r \sim \mbox{beta}(r,n-r+1)$ so its mean is

$$E(Z_r)=\frac{r}{n+1}$$ (5.12)

Note: The Beta density is given by

$$f(x;a,b) = {{1} \over{B(a,b)}} \times x^{a-1}(1-x)^{b-1} \tim... ...x{where } B(a,b) = {{\Gamma(a) \Gamma(b)} \over {\Gamma(a+b)}}$$

and you may need to revise the Gamma function.

Theorem 5..5
The joint pdf of $Z_r$ and $Z_s$ $(r<s)$ is given by

$$f_{Z_r,Z_s} (z_r,z_s) = \frac{n!}{(r-1)!(s-r-1)!(n- s)!}\, z^{r-1}_r(z_s-z_r)^{s-r-1}(1-z_s)^{n-s}$$

$$\quad\quad\quad \mbox{for} \ \ 0 < z_r < z_s < 1$$ (5.13)

Proof This is left as an exercise. You need only to notice $Z_r$ and $Z_s$ have uniform distributions and use (5.10).