Dixon Statistical Test: Deriving Density Function

[mpv55]

Recently, I was reading the Dixon (Annals Math. Stat. 22, (1951) 68-78) method for extreme (outliers) values. He considered that there are n ordered values (x1, x2, ...xn) of an analytical measurement. The values belong to a normal distribution. He defined two equations:
1. For Critical value

r₀₁=\frac{x_n-x_n-1}{x_n-x₁}


2. The density function for x₁, x_n-1, x_n is

\frac{n!}{(n-3)!}f(x₁)dx₁(\oint_x₁^x_n-1f(t)dt)^n-3 f(x_n-1)dx_n-1f(x_n) dx_n

I will appreciate if someone explains the derivation of the density function or site some reference which explains it.

Thanks

 

[Stephen Tashi]

Mixing in the "SUB" tag in between LaTex expressions seems to screw up the LaTex display. I can't make out what the density function is supposed to be. Can you revise it using the underscore ? (such as using the notation x_{n-1} for x with a subscript of n-1).