Drexel dragonThe Math ForumDonate to the Math Forum

Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Math Forum » Discussions » sci.math.* » sci.stat.math.independent

Topic: The Wilson - Hilferty (1931) upside-down
Replies: 1   Last Post: Jun 16, 2013 6:17 AM

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
Luis A. Afonso

Posts: 4,743
From: LIsbon (Portugal)
Registered: 2/16/05
The Wilson - Hilferty (1931) upside-down
Posted: Jun 15, 2013 5:40 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

The Wilson - Hilferty (1931) upside-down

The direct way provides a normal standard r.v. z from a chi-square y with r degrees of freedom:
z = [(y/r)^(1/3) - (1- 2/(9*r)]/sqrt(2/(9*r))
z*sqrt(2/(9*r)) + (1-2/(9*r)) = (y/r)^(1/3)
With s=2/(9*r) we get y= r*[z*sqrt(s) + (1-s)]^3 ___(1)
y being a chi-square r.v. obtained from a z~N(0,1).

Numerical Example
With z= 1, r= 5
s=0.0444. . . y= 7.9338 (from Kalkulator)
Then p(chi,5df < y) = 0.8401
In fact p(z< 1) = 0.8413 is the exact value.

A direct application, which avoids the simulation of the samples, is that we can obtain the results from difference of normal sample means, same variance, by the C.I. semi-amplitude ssdX ,ssdY, the sum of squared deviations,
+/- t(nX+ nY-2) * k * sqrt(ssdX+ ssdY)
Here k= sqrt[1/(nX+ nY- 2)*(1/nX+ 1/nY)]
Because the quantity ssdX+ ssdY have a Chi-square distribution, nX+ nY- 2 df, the result, although approximate, do follow demands only one step.

Luis A. Afonso

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© The Math Forum at NCTM 1994-2016. All Rights Reserved.