# st: RE: trinomial confidence interval

 From "FEIVESON, ALAN H. (AL) (JSC-SK) (NASA)" <[email protected]> To "'[email protected]'" <[email protected]> Subject st: RE: trinomial confidence interval Date Fri, 10 Dec 2004 13:23:21 -0600

```One quick and dirty approach is to use the bivariate normal approximation to
the joint distribution of two of the proportion estimates (the third is
determined form the other two).

Suppose you have p1_hat = y1bar, p2_hat=y2bar based on n observations, where
(y1, y2, y3) is trinomial with cell probabilities p1, p2 and 1-p1-p2
(y1+y2+y3=1). Then

E(p1_hat)=p1
E(p2_hat)=p2
V(p1_hat)=p1*(1-p1)/n
V(p2_hat)=p2*(1-p2)/n
Cov(p1_hat, p2_hat) = -p1*p2/n

Then use the confidence ellipsoid for this bivariate normal as an
approximate confidence region for (p1, p2).

Clearly, this only makes sense if p1, p2 and 1-p1-p2 are not "close" to 0 or
1 in terms of the standard errors of p1_hat, p2_hat and p1_hat+p2_hat.

A Bayesian approach is another alternative.

Al Feiveson

-----Original Message-----
From: [email protected]
[mailto:[email protected]]On Behalf Of Paschalides,C
Sent: Friday, December 10, 2004 6:52 AM
To: [email protected]
Subject: st: trinomial confidence interval

Dear Statalisters: I have a trinomial distributions with possible events A,
B and C. I have obtained obtained results from a number of samples (or if
this complicates the situation much more, just consider one sample); I have
the marginal confidence interval for the proportion for each of the events
but I would like a joint confidence interval, say 95%, for the actual
proportions and plot the interval.
The method that I am using at the moment is trial and error (testing each
combination whether it lies in the interval) but this takes days, it is not
scientific and I am not really sure for the validity of the result.
I apologize for the non-Stata question. Could you please help? Thanks in

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```