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Topic: Prediction interval in regresion through origin
Replies: 2   Last Post: May 17, 2012 7:28 PM

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 Gaj Vidmar Posts: 21 Registered: 12/13/04
Prediction interval in regresion through origin
Posted: May 16, 2012 9:19 AM

Hello,

I should know this (and I've got an answer through studying and via
simulation), but a confirmation from one of the regular posters who are true
[mathematical] statisticians would really help me a lot!

So, what is the "shape" of prediction interval in simple linear regresion
through origin?

It's introductory stuff that with intercept, the confidence interval (CI)
(i.e., for mean response) is "sandglass" ("hourglass") shaped (narowest at
{mean x, mean y}), and so is the prediction interval (PI) (just often
"imperceptibly" so, i.e., seemingly parallel to the regression line, because
it is much wider and much less "concave" than the CI), right? And withouth
intercept (i.e., in regression through origin) CI is "traingle" ("fan")
shaped, right?

Now, my answer is that PI without intercept is "half-hour-glass" shaped,
i.e., like the right half of the usual PI (and also seemingly parallel to
the regression line because of being only slightly concave), but I'm not
sure any more (when you urgently need an answer like, that the more you
study and simulate the more tired and unsure you get ...)

And while we're at it, allow me another question. Suppose one wants to
estimate the simplest constrained regression model, namely linear least
squares (LSQ) just that the intercept has to be positive. How bad/good (or
perhaps the optimal, at least in some sense?) a strategy is to keep the
usual LSQ if the estimated intercept turns out to be >0 and do regression
through origin if the a_hat from LSQ <0?

Furthermore, in the seemingly trivial case of a_hat from LSQ being exactly
zero, which "inference apparatus" to use: that from regression through
origin or the usual one, or they will be equivalent anyway?? My guess is the
latter, but again, tired as I am I'm not sure.

Gaj

---
Assist Prof Gaj Vidmar, PhD
University Rehabilitation Institute, Republic of Slovenia &
Univ. of Ljubljana, Fac. of Medicine, Inst. for Biostatistics and Medical
Informatics

Date Subject Author
5/16/12 Gaj Vidmar
5/16/12 Gaj Vidmar
5/17/12 Ray Koopman