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



