PMC:5374365 / 18349-19499 JSONTXT

{"target":"https://pubannotation.org/docs/sourcedb/PMC/sourceid/5374365","sourcedb":"PMC","sourceid":"5374365","source_url":"https://www.ncbi.nlm.nih.gov/pmc/5374365","text":"Appendix A. Proof of Theorem 1\nIn this Theorem, it is shown that a gpower-normal variable becomes normal or truncated normal when its corresponding gpower transformation is applied.\nProof.  Let x=gpowery;p, then when p≠0 y=gpower−1x=px+11/p−px+1−1/p2 and it is easy to see that ∂y∂x=∂gpower−1x∂x=px+11/p+px+1−1/p2(px+1) for p\u003e0, x\u003e−1/p and the derivative is positive, which means that gpowery;p is a monotonically increasing function. Now, X=gpowerY is considered as a function of a random variable Y∼GPN(μ,σ,p). The density for X is obtained as follows: fXx=∂∂xPgpowerY;p≤x=fYpx+11/p−px+1−1/p2px+11/p+px+1−1/p2(px+1)=1K2πσ2exp−12σ2x−μX)2,−1/p\u003cx\u003c∞ that is precisely the density of a TNμX,σ2,−1/p,∞ random variable. For p\u003c0, it can be proved analogously that fXx=1K2πσ2I−∞,−1/pxexp−12σ2x−μX2,\nFor p=0, gpowerx,0=lny+y2+1=arcsinhy. The inverse transformation is gpower−1x,0=sinhx=ex−e−x2 and ∂gpower−1x,0∂x=∂∂xex−e−x2=ex+e−x2.\nAgain, the derivative is positive ∀y, meaning that the function is monotonically increasing and the density of X becomes fXx=ddyPY≤ex−e−x2dydx=12πσ2exp−12σ2x−μX)2 that is precisely the density of a NμX,σ2 random variable.  ☐","divisions":[{"label":"title","span":{"begin":0,"end":30}},{"label":"p","span":{"begin":31,"end":181}},{"label":"label","span":{"begin":182,"end":189}},{"label":"p","span":{"begin":191,"end":792}},{"label":"p","span":{"begin":793,"end":925}}],"tracks":[]}