This web appendix outlines sketch of proofs in Sections 3 5 of the paper. In this appendix we will use the following notations: c i. j=1.

Transcription

1 Web Appedix: Supplemetay Mateials fo Two-fold Nested Desigs: Thei Aalysis ad oectio with Nopaametic ANOVA by Shu-Mi Liao ad Michael G. Akitas This web appedix outlies sketch of poofs i Sectios 3 5 of the pape. I this appedix we will use the followig otatios: U c V c cu c V c, as c, a c b c ca c b c, as c, whee U c ad V c ae two sequeces of adom vectos, while a c ad b c ae two sequeces of costat vectos. oof of Theoem 3. Defie W ij Uij ij ē ij + ij σ, ij k e ijk ē ij V ij U ij W ij Ū ic Uij,, Wic W ij, Ū σ Ūic, σ Wic, W Ū, V ic ic Ū, V W. ic W Note that [ ] Ū MS + σ i ē i Ū, ad [ ci W MSE + c ] i σ Wic. It ca be easily veified that, as mi ad, ij emai fixed, ci i ē i, σ Wic Ū, ad W.

2 ombiig the above we have that, as mi ad, ij emai fixed, V M MS MSE. 3 Hece, the asymptotic joit distibutio of MS ad MSE is the same as the asymptotic joit distibutio of Ū ad W. It ca be show that, ude omality, Uij ad W ij ae idepedet, ad ij Uij χ ij, W ij χ ij. σ Usig kow esults egadig the mea ad covaiace of quadatic foms cf. Theoem i Akitas ad Aod ad the facts that Eχ aaγ a+γ, V aχ aaγ a + 4γ, we obtai ovv ij + 4 ij ij EV ij σ ij + ij ij σ ij + κ i 3 ij Let θic i ci ij ij. The, fo each class i, as c σ i, ov V ic + 4θ ic E V ic + θ ici ici ici + 4θi i + κ i 3 + κ i 3 + θi i, ij. ij ij µ i, ad 4 ici ici ici ici + ici i i Σ i i + i i. Ude the assumptio that E e ijk 4+ɛ < fo some ɛ >, Lidebeg-Felle s theoem togethe with amé-wold s theoem yield ci V ic µ i d N, Σ i. Usig the idepedece amog V ic ad the assumptio o sample sizes ad subclass levels specified as the elatio 9 i the pape, oe ca be show that V µ d N, σ 4 λ i Σ i, whee µ σ + θ. 5

3 By the asymptotic equivalece betwee V ad M show i 3, we the have M µ d N, σ 4 λ i Σ i, as mi. Note that if s, + θ/σ, s M µ [MS + θmse]/σ which, by Slutsky s theoem, is asymptotically equivalet to F + θ. Thus, by the -method, as mi, F d + θ N, σ 4 λ i s Σ i s N, Σ s, whee Σ s is as defied i Theoem 3.. oof of oollay 3. It ca be easily veified that fo lage eough, the appoximate distibutio of the classical F -test ude H : ij, ad ude the omality assumptio is: F N, +, 6 whee meas appoximately distibuted. The elatio 6 is obviously ot equivalet to the asymptotic ull distibutio specified i Theoem 3. show as the elatio i the pape, uless ij fo all i ad j so that ici i, ici i i i, ad hece both of the asymptotic ull distibutio ad the elatio 6 would become F d N, +. 7 oof of Theoem 4. Defie ew quatities Uij, Ū, W to be as the coespodig quatities i but with σi eplacig σ, ad the ew quatity W ij to be as the coespodig quatity i but with i eplacig. Fially, let Ū ic, Wic, Vij, V ic, ad V be as defied i but usig the above ew quatities. It ca the be show that Ū, W ae elated to MS, MSE via [ ] Ū MS + σi i ē i Ū, [ W MSE ci + c ] i i σ i i W ic. 3

4 Usig, ad the fact that, as mi ad, ij emai fixed, ci c i i σi i W ic, 8 we have that, as mi ad, ij emai fixed, V M MS MSE. 9 Followig the same deivatio i the poof of Theoem 3., oe ca easily get ci V ic µ i d N, Σ i, whee µ i ad Σ i ae defied by E V ic + θ ici ici i + θi µ i, whee θ ici ci ij ij σi, ad ov V ic + 4θ ic ici i i + 4θi i + κ i 3 i + κ i 3 i i ici i ici i ici ici + ici i i i i Σ i. i i i + i By the idepedece amog V ic ad the assumptio o sample sizes ad subclass levels specified as the elatio 9 i the pape, it ca be show that V µ d N, σi 4 λ i Σ i, whee µ β + θ σ, β whee β ad θ σ ae as defied i Theoem 4.. Because V ad M ae asymptotically equivalet, as show i 9, we the have M µ d N, σi 4 λ i Σ i, as mi. Fially, by the -method with s, + θ /β, whee θ θ σ /β, it ca be easily veified that, as mi, F + θ d N, σi 4 λ i s Σ i s N, Σ s, whee Σ s is as defied i Theoem 4.. 4

5 oof of oollay 4. The fact that, whe the desig is balaced, the uweighted statistic F equals the classical F -statistis clea. Next, the asymptotic ull distibutio of oollay 4. show as the elatio 5 i the pape follows diectly fom Theoem 4.. Fially, the fact that the classical F -test pocedue is ot valid follows by compaig the elatio 7 above ad the elatio 5 i the pape. oof of Theoem 5. Defie V ij U ij, W ij, V ic Ū ic, Wic, ad V Ū, W, whee Note that U ij σ ij ij ē ij + ij σ ij, W ij σ ij ij ij Ū ic Uij, Ū e ijk ē ij, Wic W ij, k Ū MS + W MSE i σ ij ij ē ij S ij + Ūic, W Wic. Ū, i ij S ij. Ude the assumptios specified i Theoem 5., it ca be easily veified that, as mi, σ ij ij ē ij, Ū, i Sij, i ij S ij. ombiig the above we have that, as mi ad, ij emai fixed, V M MS MSE. Followig the same deivatio i the poof of Theoem 3., oe ca easily get the asymptotic distibutio of V ic as V ic µ i d N, Σ i, 5 ad

6 whee µ i ad Σ i ae E V ic j σ ij + j ijij j σ ij ai + θ i a i µ i, ad ov V ic j σ4 ij + 4 j ijσij ij j bi + 4θ i + b b 3i i σ 4 ij ij Σ i. + σij 4 κ ij 3 j ij By the idepedece amog V ic, the assumptios i Theoem 5. ad the asymptotic equivalece show i, we the have M a + θ a d N, λ i Σ i whee a ad θ ae as defied i the theoem above. Fially, usig the -method with s, + θ /a, θ θ /a, oe ca easily get the limitig distibutio of F as show i Theoem 5. ad complete the poof., oof of oollay 5. The poof follows diectly fom Theoem 5.. 6