Supplementary Material for: Classical Testing in Functional Linear Models

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1 To appear i the Joural of Noparametri Statistis Vol. 00, No. 00, Moth 20XX, 1 16 Supplemetary Material for: Classial Testig i utioal Liear Models Deha Kog a Aa-Maria Staiu b ad Arab Maity b a Departmet of Biostatistis, Uiversity of North Carolia, Chapel Hill, NC, 27599, U.S.A. ; b Departmet of Statistis, North Carolia State Uiversity, Raleigh, NC 27695, U.S.A. ; (v4.0 released Jue 2015) The Supplemet Material otais five setios. Setio 1 reviews the estimatio of the futioal priipal ompoet sores. Setio 2 presets importat results used i the proof of the mai theorems. Setio 3 presets the proofs of all the theorems. Setio 4 disusses the mathematial expressios of the testig proedures i the otext of partial futioal liear model. Setio 5 otais additioal simulatio results for the settig whe the futioal ovariate is observed sparsely, whe we ompare with bootstrap testig, the miimax adaptive testig ad the restrited likelihood ratio testig methods, whe the latet proess is ot Gaussia ad whe X i (t) is geerated from a large umber of basis futios. 1. Estimatio of the futioal priipal ompoet sores ad seletio of umber of futioal priipal ompoets I this setio we fous o the estimatio of the futioal priipal ompoet sores; the methods have bee previously disussed i the literature; see for example Yao, Müller, ad Wag (2005), Zhu, Yao, ad Zhag (2014). We summarize them here for ompleteess of our proposed methodology. To fix ideas, reall that the true futios X i ( ) are observed at a grid of poits t i1,..., t imi } ad with measuremet error; W i1,..., W imi } are the orrespodig observatios assoiated with X i ( ). Assume more geerally, that the uderlyig urve X i ( ) has mea futio η( ) ad ovariae futio K(, ) ad furthermore that X i a be expaded as X i (t) = η(t) + j 1 ξ ijφ j (t). The futioal priipal ompoet sores ξ ij are uorrelated with mea 0 ad variae λ j, ad represet the mai objet of iterest i this setio. There are differet approahes to estimate the sores ξ ij s; we briefly summarize them separately. Cosider first a dese samplig desig. Oe simple estimatio approah relies o the ideas of first smoothig eah separate urve usig loal liear kerel smoothig metioed ad the treatig the estimated urves, Xi ( ) as the true urves, see Zhu et al. (2014) for detail. The mea ad the ovariae futio a be estimated osistetly usig the sample mea ad the sample ovariae, respetively, of the estimated urves X i. Eigeaalysis of the estimated ovariae futio, provides o- Correspodig author. kogdehastat@gmail.om

2 sistet estimates of the eigevalues ad eigefutios, λ j, φ j ( )} j. The the futioal priipal ompoet sores a be estimated usig umerial itegratio, ξ ij = mi k=2 X i (t ik ) η(t ik )} φ j (t ik )(t ik t i(k 1) ), where η is the estimated mea futio. The umber of futioal priipal ompoet sores is estimated based o the umulative peretage of explaied variae; typial threshold values are 90%, 95%, 99%. Next, osider the ase of a sparse samplig desig. The ommo tehique of estimatig ξ ij is through oditioal expetatio method proposed by Yao et al. (2005). The mea futio is estimated usig loal liear regressio of the observed data W ij : j} i. The the ovariae futio is estimated i two steps. irst a twodimesioal satterplot smoothig is used o W ij W ik : j k} i to obtai ϖ(s, t), a estimate of EX i (s)x i (t)}; ad the the estimated ovariae futio is give by K(s, t) = ϖ(s, t) η(s) η(t). urthermore the error variae a be estimated usig the differee betwee a poitwise variae estimate, obtaied via loal liear smoothig of Wij 2 : j} i, say Ĝ(t, t) ad K(t, t). Yao et al. (2005) disuss a simple estimate of the oise variae as σ e 2 = Ĝ(t, t) K(t, t)}dt. Although these loal smoothig-based estimators have bee proved to have good theoretial properties, i pratie, splie-based estimators of the mea ad ovariae has also bee reported very suessful. Eigeaalysis of K provides osistet estimates of the eigevalues ad eigefutios λ j, φ j ( )} j. The truatio s of the eigevalues/eigefutios a be seleted via iformatio riteria, suh as Akaike iformatio riterio. However the umulative peretage of the explaied variae riterio remais the most popular oe. The ideas behid estimatig the futioal priipal ompoet sores rely o represetig the model for the observed data usig a mixed effets model formulatio, oditioal o the truatio level. It follows that the futioal priipal ompoet sores a be estimated usig oditioal expetatio. Assumig Gaussia assumptio, the oditioal expetatio has simple aalytial expressio, ξ ij = λ j φ i Σ 1 W i (W i η i ), where φ ij is the m i vetor with elemets φ j (t ij ), Σ Wi is the m i m i matrix with the (j, k) elemet equal to K(t ij, t ik )+σei(j 2 = k), ad η i is the m i dimesioal vetor with the jth elemet equal to η(t ij ). Therefore the estimates of the sores ξ ij a be obtaied from expressio of ξ ij, by substitutig the estimates of η( ), λ j, φ j ( ), ad σe. 2 This method is also appliable whe the samplig desig is dese, ad we used it i the simulatios ad data appliatio. 2. Asymptotis for dese futioal ovariate We begi with some otatios. Defie a as the l 2 orm of a vetor a. or a proess X(t), deote X( ) L2 = T X(t)}2 dt} 1/2. We also deote K K = [ T K(t 2 1, t 2 ) K(t 1, t 2 )} 2 dt 1 dt 2 ] 1/2, where K is the ovariae futio of the proess X( ) ad K is a estimate of K usig the smoothed trajetories ˆX i ( ). Suppose β a (t) = k=1 β jaφ j (t), where β ja = β a (t)φ j (t)dt. The followig lemma is take from Lemma 1 of Zhu, Yao, ad Zhag (2014). Lemma 2.1 Uder Coditios (C1)-(C3), we have K K = O P ( 1/2 ) φ j ( ) φ j ( ) L2 8 1/2 δj 1 K K. E( X i ( ) X i ( ) 2 L 2 ) = O( 1 ). 2

3 Similar to Li, Wag, ad Carroll (2010), without loss of geerality, we assume the X i ( ) : 1 i } have bee etered to have zero poitwise mea futio. Oe a estimate the eigefutios by a eige-deompositio of K, ad the futioal priipal ompoets sores by ξ ij = Xi (t) φ j (t)dt. Deote ξ ij = X i (t) φ j (t)dt the aalogue of ξ ij usig the true trajetories X i (t). Lemma 2.2 Let ij = ξ ij ξ ij. Assume the oditios (C1)-(C3) hold true, we have E( s 2 ij ) = O(s 1 ). Proof. Sie ij = ξ ij ξ ij ad φ j ( ) : j 1} are a set of orthoormal basis, we have 2 ij = [ X i (t) X i (t)} φ j (t)dt] 2 [ X i (t) X i (t)} 2 dt] φ 2 j(t)dt} = X i (t) X i (t)} 2 dt. By Lemma 1, E( s 2 ij ) s E( X i ( ) X i ( ) 2 L 2 ) = O(s 1 ). Lemma 2.3 Assume the oditios (A),(B1)(B2) ad (C1)-(C3) are met. The k=1 ( ξ ijξ ik ξ ij ξik ) 2 = O P (s 2 δs 2 ) ad s β ja (ˆξ ij ξ ij )} 2 = O P (s 2 δs 2 ). Proof. Notie that (ξ ijξ ik ξ ij ξik ) = (ξ ij ξ ij )ξ ik + (ξ ik ξ ik )ξ ij (ξ ik ξ ik )(ξ ij ξ ij ), we have 3[ s s k=1 s s + (ξ ij ξ ik ξ ij ξik )} 2 k=1 s s k=1 = 3(J 1 + J 2 + J 3 ). (ξ ij ξ ij )ξ ik } 2 + s s k=1 (ξ ik ξ ik )(ξ ij ξ ij )} 2 ] (ξ ik ξ ik )ξ ij } 2 3

4 or the first term J 1, we have J 1 = 2 s s ( [ k=1 s s [ k=1 s s 2 = 2( k=1 s k=1 = I 1 + I 2. X i (t) φ j (t) φ j (t)}dt + ij ]ξ ik ) 2 X i (t)ξ ik φ j (t) φ j (t)}dt] 2 + ( X i ( )ξ ik 2 L 2 φ j ( ) φ j ( ) 2 L X i ( )ξ ik 2 L 2 ) s φ j ( ) φ j ( ) 2 L 2 + 2( ij ξ ik ) 2 } s s ( k=1 s 2 ij)( ξik 2 ) 2 ij s k=1 By Lemma 1, oe has s φ j ( ) φ j ( ) 2 L 2 8s δs 2 ˆK K 2 = O P (s δs 2 1 ). Also otie that X i ( )ξ ik 2 L 2 = Xi 2 (t)ξik 2 dt = Xi 2 (t) X i (s)φ k (s)ds} 2 dt Xi 2 (t)dt X i ( ) 2 L 2 φ k ( ) 2 L 2 X i ( ) 4 L 2. (1) ξ 2 ik ) By oditio (C1) ad the ubii s Theorem, E( X i ( ) 4 L 2 ) E X i (t)} 4 dt <. Cosequetly, E s k=1 X i( )ξ ik 2 L 2 } = O(s ). By Markov s iequality, k=1 X i( )ξ ik 2 L 2 = O P (s ). Combiig with the previous result, we obtai I 1 = O P (s 2 δs 2 ). or I 2, oe has 2 ij = O P (s ) by Lemma 2.2 ad Markov s iequality. As E( k=1 ξik 2 ) = s k=1 λ k k=1 λ k = O(), by Markov s iequality, we obtai k=1 ξik 2 = O P (), whih idiates I 2 = O P (s ) = o P (s 2 δs 2 ). Thus J 1 = O P (s 2 δs 2 ). Similarly, oe has J 2 = O P (s 2 δs 2 ). or the third term, we have J 3 = = 4 s s k=1 s s ( k=1 s s (ξ ik ξ ik )(ξ ij ξ ij )} 2 [ k=1 s s k=1 s s k=1 X i (t) φ j (t) φ j (t)}dt + ij ][ = I 3 + I 4 + I 5 + I 6. X i (t) φ j (t) φ j (t)}dt X i (t) φ j (t) φ j (t)}dt ik } 2 X i (t) φ k (t) φ k (t)}dt ij } X i (t) φ k (s) φ k (s)}ds + ik ]) 2 X i (s) φ k (s) φ k (s)}ds} 2 s s k=1 ik ij } 2 4

5 or I 3, oe has I 3 4 s s ( k=1 4 X i ( ) 4 L 2 X i ( ) 2 L 2 φ k ( ) φ k ( ) L2 φ j ( ) φ j ( ) L2 ) 2 (2) s φ j ( ) φ j ( ) 2 L 2 s k=1 φ k ( ) φ k ( ) 2 L 2. By Lemma 1, we have s φ j ( ) φ j ( ) 2 L 2 = s k=1 φ k ( ) φ k ( ) 2 L 2 = O P (s δs 2 1 ). By oditio (C1), E( X i ( ) 4 L 2 ) < whih idiates X i( ) 4 L 2 = O P () by Markov s iequality. As a result, I 3 = O P (s 2 δs 4 ). By oditio (A), oe obtais I 3 = O P (s 2 δs 4 ) = o P (s 2 δs 2 ). or I 4, oe has I ( s s ( k=1 s s k=1 X i ( ) L2 φ j ( ) φ j ( ) L2 ik ) 2 (3) ( X i ( ) L2 φ j ( ) φ j ( ) L2 ) 2 }( 2 ik ) X i ( ) 2 L 2 )( s φ j ( ) φ j ( ) 2 L 2 )( s k=1 2 ik ). By Lemma 2.2, oe has k=1 2 ik = O P (s ) by Markov s iequality. Usig similar argumets whe we otrol the terms i I 3, we a get X i( ) 2 L 2 = O P () ad φ j ( ) φ j ( ) 2 L 2 = O P (s δs 2 1 ). Thus, I 4 = O P (s 2 δs 2 ) = o P (s 2 δs 2 ). Usig similar tehique, we obtai I 5 = O P (s 2 δs 2 ) = o P (s 2 δs 2 ). or I 6, we a easily see I 6 4( k=1 2 ik )( 2 ij ) = O P (s 2 ) = o P (s 2 δs 2 ) by Lemma 2.2. Combiig with the previous results, J 3 = o P (s 2 δs 2 ). Cosequetly, k=1 (ξ ijξ ik ξ ij ξik ) 2 = O P (s 2 δs 2 ). Now, we otrol the order of s β ja (ˆξ ij ξ ij )} 2. We have = 2 2 [ s s s [ β ja (ˆξ ij ξ ij )} 2 X i ( ) 2 L 2 = J 4 + J 5. X i (t)β ja φ j (t) φ j (t)}dt + β ja ij ] 2 X i (t)β ja φ j (t) φ j (t)}dt] s s β ja ij } 2 β ja φ j ( ) φ j ( ) L2 } β a ( ) 2 L 2 s s 2 ij (4) By Markov s iequality, we obtai X i( ) 2 L 2 = O P () as E( X i( ) 2 L 2 ) = O(). By Lemma 1, s β ja φ j ( ) φ j ( ) L2 s β ja δj 1 ˆK K 5

6 8 β a (t) s δs 1 ˆK K = O P (s δs 1 1/2 ). Combiig the above results, J 4 = O P (s 2 δs 2 ). or J 5, by Lemma 1, oe has J 5 = O P (s 2 ). Thus, s β ja (ˆξ ij ξ ij )} 2 = O P (s 2 δs 2 ). 3. Proof of the Theorems Without loss of geerality, we assume the oise ɛ follows a stadard ormal distributio. If var(ɛ) = σ 2, we a simply trasform Y = Y/σ, ad the proof follows for Y istead of Y. We also assume that the urves X i ( ) has bee etered. or the distributios disussed i the theorem, they are oditioal o the origial urve X i ( ) ad the observed data poits W i1,..., W imi } for i = 1,.... We deote by A ad A 2, the robeius orm ad the spetral orm of a matrix A Proof of Theorem 1 The proof of Theorem 1 does ot rely o ay of the Lemmas itrodued i the previous setio. Uder the ull, we have Y = ɛ, where ɛ follows a stadard ormal distributio. Sie P B P 1 is idempotet with rak s, Y (P B P 1)Y follows a hi-square distributio with degrees of freedom s. Thus, we a write Y (P B P 1)Y = s A 2 1j, where A 1j s are i.i.d. stadard ormal radom variables. By the etral limit theorem (CLT), we have ( s A 2 1j s )/ 2s d N(0, 1) whe s. Thus Y (P B P 1)Y = O P (s ). Notie that σ 2 = Y (I P 1 )Y/, ad I P 1 is idempotet with rak 1, we a write Y (I P 1 )Y = 1 A2 2j, where A 2j s are i.i.d. stadard ormal radom variables. By the CLT, oe has Y (I P 1 )Y ( 1)}/ 2( 1) d N(0, 1), whih idiates that Y (I P 1 )Y = 1 + O P ( 1/2 ). Thus, σ 2 = 1 + O P ( 1/2 ). Combiig the above results, oe has (T S s )/ 2s = s A 2 1j (1+O P ( 1/2 )) s }/ 2s = ( s A 2 1j s )/ 2s + O P (s 1/2 1/2 ). Uder the assumptio that s = o(), by Slutsky s theorem, oe has (T S s )/ 2s d N(0, 1). or T, sie I P ˆB follows a hi-square distributio with degrees of freedom s 1, oe has Y (I P ˆB)Y = s 1 A 2 3j, where A 3j s are i.i.d. stadard ormal radom variables. By CLT, we have Y (I P ˆB)Y = s 1 + O P ( 1/2 ). Notie that T = Y (P B P 1)Y/s }/Y (I P ˆB)Y/( s 1)}, we have (s T s )/ 2s = s A 2 1j (1 + O P ( 1/2 )) s }/ 2s = ( s A 2 1j s )/ 2s + O P (s 1/2 1/2 ). Uder the assumptio that s = o(), by Slutsky s theorem, oe has (s T s )/ 2s d N(0, 1) Proof of Theorem 2 Let ν = (ν 1,..., ν ), where ν i = j=s ξ +1 ijβ ja. Defie µ i,α = EY i X i ( )} = α + Xi (t)β a (t)dt, ad deote µ i = X i (t)β a (t)dt. We defie µ α = (µ 1,α,..., µ,α ) ad µ = (µ 1,..., µ ). It is easy to see that µ α = µ + α 1, where 1 is a dimesioal olum vetor with every elemet 1. Let β a (t) = β jaφ j (t), ad deote β a,s = (β 1a,..., β sa), whih is a s dimesioal vetor, ad let η a,s = (α, βa,s ). 6

7 We a write Y (P B P 1)Y = µ α (P B P 1)µ α + µ α (P B P 1)ɛ + ɛ (P B P 1)ɛ = µ (P B P 1)µ + µ (P B P 1)ɛ + ɛ (P B P 1)ɛ. The seod equality holds beause (P B P 1)1 = 0 ad µ α = µ + α 1. Notie that µ = Mβ a,s + ν, oe has µ (P B P 1)µ = β a,s M (P B P 1)Mβ a,s + 2ν (P B P 1)Mβ a,s + ν (P B P 1)ν = β a,s M P B Mβ a,s β a,s M P 1 Mβ a,s + 2ν (P B P 1)Mβ a,s +ν (P B P 1)ν = D 1 D 2 + 2D 3 + D 4. or the first term, otie that P B M = M, we a write D 1 = β a,s M Mβa,s + 2β a,s M P B (M M)β a,s β a,s (M M) P B (M M)β a,s = β a,s M Mβ a,s + β a,s ( M M M M)β a,s + 2β a,s M P B (M M)β a,s β a,s (M M) P B (M M)β a,s = E 1 + E 2 + 2E 3 E 4. By simple alulatio, oe has E 1 = β a (t 1 )β a (t 2 )K s (t 1, t 2 )dt 1 dt 2 (1 + o P (1)) = O P (), where K s (, ) is the projetio of the ovariae futio K(, ) over the first s eigefutios of K(, ). Notie that E 1 = Mβ a,s 2, we a also obtai Mβ a,s = O P ( 1/2 ). or the seod term, we have E 2 β a,s 2 M M M M 2 β a ( ) 2 L 2 M M M M. By Lemma 3, we have M M M M = s k=1 ( ξ ijξ ik ξ ij ξik ) 2 } 1/2 = O P ( 1/2 s δs 1 ). Uder oditio (A), we obtai that E 2 = o P (). Notie that P B is a projetio matrix, ad the eigevalue of it a oly be 0 ad 1. Thus, we obtai E 3 Mβ a,s (M M)β a,s ad E 4 (M M)β a,s 2. By Lemma 3, (M M)β a,s 2 = s β ja (ˆξ ij ξ ij )} 2 = O P (s 2 δs 2 ) = o P () by oditio (A), whih idiates that E 3 = o P () ad E 4 = o P (). Sie E 2, E 3 ad E 4 are domiated by E 1, we a write D 1 = β a (t 1 )β a (t 2 )K s (t 1, t 2 )dt 1 dt 2 (1 + o P (1)). Now we take a look at the term D 2. We a write D 2 = 1 ( E( s β ja ξ ij ) = 0 ad s β ja ξ ij s are i.i.d for eah 1 i, we have E( = s s β ja ξ ij ) 2 = E( β 2 jaλ j β a ( ) 2 L 2 By Markov s iequality, oe has D 2 = O P (1) = o P (). s β ja ξ ij ) 2 λ j = O(). β ja ξ ij ) 2. Sie 7

8 Notie that P B P 1 is a projetio matrix, oe has D 3 D 4 ν 2. Sie ν 2 = ( j=s β +1 jaξ ij ) 2 ad ν Mβ a,s ad E ( j=s +1 β ja ξ ij )} 2 = j=s +1 β 2 jaλ j β a ( ) 2 L 2 j=s +1 Sie λ j < ad s is divergig, we a obtai j=s λ +1 j = o(1), whih idiates that E ν 2 = o(). By Markov s iequality, we have ν 2 = o P (). Sie Mβ a,s = O P ( 1/2 ), we a see D 3 = o P () ad D 4 = o P (). As a result, D 2, D 3 ad D 4 are domiated by D 1, whih idiates µ (P B P 1)µ = µ(p B P 1)µ = β a (t 1 )β a (t 2 )K s (t 1, t 2 )dt 1 dt 2 (1 + o P (1)). Notie that β a (t 1 )β a (t 2 )K(t 1, t 2 )dt 1 dt 2 β a (t 1 )β a (t 2 )K s (t 1, t 2 )dt 1 dt 2 λ j. = βjaλ 2 j β a ( ) 2 L 2 j=s +1 j=s +1 λ j = o(1), ombiig with the previous argumets, oe has µ (P B P 1)µ = β a (t 1 )β a (t 2 )K(t 1, t 2 )dt 1 dt o P (1)}. Notie that ɛ ad µ (P B P 1) are idepedet, ad ɛ i has fiite seod momet, we have Eµ (P B P 1)ɛ} = 0 ad E[µ (P B P 1)ɛ} 2 ] = O P (µ (P B P 1)µ) = O P (). Thus, we have µ (P B P 1)ɛ = O P ( ). Thus, µ (P B P 1)µ + µ (P B P 1)ɛ = β a (t 1 )β a (t 2 )K(t 1, t 2 )dt 1 dt o P (1)}. rom the proof of theorem 1, we kow that ɛ (P B P 1)ɛ = s A 2 1j, where A 1j s are i.i.d. stadard ormal radom variables. Thus, Y (P B P 1)Y = s A 2 1j + β a (t 1 )β a (t 2 )K(t 1, t 2 )dt 1 dt o P (1)}. Meawhile, otie that σ 2 = Y (I P 1 )Y/( 1), we have Y (I P 1 )Y = µ α (I P 1 )µ α + µ α (I P 1 )ɛ + ɛ (I P 1 )ɛ = µ (I P 1 )µ + µ (I P 1 )ɛ + ɛ (I P 1 )ɛ. Notie that µ (I P 1 )µ = Mβ a,s 2 βa,s M P 1 Mβ a,s + 2ν (I P 1 )Mβ a,s + ν (I P 1 )ν. Sie βa,s M P 1 Mβ a,s = D 2 = o P (), ν (I P 1 )ν ν 2 = o P () ad ν (I P 1 )Mβ a,s ν Mβ a,s = o P (), oe has µ (I P 1 )µ = β a (t 1 )β a (t 2 )K s (t 1, t 2 )dt 1 dt 2 (1 + o P (1)) = β a (t 1 )β a (t 2 )K(t 1, t 2 )dt 1 dt 2 (1 + o P (1)). By the proof of theorem 1, ɛ (I P 1 )ɛ/ = 1 + O P ( 1/2 ). or the seod term, sie ɛ ad µ (I P 1 ) are idepedet, we a see that Eµ (I P 1 )ɛ} = 0 ad E[µ (I P 1 )ɛ} 2 ] = O P (µ (I P 1 )µ) = O P (), we have µ (I P 1 )ɛ = O P ( ). Thus, we a Y (I P 1 )Y/ = (1 + β a (t 1 )β a (t 2 )K(t 1, t 2 )dt 1 dt 2 )(1 + o P (1)). As ( s A 2 1j s )/ 2s d N(0, 1) whe s, ombiig the above results, oe has (1 + β a (t 1 )β a (t 2 )K(t 1, t 2 )dt 1 dt 2 )T S s Λ }/ 2s d N(0, 1) whe s, where Λ = β a (t 1 )β a (t 2 )K(t 1, t 2 )dt 1 dt o P (1)}. or T, we a write Y (I P B )Y = µ α (I P B )µ α + 2µ α (I P B )ɛ + ɛ (I P B )ɛ. 8

9 Sie ɛ (I P B )ɛ = s 1 A 2 3j, osequetly, we have E(ɛ (I P B )ɛ) = E( (I P B )ɛ 2 ) = O(), whih idiates ɛ (I P B )ɛ = (I P B )ɛ 2 = O P (). or the first term, we a write µ α (I P B )µ α = (Bη a,s + ν) (I P B )(Bη a,s + ν) = η a,s B (I P B )Bη a,s + 2ν (I P B )Bη a,s + ν (I P B )ν = D 5 + D 6 + D 7. or D 5, otie that (I P B ) B = 0 ad I P B is a projetio matrix, we have D 5 = η a,s (B B) (I P B )(B B)η a,s = β s,s (M M) (I P B )(M M)β s,s (M M)β s,s 2. We a also obtai D 6 ν (M M)β s,s ad D 7 ν 2. Sie (M M)β s,s 2 = o P () ad ν 2 = o P () by the previous statemet, we obtai µ α (I P B )µ α = (I P B )µ α 2 = D 5 + D 6 + D 7 = o P (). As µ α (I P B )ɛ (I P B )µ α (I P B )ɛ, we obtai µ α (I P B )ɛ = o P (). As a result, Y (I P B )Y = ɛ (I P B )ɛ + o P (). By the proof of Theorem 1, ɛ (I P B )ɛ/( s 1) = 1+o P (1). Thus, Y (I P B )Y/( s 1) = 1 + o P (1). Notie that Y (P B P 1)Y = s A 2 1j + β a (t 1 )β a (t 2 )K(t 1, t 2 )dt 1 dt o P (1)} ad ( s A 2 1j s )/ 2s d N(0, 1) whe s, ombiig the above results, oe has s T s Λ }/ 2s d N(0, 1) whe s, where Λ = β a (t 1 )β a (t 2 )K(t 1, t 2 )dt 1 dt o P (1)} Proof of Corollary 1 ollowig similar argumets as the proof of Theorem 2, we have Y (P B P 1)Y = s Y (I P 1 )Y/ = (1 + ρ 2 A 2 1j + ρ 2 Y (I P B )Y/( s 1) = 1 + o P (1). β a (t 1 )β a (t 2 )K(t 1, t 2 )dt 1 dt o P (1)}, β a (t 1 )β a (t 2 )K(t 1, t 2 )dt 1 dt 2 )1 + o P (1)}, Thus, we obtai (1 + ρ 2 βa (t 1 )β a (t 2 )K(t 1, t 2 )dt 1 dt 2 )T S s Λ }/ 2s d N(0, 1) ad s T s Λ }/ 2s d N(0, 1) whe s, where Λ = ρ 2 β a (t 1 )β a (t 2 )K(t 1, t 2 )dt 1 dt o P (1)}. Our test statistis would have loal power if Λ / s has order o smaller tha 1. Sie the order of Λ is exatly ρ 2, this is equivalet to require s /(ρ 2 ) = O(1). 9

10 3.4. Proof of Corollary 2 The proof of Corollary 2 is exatly the same as the proof of Theorem 1, sie the ull distributio of the tests is derived usig the true model, i.e. β( ) 0, ad thus it is ot affeted if the estimated futioal priipal ompoet sores are used istead of the true oes. 4. Testig proedures i partial futioal liear model I this setio we provide the expressios for the, (modified) likelihood ratio ad test statistis i the otext of partial futioal liear model disussed i Setio 4. The expressio for the sore test statistis is iluded i the mausript. The test statisti is defied as T W = Y B(B B) 1 L M (I P Z )ML(B B) 1 B Y/ σ 2, where L is a s (s + p + 1) matrix with L = (0 s (p+1), I s s ). The likelihood ratio test statisti is T L = Y (I P Z )Y/ σ 2 Y (I P B )Y/ σ 2 + log( σ 2 / σ 2 ). The test statisti is T = Y (P B P Z )Y/s }/Y (I P B )Y/( s p 1)}. or σ 2 ad σ 2 i test ad the modified likelihood ratio test, we would still use the restrited maximum likelihood estimates versio by pluggig the restrited maximum likelihood estimates σ 2 REML = Y (I P Z )Y/( 1 p) ad σ 2 REML = Y (I P B )Y/( s p 1) to improve the performae for small sample ases. 5. Additioal simulatio results 5.1. Simulatio results for moderate ad sparse samplig desigs I this setio, we provide additioal simulatio results for the settig whe the futioal ovariate is observed sparsely ad with measuremet error. Speifially igure S1 depits the power urves for testig H 0 : β( ) = 0 for the, sore, likelihood ratio ad testig proedures, whe the futioal ovariate is observed at samplig desigs 2 ad 3, ad for sample sizes 50, 100, 500. The results idiate that the power is similar aross the tests, eve whe the samplig desig is sparse, with differet degrees of sparseess. i additio, the power dereases with the level of sparseess of the desig: ompare the power urves for Desig 1, 2, ad 3. This is expeted due to the redued umber of observatios i the ase of sparse desigs. 10

11 (d) (e) (f) igure. S1: urves orrespodig to Desig 2 (top paels) ad Desig 3 (bottom paels) for varyig sample sizes: 50 (left paels), 100 (middle paels) ad 500 (right paels) Compariso with bootstrap testig, the miimax adaptive testig ad the restrited likelihood ratio testig methods I this setio, we evaluate our proposed methods relative to the oes disussed i Gozález-Mateiga, Gozález-Rodríguez, Martíez-Calvo, ad Garía-Portuguéss (2014), Hilgert, Mas, ad Verzele (2013) ad Swihart, Goldsmith, ad Craiieau (2014). Speifially, we osider the proposed test alog with its asymptoti ull distributio approximated as i Remark 2 ad ompare its performae with the test with its ull distributio approximated by the wild bootstrappig method proposed i Gozález-Mateiga et al. (2014), the miimax adaptive testig itrodued i Hilgert et al. (2013) ad the restrited likelihood ratio test disussed i Swihart et al. (2014); beause the restrited likelihood ratio test beomes a likelihood ratio test whe there is a sigle futioal ovariate (the settig we ivestigate) we will refer to the latter test by likelihood ratio test. As the methods i Gozález-Mateiga et al. (2014) ad Hilgert et al. (2013) are ot appliable to the ase whe the futioal ovariates are measured at a irregular sparse desig, or orrupted with measuremet error, we restrit the ompariso to the seario whe the samplig desig is dese ad the ovariates are measured without oise. or simpliity, we geerate data followig the same settig as 11

12 restrited bootstrap miimax1 miimax2 restrited bootstrap miimax1 miimax2 restrited bootstrap miimax1 miimax2 igure. S2: urves ompariso of the level 0.05 tests usig: test, bootstrap test, likelihood ratio test (Swihart et al. 2014) ad the miimax tests, for varyig sample sizes: 50 (left paels), 100 (middle paels) ad 500 (right paels). i Desig 1 desribed i Setio 6.1 of the paper, ad set the measuremet error proess e i ( ) = 0; the results are ompared for sample size varyig from = 50, 100 to = 500. Table S1 ad igure S2 show the results of the Type I error rate ad power performae of the methods. The results are based o T 3, of Gozález-Mateiga et al. (2014), whih is referred by bootstrap, ad the tests T α (1) ad T α (2) of Hilgert et al. (2013), whih are referred by miimax1 ad miimax2, respetively. The test statisti used i Swihart et al. (2014) is referred as restrited. I terms of Type I error rate, Table S1 reveals that our test, the bootstrap ad miimax tests maitai the omial level very well: similar Type I error rate is obtaied usig our test, the bootstrap ad miimax2 proedures. Our ivestigatio poits out that the miimax1 test is slightly oservative. The likelihood ratio test (Swihart et al. 2014) has greatly iflated Type I error rate, whih oiides with our ow experiee regardig the performae of the likelihood ratio test. The power performae is illustrated i igure S2 ad reveals that our test, the bootstrap ad miimax tests are omparable. The likelihood ratio test has the greatest power, whih is as expeted, beause it has sigifiatly iflated Type I error. rom the results, we a see our test, the bootstrap test ad miimax tests, espeially miimax2, have ompatible performae, whe the samplig desig is dese ad the ovariates are measured without oise. Uder this seario, all the tests have good performae, exept the likelihood ratio test. Nevertheless, i otrast to the ompetig alteratives, i.e. the miimax tests ad the bootstrap test, our proposed approah a aout for measuremet error i the futioal ovariates, as well as irregular or sparse samplig desig. Table. S1: The Type I error rate ompariso of the level 0.05 tests usig: the proposed test, the bootstrap test, ad the miimax tests for varyig sample sizes. bootstrap miimax1 miimax2 restrited Simulatio results whe the latet proess is ot Gaussia I this setio, we osider the ase whe the PC sores are o-ormal, i.e. the X i ( ) is ot a Gaussia proess. We oly osider the dese ase here, i.e. the settigs are 12

13 Type I error (d) igure. S3: urves whe the latet proess is ot Gaussia, for varyig sample sizes: 50 (left paels), 100 (middle paels) ad 500 (right paels). exatly the same as the Desig 1 i setio 6.1 i the paper exept that the PC sore ξ ij are geerated i.i.d. from ( λ j / 3)t 3. The results are preseted i igure S3, ad orrespod to fixig the level of sigifiae at 5%. igure S3 shows the performae of the tests with respet to Type I error rate as the sample size ireases from 50 to 500. I partiular, test gives reasoable Type I error rates for various sample sizes. The sore test seems to be somewhat oservative for small samples for all the samplig desigs, while ad the modified likelihood ratio test idiate a iflated Type I error rates for small ad moderate sample sizes ( = 50 or = 100). or large sample size ( = 500), all of the tests give Type I error rates lose to the omial level. igure S3 -(d) display the power performae of the tests for various sample sizes. The tests have omparable power for all sample sizes ivestigated. The performae of all the tests is similar as that whe the PC sores are ormal Simulatio results whe X i (t) is geerated from a large umber of basis futios I this subsetio, we oduted simulatios for the ase whe X i (t) is geerated from a large umber of basis. I partiular, the uderlyig geeratig proess for the ith futioal ovariate is X i (t) = j 1 ξ ijφ j (t), where ξ ij are geerated idepedetly as N(0, λ j ), ad λ j = 11 j for 1 j 10, λ j = (20 j)/10 for 11 j 19, ad λ 20 = Also φ j s are ourier basis futios o [0, 10] defied as φ 2j 1 (t) = os(2j 1)πt/10}/ 5 ad φ 2j (t) = si(2j 1)πt/10}/ 5 for 1 j 10 ad 0 t 10. or the other settigs, they are the same as Setio 7.1. We osider three desigs 1, 2, 3 ad for power performae, we osider β( ) = β ( ) orrespodig to > 0 for takig values i grid of 20 equally spaed poits i [0.01, 0.2]. We also osider three differet thresholds 85%, 90% ad 99%. The omial level of the test is set as igure S4 show the performae of the tests with respet to Type I error 13

14 Type I error Desig 1 Desig 2 Desig Type I error Desig 1 Desig 2 Desig Type I error Desig 1 Desig 2 Desig igure. S4: Pael show the estimated Type I error (depited as the height of the bars) for all the four tests i ie settigs obtaied from ombiig three samplig desigs ad sample sizes whe the thresholds 85%, 90% ad 99%. rate with differet thresholds for various samplig desigs as the sample size ireases from 50 to 500. I partiular, test gives reasoable Type I errors for all the desigs ad various sample sizes. The sore test seems to be somewhat oservative for small samples for all the samplig desigs, while ad the modified likelihood ratio test idiate a iflated Type I error for small ad moderate sample sizes ( = 50 or = 100). or large sample size ( = 500), all of the tests give Type I error rates lose to the omial level. or differet thresholds of peretages of variae explaied, the Type I error performae is quite similar. igures S5 S13 show the power performae of the tests with differet thresholds for various samplig desigs as the sample size ireases from 50 to 500. Whe sample size is small ( = 50), there would be a little bit power loss whe we use a larger threshold 99% ompared with smaller thresholds 85%, 90%, but whe the sample size beomes large ( = 500), the power performae is quite robust to the hoie of the thresholds. Referees Gozález-Mateiga, W., Gozález-Rodríguez, G., Martíez-Calvo, A., ad Garía-Portuguéss, E. (2014), Bootstrap idepedee test for futioal liear models, upublished mausript. Hilgert, N., Mas, A., ad Verzele, N. (2013), Miimax adaptive tests for the futioal liear model, The Aals of Statistis, 41, Li, Y., Wag, N., ad Carroll, R.J. (2010), Geeralized futioal liear models with semiparametri sigle-idex iteratios, Joural of the Ameria Statistial Assoiatio, 105, Supplemetary materials available olie. 14

15 Desig 1, =50, Variae=0.85 Desig 1, =50, Variae=0.9 Desig 1, =50, Variae= igure. S5: Pael, ad orrespod to the hages of the power for Desig 1, sample size 50, ad thresholds 85%, 90% ad 99% respetively. Desig 1, =100, Variae=0.85 Desig 1, =100, Variae=0.9 Desig 1, =100, Variae= igure. S6: Pael, ad orrespod to the hages of the power for Desig 1, sample size 100, ad thresholds 85%, 90% ad 99% respetively. Desig 1, =500, Variae=0.85 Desig 1, =500, Variae=0.9 Desig 1, =500, Variae= igure. S7: Pael, ad orrespod to the hages of the power for Desig 1, sample size 500, ad thresholds 85%, 90% ad 99% respetively. Desig 2, =50, Variae=0.85 Desig 2, =50, Variae=0.9 Desig 2, =50, Variae= igure. S8: Pael, ad orrespod to the hages of the power for Desig 2, sample size 50, ad thresholds 85%, 90% ad 99% respetively. Swihart, B., Goldsmith, J., ad Craiieau, C. (2014), Restrited Likelihood Ratio Tests for utioal Effets i the utioal Liear Model, Tehometris, p. to appear. Yao,., Müller, H.G., ad Wag, J.L. (2005), utioal data aalysis for sparse logitudial data, Joural of the Ameria Statistial Assoiatio, 100, Zhu, H., Yao,., ad Zhag, H.H. (2014), Strutured futioal additive regressio i repro- 15

16 Desig 2, =100, Variae=0.85 Desig 2, =100, Variae=0.9 Desig 2, =100, Variae= igure. S9: Pael, ad orrespod to the hages of the power for Desig 2, sample size 100, ad thresholds 85%, 90% ad 99% respetively. Desig 2, =500, Variae=0.85 Desig 2, =500, Variae=0.9 Desig 2, =500, Variae= igure. S10: Pael, ad orrespod to the hages of the power for Desig 2, sample size 500, ad thresholds 85%, 90% ad 99% respetively. Desig 3, =50, Variae=0.85 Desig 3, =50, Variae=0.9 Desig 3, =50, Variae= igure. S11: Pael, ad orrespod to the hages of the power for Desig 3, sample size 50, ad thresholds 85%, 90% ad 99% respetively. Desig 3, =100, Variae=0.85 Desig 3, =100, Variae=0.9 Desig 3, =100, Variae= igure. S12: Pael, ad orrespod to the hages of the power for Desig 3, sample size 100, ad thresholds 85%, 90% ad 99% respetively. duig kerel Hilbert spaes, Joural of the Royal Statistial Soiety. Series B., 76,

17 Desig 3, =500, Variae=0.85 Desig 3, =500, Variae=0.9 Desig 3, =500, Variae= igure. S13: Pael, ad orrespod to the hages of the power for Desig 3, sample size 500, ad thresholds 85%, 90% ad 99% respetively. 17