r = r r| r|

≤ C 1ς C γ w h|1 − h h ′ | γ c w c b −c w c b |v| γ dv = C 1ς C γ w h |b ′ − bb ′ | γ c w c b −c w c b |v| γ dv ≤ C 2ς h|b ′ − b| γ , where C sς is a finite constant that depends on ς i ; for example, C 1ς ≡ sup u c ∈U c u d i ∈U d m ς u c + hv, u d i f u c + hv, u d i dv ∞. Similarly, K 2 = u d i ∈U d ,u d i =u d ,, , ,w u c i − u c h ,, , , γ |λ ′ − λ| γ m ς u c i , u d i × f u c i , u d i du c i = h|λ ′ − λ| γ u d i ∈U d c w c b −c w c b w v γ m ς u c + hv, u d i × f u c + hv, u d i dv ≤ C 3ς h|λ ′ − λ| γ ≤ C 3ς hn −γ σ |r ′ − r| γ . It follows that E [|K b ′ r ′ ,iu − K br,iu | γ ς i ] ≤ c γ C 2ς ∨ C 3ς h|b ′ − b| γ + |r ′ − r| γ , A.3 where a ∨ b = max a, b . Then by the C r inequality E [|K b ′ r ′ ,iu | γ ς i ] ≤ c γ E [|h ′ K h ′ λ ′ ,iu − hK hλ,iu | γ ς i ] + c γ E [|hK hλ,iu | γ ς i ] ≤ c γ C 2ς ∨ C 3ς h|b ′ − b| γ + |λ ′ − λ| γ + C 4ς h ≤ C 5ς h. A.4 Note that |h ′ α − h α | = |b ′ α − b α |n −αδ = αn −αδ b ∗ α−1 |b ′ − b| ≤ C 6 h α |b ′ − b|, A.5 where the second equality follows from the intermediate value theory and b ∗ lies between b ′ and b. Then by the C r inequality, and A.3 – A.5 , we have that for any α 0 and γ 0, E [|h ′ −α K b ′ r ′ ,iu − h α K br,iu | γ ς i ] ≤ c γ E [|h −α K b ′ r ′ ,iu − K br,iu | γ ς i ] + c γ E [|[h ′ −α − h −α ]K b ′ r ′ ,iu | γ ς i ] ≤ c γ h −αγ E [|K b ′ r ′ ,iu − K br,iu | γ ς i ] + c γ |h ′ −α − h −α | γ E [|K b ′ r ′ ,iu | γ ς i ] ≤ c 2 γ h 1−αγ |b ′ − b| γ +|λ ′ − λ| γ + c γ C 6 h −αγ |b ′ − b|C 5ς h ≤ C 7ς h 1−αγ |b ′ − b| γ + |λ ′ − λ| γ . In the general case where p c 1 or p d 1, with a lit- tle bit abuse of notation we use row vectors to denote the bandwidth parameters. We can write h s = b s n −δ s and λ t = r t n −σ t for some b s , r t , δ s , σ t 0, s = 1, . . . , p c , and t = 1, . . . , p d . Let b = b 1 , . . . , b p c and r = r 1 , . . . , r p d . Similarly, h ′ , λ ′ = h ′ 1 , . . . , h ′ p c , λ ′ 1 , . . . , λ ′ p d and b ′ , r ′ = b ′ 1 , . . . , b ′ p c , r ′ 1 , . . . , r ′ p d are connected through h ′ s = b ′ s n −δ s and λ ′ t = r ′ t n −σ t for b ′ s , r ′ t 0. Then using the fact that our multivariate kernel function is a product of univariate kernel functions, we can follow the above arguments and readily show that E [|h ′ −α K b ′ r ′ ,iu − h α K br,iu | γ ς i ] ≤ C 8ς h 1−αγ b ′ − b γ + λ ′ − λ γ , A.6 where C 8ς is a finite constant depending on ς i . Below we use C to denote a generic constant that can vary from equation to equation. Now we prove i. Let J 1n,st

b, r =

n,st u;h, λ for s, t = 1, 2, where n,st ’s are defined in the proof of Lemma 1. By Theorem 3.1 in Li and Li 2010 , it suffices to show that for any b ′ , r ′ and b, r that lie in a compact set e.g., [b, ¯ b ] × [r, ¯r] for the case p c = p d = 1, there exist α 0 and γ 1 such that E{J 1n,st b ′ , r ′ − J 1n,st

b, r

α } ≤ C{b ′ − b γ + λ ′ − λ γ } for some C ∞. A.7 We only show A.7 for the case s, t = 1, 1 as the other cases are similar. Let ς i,j l denote the j, lth element of Q i X ′ i for j = 1, . . . , k and l = 1, . . . , d. Let J j,l 1n,st b, r denote the j, lth element of J 1n,st b, r . Then E|J j,l 1n,11 b ′ , r ′ − J j,l 1n,11 b, r| 2 is bounded above by 2E , , , , , , 1 n n i=1 {[h ′ −1 K b ′ r ′ ,iu − h −1 K br,iu ]ς i,j l − E{[h ′ −1 K b ′ r ′ ,iu − h −1 K br,iu ]ς i,j l }} , , , , , , 2 + 2|E{[h ′ −1 K b ′ r ′ ,iu − h −1 K br,iu ]ς i,j l }| 2 ≡ 2S 1 + 2S 2 , say. By Assumption A1, Jensen’s inequality, and A.6 , for suffi- ciently large n S 1 = 1 n 2 n i=1 var[h ′ −1 K b ′ r ′ ,iu − h −1 K br,iu ]ς i,j l ≤ n −1 E{[h ′ −1 K b ′ r ′ ,iu − h −1 K br,iu ]ς i,j l } 2 ≤ n −1 Ch 1−2 b ′ − b 2 + λ ′ − λ 2 ≤ Cb ′ − b 2 + λ ′ − λ 2 as nh → ∞ implies that for sufficiently large n, n −1 h −1 can be bounded by the constant 1. By A.6 , S 2 = |E{[h ′ −1 K b ′ r ′ ,iu − h −1 K br,iu ]ς i,j l }| 2 ≤ [Cb ′ − b + λ ′ − λ] 2 ≤ Cb ′ − b 2 + λ ′ − λ 2 . It follows that E{|J j,l 1n,11 b ′ , r ′ − J j,l 1n,11 b, r| 2 } ≤ Cb ′ − b 2 + λ ′ − λ 2 and A.7 holds for α = γ = 2 and s, t = 1, 1 . Analogously, we can show that it also holds for α = γ = 2 and other values of s and t. This completes the proof of i . Next we prove ii. Let J 2n

b, r ≡