A Non-Uniform Bound on Normal Approximation of Randomized Orthogonal Array Sampling Designs

Transcription

1 International Mathematical Forum,, 007, no. 48, A Non-Uniform Bound on Normal Approximation of Randomized Orthogonal Array Sampling Designs K. Laipaporn Department of Mathematics School of Science, Walailak University Nakhon Si Thammarat 8060, Thailand lkittipo@wu.ac.th K. Neammanee Department of Mathematics Faculty of Science, Chulalongkorn University Bangkok 0330, Thailand Kritsana.N@chula.ac.th Abstract This paper gave a non-uniform bound on normal approximation of randomized orthogonal array sampling designs which is proposed by Owen and Tang in 99 and 993. Our method is Stein s method. Mathematics Subject Classification: Primary 6E0, secondary 60F05 Keywords: non-uniform bound, normal approximation, computer experiment, orthogonal array sampling designs, Stein s method Introduction The problem of computing a value of integral over a high dimension is important in many scientific fields. Consider a deterministic function Y = f(x) R where x [0, ] d and f is known but hard to calculate. Then our aim is to estimate [0,] d f(x)dx,

2 348 K. Laipaporn and K. Neammanee that is the expectation of f X, μ = E(f X), where X is a random vector uniformly distributed on a unit hypercube [0, ] d. The simplest way is to draw the samplings X,X,..., X n independently and uniformly distributed from [0, ] d and use ˆμ = n n f X i (.) i= as an estimator of μ. This method is called the Monte Carlo method.(see[0],[4] and [6] for more details) Beside the uniform sampling, there are various alternative ways to sellect the points X i s for ˆμ. For examples, lattice sampling([9]), Latin hypercube sampling([],[3],[5],[8]), the orthogonal arrays([],[6]) and scrambled net([3],[4]). In 996, Loh considered some classes of the orthogonal arrays and he gave a uniform bound on normal approximation. In this work, we shall establish a non-uniform bound by using the Stein s method. The organization of this paper is as follows. In section, we give some definitions, notations and state the main result. Some useful lemmas are proved in section 3 and we show the main result in section 4. Preliminary Notes and Main Result Definition.. An orthogonal array of strength t with index λ (λ ) is an n d matrix with elements taken from the set {0,,..., } such that for any n t submatrix, each of the t possible rows appears the same number λ of times where d, n, and t are positive integers with t d and. Of course, n = λ t. A class of this arrays is denoted by OA(n, d,, t)(see [8] for more details). In 996, Loh considered the class OA(n, 3,,) when n = and constructed the sampling X,X,..., X on the unit cube [0, ] 3 as follows: Let (a) π,π,π 3 be random permutations of {0,,..., }, (b) U i,i,i 3,j be [0, ] uniform random variables where i,i,i 3 {0,,..., }, j {,, 3}, and (c) U i,i,i 3,j s and π k s be all stochastically independent. An orthogonal array-based sample of size, {X,X,..., X }, is defined to be {X(π (a i, ),π (a i, ),π 3 (a i,3 )) : i }, where, for each i,i,i 3 {0,,..., } and j {,, 3}, X ( i,i,i 3 ) = ( X (i,i,i 3 ),X (i,i,i 3 ),X 3 (i,i,i 3 ) ),

3 Normal approximation of orthogonal array 349 X j (i,i,i 3 )= i j + U i,i,i 3,j, and a i,j is the (i, j) th element of some arbitary but fixed A OA(, 3,,). So the estimator ˆμ of μ in (.) can be express in the form of ˆμ = f X(π (a i, ),π (a i, ),π 3 (a i,3 )). i= Owen([]) gave an expression for the asymptotic variance Var(ˆμ) ofˆμ. In this paper, we assume that Var(ˆμ) > 0 and define W = (ˆμ μ). (.) Var(ˆμ) Loh([8]) gave a uniform bound on the normal approximation of W and Theorem. is his result. Theorem.. Let Φ be the stantdard normal distribution function, i.e., Φ(x) = x e t dt. Suppose that E(f X) r < for some even integer r 4. π Then sup { P (W w) Φ(w) : <w< } = O( r r ) as. In 006, Laipaporn and Neammanee([]) improved the order in Theorem. to the rate O( ) under the assumption that E(f X) 6 <. In this work, we investigate a non-uniform bound. The following theorem is our main result. Theorem.. Suppose that E(f X) r < for even number r 0. Then, as, for z R, P (W z) Φ(z) max ( O( ( + z ) r r 8 r ), ( + z ) O( ) ). 6 3 Auxiliary Results In this section, we will give some lemmas which used in proving the main result in section 4. In 996, Loh defined a random function ρ π be such that ( i,i,ρ π (i,i ) ) = ( π (a i, ),π (a i, ),π 3 (a i,3 ) ) (3.) for some i {,..., } and showed that W in (.) can be rewritten again as the form W = Y ( i,i,ρ π (i,i ) ) (3.) i =0 i =0

4 350 K. Laipaporn and K. Neammanee [ where Y (i,i,i 3 )= f X(i,i,i 3 ) μ Var(ˆμ) ] μ k,l (i k,i l ), and k<l 3 3 μ j (i j ) μ(i,i,i 3 )=Ef X(i,i,i 3 ),μ j (i j )= [μ(i,i,i 3 ) μ], μ k,l (i k,i l )= i k =0 k j j= [μ(i,i,i 3 ) μ μ k (i k ) μ l (i l )]. i j =0 j k,l Let I and K be uniformly distributed random variables on {0,,..., }, (I,K) uniformly distributed on { (i, k) i, k =0,,...,,i k } and assume that they are indenpendent of all π,π,π 3 and U i,i,i 3,j s defined previously. Define W = W S S + S 3 + S 4 where S = Y ( I,i,ρ π (I,i ) ), S = Y ( K, i,ρ π (K, i ) ), i =0 i =0 S 3 = Y ( I,i,ρ π (K, i ) ), S 4 = Y ( K, i,ρ π (I,i ) ). i =0 i =0 Moreover, for each i,i and i 3 {0,,,..., }, and z 0, we also let and μ(i,i,i 3 )=EY (i,i,i 3 ) Y z (i,i,i 3 )=Y(i,i,i 3 )I ( Y (i,i,i 3 ) > +z ), Ŷ z (i,i,i 3 )=Y (i,i,i 3 )I ( Y (i,i,i 3 ) +z ), Ŷ = Ŷ z (i,i,ρ π (i,i )) i =0 i =0 Ỹ = Ŷ Ŝ,z Ŝ,z + Ŝ3,z + Ŝ4,z where Ŝ,z = Ŷ z (I,i,ρ π (I,i )), Ŝ,z = Ŷ z (K, i,ρ π (K, i )), i =0 i =0 Ŝ 3,z = Ŷ z (I,i,ρ π (K, i )), Ŝ 4,z = Ŷ z (K, i,ρ π (I,i )), i =0 i =0

5 Normal approximation of orthogonal array 35 and I is an indicator function. Loh([8]) showed that ( W, W ) is an exchangeable pair, S,S,S 3,S 4 are identically distributed and if E(f X) r < for a positive even integer r, then ESi r = O( r ) for i =,, 3, 4, (3.3) and By the fact that EY r (i,i,i 3 )=O( 3 r ), (3.4) i =0 i =0 i 3 =0 EY (i,i,i 3 )=+O( ). i =0 i =0 i 3 =0 for every positive even number r([]) we have (3.5) E( W W ) r O( r ) (3.6) i =0 i =0 i 3 =0 E Y (i,i,i 3 ) m Y z (i,i,i 3 ) n E Y (i,i,i 3 ) m+n+t i =0 i =0 i 3 =0 ( + z) t O(3 m n t ) (3.7) ( + z) t for any integers m, n and t which m 0,n,t > 0 and m + n + t is an even number. The following lemmas are used in section 4 and in what follows, C denotes a generic absolute constant whose value may be different at each appearance. Lemma 3... If E(f X) <, then EŶ O( ) as.. If E(f X) r <, then, for any even number r, E Ỹ Ŷ r O( ) r as. 3. If E(f X) r+ <, then, for any odd number r 3, E Ỹ Ŷ r O( ) r as. Proof. see [] on p.0. Lemma 3... If E(f X) <, then E ( Y (i,i,i 3 ) ) O() as. i =0 i =0 i 3 =0. If E(f X) 4 <, then E ( Y (i,i,i 3 ) )4 O( ) as. i =0 i =0 i 3 =0

6 35 K. Laipaporn and K. Neammanee Proof.. By the fact that i j =0 (see [8], p.) and (3.5), we have E ( Y (i,i,i 3 ) ) i =0 i =0 i 3 =0 =. Note that E ( where M = i,i,i 3 EY (i,i,i 3 )+ μ(i,i,i 3 ) = 0 for j =,, 3, (3.8) i,i,i 3 j,j,j 3 (j,j,j 3 ) (i,i,i 3 ) = EY (i,i,i 3 ) μ (i,i,i 3 ) i,i,i 3 i,i,i 3 O(). i =0 i =0 i 3 =0 μ(i,i,i 3 ) μ(j,j,j 3 ) Y (i,i,i 3 ) )4 = C M + C M + C 3 M 3 + C 4 M 4 + C 5 M 5 EY 4 (i,i,i 3 ), i,i,i 3 M = EY (i,i,i 3 )Y 3 (j,j,j 3 ), M 3 = M 4 = M 5 = i,i,i 3 j,j,j 3 (j,j,j 3 ) (i,i,i 3 ) i,i,i 3 j,j,j 3 (j,j,j 3 ) (i,i,i 3 ) EY (i,i,i 3 )Y (j,j,j 3 ), i,i,i 3 j,j,j 3 k,k,k 3 (j,j,j 3 ) (i,i,i 3 ) (k,k,k 3 ) (i,i,i 3 ) (k,k,k 3 ) (j,j,j 3 ) i,i,i 3 j,j,j 3 k,k,k 3 l,l,l 3 (j,j,j 3 ) (i,i,i 3 ) (k,k,k 3 ) (i,i,i 3 ) (l,l,l 3 ) (i,i,i 3 ) (l,l,l 3 ) (k,k,k 3 ) (3.9) EY (i,i,i 3 )Y (j,j,j 3 )Y (k,k,k 3 ), EY (i,i,i 3 )Y (j,j,j 3 )Y (k,k,k 3 )Y (l,l,l 3 ) and C,C,C 3,C 4,C 5 are constants. From (3.4) and (3.8), we know that

7 Normal approximation of orthogonal array 353 M = O( ), (3.0) M = μ(i,i,i 3 )EY 3 (j,j,j 3 ) i,i,i 3 j,j,j 3 (j,j,j 3 ) (i,i,i 3 ) = μ(j,j,j 3 )EY 3 (j,j,j 3 ) j,j,j 3 EY 4 (j,j,j 3 ) j,j,j 3 = O( ), (3.) M 3 = EY (i,i,i 3 )EY (j,j,j 3 ) i,i,i 3 j,j,j 3 (j,j,j 3 ) (i,i,i 3 ) ( i,i,i 3 EY (i,i,i 3 ) ) O( ), (3.) M 4 = EY (i,i,i 3 ) μ(j,j,j 3 ) μ(l,l,l 3 ) i,i,i 3 j,j,j 3 l,l,l 3 (j,j,j 3 ) (i,i,i 3 ) (l,l,l 3 ) (i,i,i 3 ) (l,l,l 3 ) (j,j,j 3 ) EY (i,i,i 3 ) μ(j,j,j 3 ) μ(i,i,i 3 ) (3.3) i,i,i 3 j,j,j 3 (j,j,j 3 ) (i,i,i 3 ) + i,i,i 3 j,j,j 3 (j,j,j 3 ) (i,i,i 3 ) i,i,i 3 EY 4 (i,i,i 3 )+ ( EY (i,i,i 3 ) μ (j,j,j 3 ) i,i,i 3 EY (i,i,i 3 ) ) O( ), (3.4) and M5 = i,i,i 3 j,j,j 3 k,k,k 3 (j,j,j 3 ) (i,i,i 3 ) (k,k,k 3 ) (i,i,i 3 ) (k,k,k 3 ) (j,j,j 3 ) l,l,l 3 (l,l,l 3 ) (i,i,i 3 ) (l,l,l 3 ) (j,j,j 3 ) (l,l,l 3 ) (k,k,k 3 ) μ(i,i,i 3 ) μ(j,j,j 3 ) μ(k,k,k 3 ) μ(l,l,l 3 )

8 354 K. Laipaporn and K. Neammanee =3 μ (i,i,i 3 ) μ(j,j,j 3 ) μ(k,k,k 3 ) i,i,i 3 j,j,j 3 k,k,k 3 (j,j,j 3 ) (i,i,i 3 ) (k,k,k 3 ) (i,i,i 3 ) (k,k,k 3 ) (j,j,j 3 ) =3 μ (i,i,i 3 ) μ(j,j,j 3 ) μ(i,i,i 3 ) i,i,i 3 j,j,j 3 (j,j,j 3 ) (i,i,i 3 ) + i,i,i 3 j,j,j 3 (j,j,j 3 ) (i,i,i 3 ) 3 ( O( ). μ (i,i,i 3 ) μ(j,j,j 3 ) μ(j,j,j 3 ) EY 4 (i,i,i 3 )+ ( EY (i,i,i 3 ) ) ) i,i,i 3 i,i,i 3 (3.5) Hence from (3.9)-(3.5), we have. Lemma 3.3. Suppose that E(f X) 4 <. Then. E ( Y z (i, j, k) ) k=0. E ( Y z (i, j, k) )4 k=0 ( + z) O( ) as. Proof.. It follows from (3.4) and (3.7) that E ( i =0 i =0 i 3 =0 Y z (i,i,i 3 ) ) = EYz (i,i,i 3 )+ i,i,i 3 ( + z) ( + z) O( ). ( + z) O( ) as. i,i,i 3 i,i,i 3 EY 4 (i,i,i 3 )+ j,j,j 3 (j,j,j 3 ) (i,i,i 3 ). By the same argument as (3.9) we see that EYz (i,i,i 3 ) i,i,i 3 EY z (i,i,i 3 )Y z (j,j,j 3 ) E ( Y z (i,i,i 3 ) )4 = C M,z + C M,z + C 3 M 3,z + C 4 M 4,z + C 5 M 5,z i =0 i =0 i 3 =0 (3.6)

9 Normal approximation of orthogonal array 355 where M,z = EYz 4 (i,i,i 3 ), i,i,i 3 M,z = M 3,z = M 4,z = i,i,i 3 j,j,j 3 (j,j,j 3 ) (i,i,i 3 ) i,i,i 3 j,j,j 3 (j,j,j 3 ) (i,i,i 3 ) EY z (i,i,i 3 )Y 3 z (j,j,j 3 ), EY z (i,i,i 3 )Y z (j,j,j 3 ), i,i,i 3 j,j,j 3 k,k,k 3 (j,j,j 3 ) (i,i,i 3 ) (k,k,k 3 ) (i,i,i 3 ) (k,k,k 3 ) (j,j,j 3 ) EYz (i,i,i 3 )Y z (j,j,j 3 )Y z (k,k,k 3 ), M 5,z = i,i,i 3 j,j,j 3 k,k,k 3 l,l,l 3 (j,j,j 3 ) (i,i,i 3 ) (k,k,k 3 ) (i,i,i 3 ) (l,l,l 3 ) (i,i,i 3 ) (k,k,k 3 ) (j,j,j 3 ) (l,l,l 3 ) (j,j,j 3 ) (l,l,l 3 ) (k,k,k 3 ) EY z (i,i,i 3 )Y z (j,j,j 3 )Y z (k,k,k 3 )Y z (l,l,l 3 ) and C,C,C 3,C 4,C 5 are some constants. From (3.7), M,z ( + z) O( 3 ), (3.7) { M,z E Y z (i,i,i 3 ) }{ E Yz 3 (j,j,j 3 ) } i,i,i 3 j,j,j 3 = ( + z) O( ), (3.8) 4 { M3,z EYz (i,i,i 3 ) } i,i,i 3 = ( + z) O( ), (3.9) 4 { M4,z EYz (i,i,i 3 ) }{ E Y z (j,j,j 3 ) }{ } E Y z (l,l,l 3 ) i,i,i 3 j,j,j 3 l,l,l 3 = ( + z) O( ), (3.0) 5 3 and M 5,z ( E Y z (i,i,i 3 ) )4 { ( + z) O( 3 4 )}4. (3.) i,i,i 3 Hence from (3.6)-(3.), we have.

10 356 K. Laipaporn and K. Neammanee Let g z be the solution of the Stein s euation g z(ω) ωg z (ω) =h(ω) Φ(z) where h(x) =I(x z). It is well known that { πe ω / Φ(ω)[ Φ(z)] if ω<z, g z (ω) = πe ω / Φ(z)[ Φ(ω)] if ω z. Moreover, ωg z (ω) is an increasing function of ω, and for all real ω,u, v, t and s, π 0 <g z (w) min( 4, ), for z 0 (3.) z g z(w), (3.3) and g z (w) g z (v) (3.4) s g z (w + s) g z (w + t) h(w + u)du I ( z max(s, t) <w<z min(s, t) ) ([5], p.9-0). t Lemma 3.4. For each z>0, let h : R R be defined by (3.5) h(w) =(wg z (w)) (3.6) If E(f X) r < for some positive even number r, then E 0 t h(ŷ + u)k(t)dudt O( ) as, ( + z) r where K(t) = )( (Ỹ Ŷ I ( 0 t 4 Ỹ Ŷ ) I ( Ỹ Ŷ t<0)). Proof. By the fact that 4( + z )e z 8 ( Φ(z)) if w z 0 h(w) 4( + z z )e z ( Φ(z)) if <w z or ω>z (3.7)

11 Normal approximation of orthogonal array 357 ([5], p.48) and Φ(z) e z for z>0 (3.8) πz ([7], p.3), we have z C( + z) if w z, h(w) C if w< z ( + z) or w>z (3.9) where C is a constant. From (3.9) and the fact that E we have E 0 t 0 t = E K(t)dudt E h(ŷ + u)k(t)dudt + E 0 t 0 t t K(t)dt E Ỹ Ŷ 3 O( ), (3.30) h(ŷ + u)k(t)i(ŷ + u<z or Ŷ + u>z)dudt h(ŷ + u)k(t)i(z Ŷ + u z)dudt C ( + z) O( )+C( + z)e 0 t K(t)I(Ŷ + u z )dudt. By Lemma 3.(,) and the fact that K(t) = 0 for t > Ŷ Ỹ, E 0 t 0 E K(t)I(Ŷ + u z )dudt t K(t)I(Ŷ + Ỹ Ŷ z )dudt E Ỹ Ŷ 3 I(Ŷ + Ỹ Ŷ z ) { E Ỹ Ŷ 3r} { r P (Ŷ + Ỹ Ŷ z r r )} O( ) { EŶ + Ỹ Ŷ z O( ). ( + z) r } r r Hence the lemma follows from this fact and (3.3). (3.3)

12 358 K. Laipaporn and K. Neammanee Lemma 3.5. Let B be the σ algebra generated by π,π,π 3 and U i,i,i 3,j s. Then. E E B { K(t)dt} O( ) as.. E ( E B { K(t)dt} ) =(+z) 6 O( ) as where E B X is the conditional expectation of X given σ algebra B. Proof.. First, we note that 3 E E B {S k } = O( ), (3.3) E E B S j S k = O( ) as (3.33) for any j, k =,, 3, 4 ([8], p.8-) and for each r, s N such that r + s is an even number, we have E S,z r ES,z = ES,z = ES 3,z = ES 4,z and E S i S j,z for i, j =,, 3, 4 where ( + z) O( ), (3.34) s s ( + z) 4 O( 4 ) (3.35) ( + z) O( ) (3.36) S,z = Y z (I,i,ρ π (I,i )), S,z = Y z (K, i,ρ π (K, i )) i =0 i =0 S 3,z = Y z (I,i,ρ π (K, i )), S 4,z = Y z (K, i,ρ π (I,i )) i =0 ([], p.-). Note from (3.35) and (3.36) that, i =0 E E B (S j,z S k ) EE B S j,z S k = E S j,z S k ( + z) O( ), (3.37) and E E B (S j,z S k,z ) E S j,z S k,z { ES j,z } { } ESk,z ( + z) 4 O( 4 ) (3.38) for any j, k =,, 3, 4. From (3.3), (3.33), (3.37) and (3.38), we have

13 Normal approximation of orthogonal array 359 E E B { K(t)dt} = E E B ( ) { (Ŝ,z + Ŝ,z Ŝ 3,z Ŝ 4,z ) } 4 4 E E B (( )Ŝ k,z ) + E E B (Ŝj,zŜk,z) = k= j<k 4 4 E E B {( )(S k S k,z ) } + k= j<k 4 4 E E B (( )Sk ) 4 + E E B S k,z S k + k= + j<k 4 k= E E B (S j S k S j,z S k S j S k,z + S j,z S k,z ) E E B (S j S j,z )(S k S k,z ) 4 E E B Sk,z k= O( ). (3.39). It follows from (3.3) and (3.34) that E ( E B { { = E E B { = ( ) K(t)dt} ) 4 ( ) 4 k=0 k i } 4 (Ŝ,z + Ŝ,z Ŝ3,z Ŝ4,z) } { E E B[ ( ) 4 Ŷ z (i, j, ρ π (i, j)) ( + Ŷ z (k, j, ρ π (k,j)) Ŷ z (i, j, ρ π (k,j)) Ŷ z (k, j, ρ π (i, j))) ]} 4 ( ) k=0 k i { E ( ) 4 [ Ŷ z (i, j, ρ π (i, j)) + Ŷ z (k, j, ρ π (k,j)) } 4 Ŷ z (i, j, ρ π (k,j)) Ŷ z (k, j, ρ π (i, j))] C ( ) { + 4 E( k=0 k i Ŷ z (i, j, ρ π (i, j))) E( } + 4 E( Ŷ z (i, j, ρ π (k,j))) E( Ŷ z (k, j, ρ π (i, j))) 8 Ŷ z (k, j, ρ π (k,j))) 8

14 360 K. Laipaporn and K. Neammanee = C + C 3( = C + C 4 EŜ 8,z = O(). Ŷ z (i, j, ρ π (i, j))) 8) + C ( E( Hence E ( E B { K(t)dt}) = E ( E B { K(t)dt} ) I ( E B ( + E ( E B { E E B { + { E ( E B { K(t)dt} ) I ( E B ( K(t)dt} 6 ( + z) 6 ( + z) 6 O( )+ E( E B { ( + z) 6 O( ). 3 3 k=0 k i Ŷ z (i, j, ρ π (k,j))) 8) E( K(t)dt) < ) 6 ( + z) 6 K(t)dt} ) 4} { P ( E B ( K(t)dt}) 4 ( 3 ( + z) 3 ) K(t)dt) ) 6 ( + z) 6 K(t)dt) )} 6 ( + z) 6 4 Proof of the main result(theorem.) To bound P (W z) Φ(z), it suffices to consider z>0 because we use the fact that Φ(z) = Φ( z) and apply the result to W when z 0. So, from now on, we assume z>0. Let γ = 4 E Ŷ Ỹ 3. We note from Lemma 3.(3) that, γ = O( ). By [] we know that sup { P (W z) Φ(z) : <z< } = O( ). Hence P (W z) Φ(z) z>. Note that ( + z) r O( ) for 0 <z. Assume that P (W z) Φ(z) P (W Ŷ )+ P(Ŷ z) Φ(z), (4.) and ( P (W Ŷ )=P ) I( Y (i, j, ρ π (i, j)) > +z)

15 Normal approximation of orthogonal array 36 ( E ) I( Y (i, j, ρ π (i, j)) > +z) = P ( Y (i, j, k) > +z) = k=0 k=0 ( + z) 4 O( ). E Y (i, j, k) 4 ( + z) 4 Thus we need to bound only the second term on the right hand side of (4.). Case ( + z)γ. By Lemma 3.() and the fact that γ, we have +z P (Ŷ >z)=p (Ŷ +> +z) E Ŷ + ( + z) C ( + z) Cγ +z. Hence P (Ŷ z) Φ(z) = P (Ŷ >z) Φ(z) = P (Ŷ > z)+( Φ(z)) +z O( ) C where we have used the fact that Φ(z) ( + z) Cγ in the last +z ineuality. Case ( + z)γ <. In view of Theorem. of [], we can show that P (Ŷ z) Φ(z) = Eg z (Ŷ ) EŶg z (Ŷ ) T + T + T 3 + T 4 (4.) where T = E {g z (Ŷ ) g z (Ŷ + t)}k(t)dt, T = Eg z (Ŷ )E K(t)dt Eg z (Ŷ ) T 3 = Eg z (Ŷ ) Eg z (Ŷ )E K(t)dt, K(t)dt, T 4 = Δg z (Ŷ ), and Δf(Ŷ )= Ef(Ŷ ) Ŷ z (i,i,i 3 ) for any function f. i =0 i =0 i 3 =0

16 36 K. Laipaporn and K. Neammanee From Lemma 3.() and Lemma 3.3() we have E( i =0 i =0 i 3 =0 And, by (3.7) and (3.8), E( i,i,i 3 = Y (i,i,i 3 )) + E( j,j,j 3 Y (i,i,i 3 )Y z (j,j,j 3 )) i =0 i =0 i 3 =0 Y z (i,i,i 3 )) O(). EY (i,i,i 3 )Y z (i,i,i 3 ) μ(i,i,i 3 )EY z (i,i,i 3 ) i,i,i 3 i,i,i 3 = O(). Hence T 4 z E z {E( = From Lemma 3.(), E g z(ŷ ) E ([], p.6), we have T 3 = Eg z (Ŷ ) E K(t)dt and T = Eg z (Ŷ )EB {E i =0 i =0 i 3 =0 i =0 i =0 i 3 =0 Ŷ z (i,i,i 3 ) Ŷ z (i,i,i 3 )) } ( + z) O( ). (4.3) C ([7], p.48) and ( + z) K(t)dt = E(Ŷ ) ΔŶ where ΔŶ O( ) K(t)dt C ( + z) E Ŷ +ΔŶ ( + z) O( ). K(t)dt} C E K(t)dt + Eg ( + z) z (Ŷ )EB{ K(t)dt } C E Ŷ + ( + z) ΔŶ + Eg z (Ŷ )I(Ŷ z )EB{ K(t)dt } = + Eg z (Ŷ )I(Ŷ < z )EB{ K(t)dt } (4.4) ( + z) O( )+T + T (4.5)

17 Normal approximation of orthogonal array 363 where and T = Eg z(ŷ )I(Ŷ z )EB{ T = Eg z(ŷ )I(Ŷ > z )EB{ K(t)dt } K(t)dt }. Note that g z(ω) =[ πωe ω / Φ(ω) + ][ Φ(z)] for ω z. Hence from this fact, Lemma 3.5(),(3.7) and the fact that e 3z 8 C, we have +z T E { π( z z z )e 8 Φ( )+}{ Φ(z) } I(Ŷ z )EB { K(t)dt} { π( z )e }{e z z } 8 + E E B { K(t)dt} z ( + z) O( ). (4.6) From Lemma 3.() and Lemma 3.5(), T EI(Ŷ >z )EB{ { P (Ŷ > z )} { E E B{ = ( + z) ( + z) Hence, by (4.5)-(4.7), we have O( )EŶ 6 K(t)dt } K(t)dt } } O( ). (4.7) 6 T ( + z) O( ). (4.8) 6 To finish the proof of Theorem., it remains to bound T. By (3.5) we have T T + T + T 3 (4.9) where T = EI( Ỹ Ŷ < +z ) I(z max(0,t) < Ŷ<z min(0,t))k(t)dt, T = EI( Ỹ Ŷ < +z 0 ) h(ŷ + u)k(t)dudt, T 3 = EI( Ỹ Ŷ +z ) t {g z(ŷ ) g z(ŷ + t)}k(t)dt

18 364 K. Laipaporn and K. Neammanee and h be defined as in Lemma 3.4. First, we consider T.Forδ>0, let f δ : R R be defined by 0 if t<z δ, f δ (t) = ( + t + δ)(t z +δ) if z δ t z +δ, 4δ( + t + δ) if t>z+δ. Note that f δ is a nondecreasing function, { f δ(t) +z δ if z δ t z +δ, 0 otherwise, (4.0) (4.) E Ŷf ey by (Ŷ ) 4E Ŷ ( + Ŷ + Ỹ Ŷ ) Ỹ Ŷ 4E Ŷ Ỹ Ŷ +4EŶ Ỹ Ŷ +4E Ŷ Ỹ Ŷ 4{EŶ } {E Ỹ Ŷ } +4{EŶ } {E Ỹ Ŷ 4 } O( )+O( ) { EŶ Ŷ r +4{E(Ŷ ) r } r r r r } r O( )+O( ) { ( ( + z)) r r EŶ } r O( ) + ( + z) r O( r 8 r (Ŷ ) and Δf Y e Y b Ef Y e Y b (Ŷ ) {E Ỹ Ŷ r } r ), (4.) i =0 i =0 i 3 =0 Ŷ z (i,i,i 3 ) 4 E( + Ŷ + Ỹ Ŷ ) Ỹ Ŷ i =0 i =0 i 3 =0 4 E( + Ŷ ) Ỹ Ŷ Y (i,i,i 3 ) i =0 i =0 i 3 =0 + 4 E( + Ŷ ) Ỹ Ŷ Y z (i,i,i 3 ) i =0 i =0 i 3 =0 + 4 E Ỹ Ŷ Y (i,i,i 3 ) i =0 i =0 i 3 =0 + 4 E Ỹ Ŷ Y z (i,i,i 3 ) i =0 i =0 i 3 =0 Ŷ z (i,i,i 3 )

19 Normal approximation of orthogonal array { E( + Ŷ ) } { E Ỹ Ŷ ( + 4 { E( + Ŷ ) } { E Ỹ Ŷ ( + 4 { E Ỹ Ŷ 4} { E( + 4 { E Ỹ Ŷ 4} { E( i =0 i =0 i 3 =0 Y (i,i,i 3 )) } i =0 i =0 i 3 =0 i =0 i =0 i 3 =0 Y z (i,i,i 3 )) } Y (i,i,i 3 )) } Y z (i,i,i 3 )) } i =0 i =0 i 3 =0 4 { E Ỹ Ŷ 4} { 4 E( Y (i,i,i 3 )) 4} 4 i =0 i =0 i 3 =0 + 4 { E Ỹ Ŷ 4} { 4 E( i =0 i =0 i 3 =0 Y z (i,i,i 3 )) 4} 4 + O( ) O( ). (4.3) Using the same argument as in Lemma.3 of [],we can show that E f δ (Ŷ + t)k(t)dt = EŶf δ (Ŷ ) Δf δ (Ŷ ). (4.4) Thus, from (4.)-(4.4), we have T = E I( Ỹ Ŷ < +z )I(z max(0,t) < Ŷ<z min(0,t))k(t)dt t ey by E I( Ỹ Ŷ < +z )I(z Ỹ Ŷ < Ŷ<z+ Ỹ Ŷ )K(t)dt t Y e Y b +z E I( Ỹ Ŷ < +z )( + z Ỹ Ŷ ) t ey by I(z Ỹ Ŷ < Ŷ<z+ Ỹ Ŷ )K(t)dt +z E f (Ŷ + t)i(z Ỹ Ŷ < Ŷ<z+ Ỹ Ŷ )K(t)dt ey by t ey by E Ŷf ) Y e Y (Ŷ (Ŷ ) b + Δf Y e Y b O( ( + z) r By Lemma 3.4, r 8 r ). (4.5) T ( + z) r O( ). (4.6)

20 366 K. Laipaporn and K. Neammanee By (3.4) and Lemma 3.() T 3 EI( Ỹ Ŷ +z ) K(t)dt EI( Ỹ Ŷ +z ) Ỹ Ŷ { P ( Ỹ Ŷ +z )} { E Ỹ Ŷ 4} C { E Ỹ Ŷ } ( + z) = ( + z) O( ). (4.7) From (4.9), (4.5)-(4.7), we have the main theorem. References [] A.B. Owen, A central limit theorem for Latin Hypercube sampling, J.R. Statist. Soc. Ser. B, 54 (99), [] A.B. Owen, Orthogonal array for computer experiments, integration and visualization, Statist. Sinica, (99), [3] A.B. Owen, Monte-Carlo variance of scrambled net uadrature, SIAM J. Numer. Anal., 34 (997), [4] A.B. Owen, Scrambled net variance for integrals of smooth functions, Ann. Statist., 5 (997), [5] A.D. Barbour and L.H.Y. Chen, An Introduction to Stein s method, Singapore University Press, 005. [6] B. Tang, Orthogonal array-based Latin hypercubes, J. Amer. Statist. Assoc., 88 (993), [7] C.M. Stein, Approximate computation of expectations, IMS, Hayward, CA., 986. [8] D. Raghavarao, Constructions and combinatorial problems in design of experiments, John Wiley, New York, 97. [9] H.D. Patterson, The errors of lattice sampling, J.R. Statist. Soc. Ser. B, 6 (954),

21 Normal approximation of orthogonal array 367 [0] H. Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods, SIAM, Philadelphia, 99. [] K. Laipaporn and K. Neammanee, A non-uniform concentration in euality for randomized orthogonal array sampling designs, Thai. J. Math., 4, No. (006), [] K. Laipaporn and K. Neammanee, A uniform bound on a combinatorial central limit theorem for randomized orthogonal array sampling designs, (to appear in) Journal of stochastic analysis and application. [3] M.D. McKay, W.J. Conover and R.J. Beckman, A comparison of three methods for selecting values of input variables in the analysis of output from a computer code, Technometrics, (979), [4] M. Evans and T.Swartz, Approximating Integrals via Monte-Carlo and Deterministic Methods, Oxford Univ. Press, 000. [5] M.L. Stein, Large sample properties of simulations using Latin hypercube sampling, Ann. Statist., 9 (987), [6] P.J. Davis and P. Rabinowitz, Methods of Numerical Integration, nd ed. Academic Press, Orlando, FL., 984. [7] Q.A. Shao and L.H.Y. Chen, A non-uniform Berry-Esseen bound via Stein s method, Probab. Theory Relat. Fields, 0 (00), [8] W.L. Loh, A combinatorial central limit theorem for randomized orthogonal array sampling designs, Ann. Statist., 4 (996), Received: January 9, 007