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K(ζ) = 4ζ 2

x = 20

Θ = {θ i } Θ i=1 M = {m i} M i=1 A = {a i } A i=1 M A π = (π i ) n i=1 (Θ) n := Θ

Θ (a, θ) u(a, θ) E γq [ E π m [u(a, θ)] ] C(π, Q) Q γ Q π m m Q m π m a Π = (π) U M A Θ (R) u i,j := u(a i, θ j ) i j U Q Θ M D M A C : (Θ) Q R Q (Θ) (x) x R := R {, } π Q C( π, Q) = Q π Q ˆ C(π, Q) Q Q ˆ Q ˆ

(QDUΠ) C(π, Q) Q Q,D D Q q i,j = Pr(m j θ i ) m j θ i D d i,j = Pr(a j m i ) a j m i Q γ Q ( (Θ)) i Q θ i Q D (π, U) π U {U i } θ i a i {(U i, θ i, a i )} C(, ) (X) X

Q Q,D D n i=1 π i A j=1 u j,i M q i,k d k,j C(π, Q) k=1 (i, j) j=1 M k=1 q i,k d k,j = Pr(a j θ i ) j i [ A M u j,i q i,k d k,j ] i k=1 [ [ n A M π i u j,i q i,k d k,j ]] i {1,..., n} j {1,..., A } π i u j,i i {1..., n}, j {1,..., A } k {1,..., M } q i,k d k,j q i,k d k,j q i,k d k,j i=1 j=1 k=1

π C(π, Q 1 ) = C(π, Q 2 ) Q 1 Q 2 {(U i, θ i, a i )} C(π, Q) Q γq γ Q C(π, Q) := C(π, R) R Q γq C

π U 0, U 1,..., U J 1 Q 0, Q,..., Q J 1 D 0, D 1,..., D J 1 D j i U i Q j J 1 J 1 ( (Q j D j U j Π) Q (j+1) mod J D j=0 j=0 (j+1) mod J j ) U j Π π U Q D k {1,..., A } k

D d,k l {1,..., A } u k, ΠQ d,k u l, ΠQ d,k u k, u l, k l U U = {U 1, U 2,... U J } U, Ũ U i {1,... n} τ i u i,j i {1,... n} ũ i,j u i,j j = τ i ũ i,j u i,j j τ i j {1,..., A }

( Pr ) Pr ( a u(z, θ) z A a z A ) ũ(z, θ) > C(π, Q) R C R

π (Θ), λ (0, 1), Q 1, Q 2 Q C(π, λq 1 + (1 λ)q 2 ) λc(π, Q 1 ) + (1 λ)c(π, Q 2 ) π (Θ), λ (0, 1), Q 1, Q 2 Q C(π, λq 1 + (1 λ)q 2 ) λc(π, Q 1 ) + (1 λ)c(π, Q 2 ) Q 1 Q 2 2 2 Q 1 = Q 3 = 0.5Q 1 + 0.5Q 2 = 0.5 0.5 0.5 0.5 0.25 0.75 0.25 0.75 Q 2 = 0.75 0.25 0.75 0.25

Q 3 Q 1 Q 2 Q 1 Q 2 Q 3 Q 1 Q 2 R M M π (Θ) Q Q C(π, Q) C(π, QR) Q R Q QR QR Q Q QR Q QR Q 1 Q 2 R Q 1 R = Q 2 Q 2 R = Q 1

C C R M M Q Q, C(π, Q) = C(π, QR) π C U

{c i,j (, )} i {1,..., n}, j {1,..., M } (Θ) [0, 1] π c i,j (π, ) R 2 c i,j (π, ) q 2 > 0 q (0, 1), i {1,..., n}, j {1,..., M } C(π, Q) := i,j c i,j(π, q i,j ) n r

r r n Pr(a = x θ = x) C(π, Q) x θ a C(π, Q) Q C(π, Q)

A = Θ U = ri n r > 0 r(qdπ) P := (QDΠ) = 1 (QD) n r Q D P (r) P (r) Q D D Q D P (r) = 1 n M q i,j i {1,...,n} j=1 P (r) P (r)

x A, y Θ, Pr(θ = x a = x) Pr(θ = y a = x)

C P (r) rp (r) Θ = {1, 2, 3, 4, 5} θ = 2 Θ

ρ Θ x, y, z Θ, ρ(x, y) > ρ(x, z) = Pr(a = y θ = x) < Pr(a = z θ = x) ρ Θ Θ ρ(x, y) = x y C : (Θ) Q R

I H I(π, Q) := α(h(π) E[H(π Q)]) n = α π i ln π i n π i M i=1 i=1 j=1 ( ) π i q i,j q i,j ln n k=1 p kq k,j α > 0 0 ln 0 = 0 0 ln 0 0 = 0 π i = 1 i {1,..., n} Θ n C(π, Q) = I(π, Q) := α(h(p) E[H(π Q)]) α > 0 C P (r) = exp ( α) r n 1+exp( α) r 1 n 1+exp( r α) Q 1 = 1 4 1 4 1 4 1 4 1 16 1 8 3 16 1 4 1 8 3 16 1 4 1 16 9 16 7 16 5 16 7 16 Q 2 = 1 8 1 8 1 8 1 8 7 96 7 48 7 32 7 24 7 48 7 32 7 24 7 96 21 32 49 96 35 96 49 96 Q λ := λq 1 + (1 λ)q 2, λ [0, 1] I(π, Q λ ) = λi(π, Q 1 + (1 λ)i(π, Q 2 )

α θ Q, Q Q = QR R Q Q C(Q) = 0, Q = Q κ, Q = Q, Q

κ = 30 q = 0.3 q = 0.8 κ Q Q Q q q κ q κ r q κ rq r κ q q q r > κ q q q

Θ R θ 1 < θ 2 <... < θ n ˆm N(θ, σ 2 ) θ ζ 2 := σ 2 K(ζ) K ˆm θ Pr(θ ˆm) = ϕ ( ) ˆm θ σ σ ϕ ( ) ˆm θ i = σ σ 1 1 n n 1 1 i=1 n n i=1 1 ϕ ( ) ˆm θ σ σ 1 ϕ ( ) ˆm θ i σ σ ϕ( ) q q Q Q Θ θ 1 θ n K

ˆm θ θ θ 1 ˆm 1 2 (θ 1 + θ 2 ) θ i ˆm [ 1 2 (θ i 1 + θ i ), 1 2 (θ i + θ i+1 ) ] i {2, 3,..., n 1} θ n ˆm 1 2 (θ n 1 + θ n ) i j = Pr(a = θ i θ = θ j ) Pr ( ˆm 1 2 (θ 1 + θ 2 ) θ = θj ), i = 1 Pr ( ˆm [ 1 2 (θ i 1 + θ i ), 1 2 (θ i + θ i+1 ) ] θ = θj ), i {2, 3,..., n 1} Pr ( ˆm 1 2 (θ n 1 + θ n ) θ = θ j ), i = n = Pr(a = θ i θ = θ j ) Φ ( ζ ( 1 2 (θ 1 + θ 2 ) θ j )), i = 1 Φ ( ζ ( 1 2 (θ i+1 + θ i ) θ j )) Φ ( ζ ( 1 2 (θ i + θ i 1 ) θ j )), i {2, 3,..., n 1} Φ ( ζ ( 1 2 (θ n 1 + θ n ) θ j )), i = n Φ

ζ [0, ) [ r Φ n ( ) 1 2 ζ(θ n 1 1 θ 2 ) + +1 Φ i=2 ( ( 1 Φ ( )] 1 2 ζ(θ n 1 θ n ) ) 2 ζ(θ i+1 θ i ) K(ζ) ( )) 1 Φ 2 ζ(θ i 1 θ i ) δ θ i θ i 1 = 2δ i 2 ζ [0, ) r [2Φ (ζδ) + (n 2) (2Φ (ζδ) 1)] K(ζ) n θ 1 ˆm 1(θ 2 1 + θ 2 ) Pr ( ˆm 1(θ 2 1 + θ 2 ) ) ( [ θ = θj > Pr ˆm 1 (θ 2 1 + θ 2 ), 1(θ 2 2 + θ 3 ) ] ) θ = θj j 2 θ 2 θ n 1

K(ζ) = 4ζ 2 x Θ, y, z Θ \ {θ 1, θ n }, x y > x z = Pr(a = y θ = x) < Pr(a = z θ = x) K( ) {m 1, m 2,..., m n } K

Q q i,j = Pr(m = m i θ = θ j ) = Φ ( ζ ( 1 2 (θ 1 + θ 2 ) θ j )), i = 1 Φ ( ζ ( 1 2 (θ i+1 + θ i ) θ j )) Φ ( ζ ( 1 2 (θ i + θ i 1 ) θ j )), i {2, 3,..., n 1} Φ ( ζ ( 1 2 (θ n 1 + θ n ) θ j )), i = n C(π, Q) = K(ζ) Q r(qdπ) (QDΠ) K : [0, 1] R C(π, Q) = K ( M j=1 i {1,...,n} π i q i,j ) Q q i,i = q i {1,..., n} q [ 1 n, 1] q K K ( 1 n) < r < K (1) q r = K (q) q i,j = 0 j > n q = P (r) Q D

A q q K(q) K r = K (q) P (r) = (K ) 1 (r)

n = 81 n = 80 n = 80 n = 81 n = 80 p = 0.146 t r t θ t a t y t := {θt} (a t ) y t t

Monotone Nonparametric Regression: Subject 3 Correct 0.0 0.2 0.4 0.6 0.8 1.0 0 20 40 60 80 100 Incentive Level P t := Pr(y t = 1 r t ) = Pr(a t = θ t r t ) t

2 < < <

(r 1, r 2 ) r 1 > r 2 P (r 1 ) P (r 2 ) r 1 r 2

Rejects NIAC (Lab Subject 1) Fails to reject NIAC (Lab Subject 19) Correct 0.0 0.2 0.4 0.6 0.8 1.0 Correct 0.0 0.2 0.4 0.6 0.8 1.0 0 20 40 60 80 100 0 20 40 60 80 100 Incentive Level Incentive Level

Pr(θ a)

Fails to Reject Non Responsiveness (Lab Subject 32) Rejects Non Responsiveness (Online Subject 26) Correct 0.0 0.2 0.4 0.6 0.8 1.0 Correct 0.0 0.2 0.4 0.6 0.8 1.0 0 20 40 60 80 100 0 20 40 60 80 100 Incentive Level Incentive Level >

P t = β 0 + β 1 1 δ (r t ) β 0 β 1 δ P t = β 1 1 + exp( λ(r t δ)) + β 0 λ λ = β 1

{38,..., 42} t ρ(a t, θ t ) = a t θ t {1, 2, 3, 4}

P t = β 0 + β 1 1 δ (r t )

P t = β 0 + β 1 1 δ (r t ) + β 2 r t + β 3 r t 1 δ (r t ) δ q = ˆβ 0 q = ˆβ 0 + ˆβ 1 δ κ κ = ˆδ( ˆβ q q 1 ˆβ 0 ) K r 1 r 2

r 2P K(P ) r1p P P t = β 0 + β 1 r t

K(P ) K(P ) = 1 2 ˆβ 1 (P 2 0.04) ˆβ 0 ˆβ 1 (P 0.2) K ( 1 n) = 0 P t = 1 4 exp ( r t α ) + 1 ( ( ) ) 4P K(P ) = ˆα P ln + ln(1 P ) + ln(1.25) 1 P ζ 2 K(ζ) = αζ 2 α

P t = 8 ( ) ζ 5 Φ (r t ) 3 2 5 ζ αζ = 2 ( ) ζ 5 r tϕ 2

Dots: subject 6 Binary in red Correct 0.0 0.2 0.4 0.6 0.8 1.0 0 20 40 60 80 100 Incentive Level

Dots: subject 14 Logistic in red Correct 0.0 0.2 0.4 0.6 0.8 1.0 0 20 40 60 80 100 Incentive Level

Dots: subject 66 Concave (Normal) in red Correct 0.0 0.2 0.4 0.6 0.8 1.0 0 20 40 60 80 100 Incentive Level

< < <

2 < < < 2 < < <

n x

r r (x r)p (r) r [0,x] P (r) x r

κ r κ r κn r n n 1 x < κn n 1 x κn n 1 r = κn n 1 x x κn x κn2 n n 1 x x < x κn2 (n 1) 2 κn < (n 1) 2 n 1 κn2 (n 1) 2 κn2 (n 1) 2 κn2 (n 1) 2 κn n 1 x κ = 40 α r

x r (n 1) exp ( r α ) + 1 x r x r (x) n = 5 α = 10 x [5, 100] r

x = 20 r (x) r x

r (x)

κ = 40 α = 10

µ ν λ ρ l r λ ρ m λ m ρ

m s s m s y s y

S λ ρ R R l r l r S: 10 15 15 10 R: 10 0 0 10 N λ (0.5) ρ (0.5) R R l r l r S: 10 15 15 10 R: 10 0 0 10

S: R: N µ (0.5) ν (0.5) S N λ ρ λ (0.5) ρ (0.5) S S S S mλ mρ mλ mρ mλ mρ mλ mρ R R R R R R R R l r l r l r l r l r l r l r l r 10 10 15 10 15 15 10 15 10 10 10 10 10 15 10 15 15 10 15 0 0 0 0 10 0 10 0 0 10 0 10 10

l r m λ Pr(l m λ ) > Pr(r m λ ) Pr(l m ρ ) [Pr(r m λ ), Pr(l m λ )] ρ m λ m λ ρ m ρ λ r m λ l m ρ λ ρ Pr(λ µ) > Pr(ρ µ) Pr(λ) > Pr(ρ) m x Pr(λ m x ) > Pr(ρ m x ) m λ m λ l Pr(λ m ρ ) Pr(ρ m ρ ) m ρ l ρ m λ Pr(λ m ρ ) < Pr(ρ m ρ ) m ρ r ρ l m λ

Pr(λ µ) = Pr(ρ µ) = 1 2 Pr(λ) = Pr(ρ) = 1 2 Pr(λ m λ ) = Pr(λ m ρ ) = 1 Pr(λ) = Pr(ρ) = 1 2 2 1 2 = Pr(λ m λ ) = Pr(m λ λ) Pr(λ) Pr(m λ λ) Pr(λ) + Pr(m λ ρ) Pr(ρ) Pr(m λ λ) Pr(m λ λ) + Pr(m λ ρ) Pr(m λ λ) = Pr(m λ ρ) Pr(m ρ λ) = Pr(m ρ ρ) Pr(λ) Pr(m λ λ)+pr(ρ) Pr(m ρ ρ) Pr(λ) Pr(m ρ λ)+ Pr(ρ) Pr(m λ ρ) Pr(m λ λ) = Pr(m λ ρ) Pr(m ρ λ) = Pr(m ρ ρ) m ρ Pr(λ m ρ ) = 1 2 m ρ ρ 1 2 m ρ λ 1 2

c m x y x y f(x) f(λ) := l f(ρ) := r Pr (y m x ) y m x Pr (λ m λ ) = Pr (ρ m ρ ) =: α

10α 10(1 α) α 1 2 β := Pr S ( α 1 2) α β c 10β 5 c 10β 5 β 15β + 10(1 β) c 10β + 15(1 β) c 10β 5 (15 c)β + 10(1 β) 10β + 15(1 β) c 10β 5 β

u i i {R, S} (π, π ) u S (π, π ) = π + γπ, γ > 0 u A := u(15, 0) u B := u(10, 10) u A u B β 1 u 2 A u B β 1 2 u A β + u B (1 β) u A (1 β) + u B β

u A u B β 1 2 u A u B β 1 2 u = π θ max{0, E α [π ] π } u π π α

θ 1 2 10(1 β) + (15 10θβ)β 10β + (15 10θ(1 β))(1 β) θ 1 2 c < 5 c > 5

ζ := Pr(l m λ ) = Pr(r m ρ ) Pr(λ µ) = Pr(ρ µ) = 1 2 ϕ := Pr(m λ λ, c = 0) = Pr(m ρ ρ, c = 0) ξ = Pr(c = ) ξ 1 2 ζ = 1 ϕ = 0 ξ 1 2

ζ = 1 1 2ξ ϕ = 2 2 2ξ 15ζ + 10(1 ζ) 10ζ + 15(1 ζ) ζ 1 2 10ξ + 10(1 ξ)ϕ 10(1 ξ)(1 ϕ) ξ ξϕ + ϕ 1 2 ϕ [0, 1] ξ 1 2 ϕ = 0 ξ 1 ϕ ϕ = 0 2 ξ = 1 2 ζ = 1 2 ϕ = 1 2ξ 2 2ξ

ξ < 1 2 1 2ξ 1 ξ µ ν

S knows order; R does not S knows order; R does not Round 1 Round 2 Scenario 1 Scenario 2 OR Scenario 2 Scenario 1 Scenario 1 Scenario 2 OR Scenario 2 Scenario 1 match rematch rematch switch roles rematch

n = 108 52.8% 47.2% 52.8% 13.0% 34.3% 14.8% p = 0.630

χ 2 = 91.9 p < 0.001 p = 0.0053 D = 0.052 p = 0.029

< < <

χ 2 = 78.5 p < 0.001 p = 1.000 55.4% p = 0.0048 D = 0.067 p < 0.001

< < <

p < 0.001

R 2 R 2 < < <

I i = 1,..., I J j = 1,..., J a i,j j i A I J a i := (a i,j ) J j=1

i i u i := u( a i ) := J j=1 a i,j F i,j f i,j F i,j j f i,j = f k,j f f i u i E[(u i û i ) 2 ] û i u i J E[(a i,j â i,j ) 2 ] j=1 â i,j a i,j

r i,j g i ( r i a i ) r = ((r i,j ) J j=1 C(f i, g i ) f g [ C(f i, g i ) = κ + r i f i ( a i ) ln(f i ( a i ))d a i a i ( g i ( r i a i )f i ( a i ) ln a i g i ( r i a i )f i ( a i ) bi g i ( r i b i )f i ( b i )d b i ) d r i d a i ] κ > 0 C(f i, g i ) [ J = κ f i,j (a i,j ) ln(f i,j (a i,j ))da i,j j=1 a i,j ( ) ] g i,j (r i,j a i,j )f i,j (a i,j ) + g i,j (r i,j a i,j )f i,j (a i,j ) ln dr i,j da i,j r i,j a i,j b i,j g i,j (r i,j b i,j )f i,j (b i,j )db i,j g i,j r i,j

g i E[(u i û i ) 2 ] + C(f i, g i ) â i,j h i,j (a i,j, r i,j ) := g i,j (r i,j a i,j )f i,j (a i,j ) b i,j g i,j (r i,j b i,j )f i,j (b i,j )db i,j a i,j r i,j â i,j u i û i = J j=1 â i,j û i â i,j α i,j := σ2 f i,j σ 2 h i,j σ 2 f i,j j i σ 2 q q

] σh 2 i,j [0, σ 2fi,j α i,j i u( a i ) = J j=1 a i,j u( a i, A) = J γ j ( j (A))a i,j j=1 γ j j (A) := max j a i,j min j a i,j j

Φ ϕ ( h i,j = f i,j h i,j (a i,j r i,j ) = 1 ϕ a µhi,j ) µ σ hi,j σ = κ 2 σ σ hi,j = min { κ 2, σ f i,j }

α i,j = 1 σf 2 2 i,j κ h a i,j N(µ j, σj 2 ) σ j > κ 2 i â i,j = 2σ2 j κ r 2σj 2 i,j + κ µ 2σj 2 j r i,j µ j σj 2 û i = J γ(σ j 2 )r i,j j=1 γ(x) := 2x κ 2x γ(σ2 j ) γ j ( j (A))

i u( a i ) = J j=1 a i,j u( a i, A) = J ω i,j a i,j j=1 ω i,j a i,j ā j s(, ) ϵ, η 0 a i,j > ā j s(a i,j + ϵ, ā j η) > s(a i,j, ā j ) a i,j < ā j s(a i,j ϵ, ā j + η) > s(a i,j, ā j ) a i,j, ā j 0 ϵ > 0 s(a i,j + ϵ, ā j + ϵ) > s(a i,j, ā j )

s(a i,j, ā j ) ρ i,j J ω i,j ω i,j = δ r i,j J l=1 δr i,l δ (0, 1)

j a d 1 j (1),j <... < a d 1 j (I),j d j I j f ) ad i,j (a i,j ) := f i,j (a i,j 1 j (1),j <... < a d 1 (I),j (i, j) I = 3 j f i,j d j

N(µ j, τj 2 ) τ j > κ 2 τ j = τ j V E,j j V M,j j ( V E,j = τ (1 2 9 2 3 ) V 4π M,j = τ 2 1 ) 3 V π M,j < V E,j V E,j V M,j j

γ j

{(U i, θ i, a i )} C(π, Q) C Q γq γ Q C(π, Q) := C(π, R) R Q γq γ Q Q γq C C C b : Q ( (Θ)) b ( (Θ)) {( ) n πsq (b(q)) = s,k n l=1 π lq l,k k {1,... M }, } n l=1 π lq l,k > 0 s=1 ζ (b(q)) n k Q ζ l=1 π lq l,k Q ζ Q ζ Q 1, Q 2,... Q Q lim b(q j ) = b(q) j lim b(q j )(X) = j b(q)(x) X σ (Θ) X X (b(q)) = X (b(q)) = (X) (b(q)) X (b(q)) = J N b(q j )(X) = 0 j > J lim b(q j )(X) = b(q)(x) = 0 J N, j > J j X (b(q j )) lim b(q j )(X) 0 j X X

ε > 0 J N, j > (( J ) b(q j )(X) > ε n ) πsq Q jh s,k n l=1 π lq l,k k s=1 j h (X) b(q)(x) = 0 X b(q)( X) > 0 X lim b(q j )(X) = b(q)(x) j (X) (b(q)) (Q j ) (Q j ) ((z k ) j ) := (( n ( l=1 π lq l,k ) j ) z k ((y k ) j ) := (( ) n ) ) πs q s,k n l=1 π lq l,k y k z k > 0 s=1 j z k = 0 ( ) π πs q (Q j ) (Q j ) y k = s,k n l=1 π lq l,k z k = n l=1 π lq l,k q l,k Q K {1,..., M} {((y k ) j ) k K} (X) (X) ε > 0 k K, N k j > N k, (y k ) j (X) N = N k j > N, b(q j )(X) ( k K ( n l=1 π lq l,k )) j k K b(q j )(X) k K ( n l=1 π lq l,k ) j j J N j > J (Q jh ) b(q jh )(X) k K ( n l=1 π lq l,k ) jh > 0 j h j h k K (y k ) jh (b(q jh )) M k ((y k ) jh ) (X) y k (X) k K y k X [ X ((y k ) jh ) z k = 0 lim b(qjh )(X) k K ( n l=1 π ] h lq l,k ) jh = 0 b γ ( (Θ)) b 1 ({γ}) b 1 ({γ}) Q Q R n R M b 1 ({γ}) Q γq C π Q γq C k Q jh k (y k ) j k j (z k ) j = 0 j (Q j ) j j (z k ) j (Q j ) (z k ) j 0 j, k (z k ) j

D I A 1 0 0 D := 1 0 0 I A A A D ( M A ) A Q Q D (QDUΠ) D δ : {1,..., M } {1,..., A } δ(j) j D j Q a δ(j) Q Q δ k {1,..., A } k Q j δ 1 (k) q,j q,j j Q k > A Q (Q, D) (Q, D) Pr(a θ) a A, θ Θ (i, j) Q D q i,j = k δ 1 (j) q i,k j A q i,j = Pr(a j θ i ) D j j j A k δ 1 (j) q i,k = Pr(a j θ i ) D j δ 1 (j) (QDUΠ) = (Q DUΠ) (Q, D) (Q, D) M M P p i,j p i,δ(i) = 1 i P i δ(i) Q = QP P C(π, Q ) C(π, Q) Q π Q Q Q Q D Q Q Q D Q Q (Q DUΠ) C(π, Q)

F (Q) F (Q) Q U Q Q (U) U F (Q) F (Q) U Q (U) Q 1, Q 2 Q (U) F (Q 1) = F (Q 2) Q 1 Q 2 λ (0, 1), F (λq 1+(1 λ)q 2) λf (Q 1)+(1 λ)f (Q 2) = F (Q 1) Q 1 Q 2 Q 1 Q 2 λ (0, 1) F (λq 1 + (1 λ)q 2) = λf (Q 1) + (1 λ)f (Q 2) λ Q λ Q (U) Q λ Q 1 Q 2 (Q λ DUΠ) = (Q 1 DUΠ) = (Q 2 DUΠ) (Q (U) DUΠ) U C c i,j C(π, Q) n M n M 2 c i,j (π, ) qi,j 2 C π r 1 r 2 Q i r i i = 1, 2 D j i Q i r j i, j = 1, 2 D i := Di i = D i i i = 1, 2

r 1 (Q 1 D 1 Π) + r 2 (Q 2 D 2 Π) r 2 (Q 1 D 1 Π) + r 1 (Q 2 D 2 Π) = r 1 P (r 1 ) + r 2 P (r 2 ) r 2 P (r 1 ) + r 1 P (r 2 ) = (r 1 r 2 )[P (r 1 ) P (r 2 )] 0 r 1 r 2 P (r 1 ) P (r 2 ) r 1 r 2... r N P 1 P 2... P N P i := P (r i ) P (r i, P σ1 (i)) N i=1 σ 1 σ 2 1, i = 1 σ 2 (i) := σ 1 (1), i = σ 1 1 (1) σ 1 (i), σ 2 σ 1 N r j P σ2 (i) j=1 N r j P σ1 (i) j=1 = r 1 P 1 + r σ1 1 (i)p σ1 (1) ( ) r 1 P σ1 (1) + r σ1 1 (i)p 1 = ( ( ) r 1 r σ1 (i)) 1 P1 P σ1 (1) 0, r 1 r σ1 1 (i) P 1 P σ1 (1) j 2 σ j+1 j, i = j σ j+1 (i) := σ j (j), i = σ 1 j (j) σ j (i), N N 1 σ N (i) = i (r i, P σ1 (i)) N i=1

x A y Θ Pr(θ = x a = x) Pr(θ = y a = x) r Pr(θ = x a = x) + 0 Pr(θ = z a = x) z x r Pr(θ = y a = x) + 0 Pr(θ = z a = x) z y u(x, z) Pr(θ = z a = x) u(y, z) Pr(θ = z a = x) z Θ z Θ Pr(a = x θ = z) Pr(θ = z) u(x, z) z Θ Pr(a = x) Pr(a = x θ = z) Pr(θ = z) u(y, z) z Θ Pr(a = x) u k, ΠQ d,k u l, ΠQ d,k, x y k l Θ (i, j) Q D Pr(a j θ i ) x y E[H(π Q)]) = γ Q (ˆπ)H(ˆπ) ˆπ (γ Q ) γ Q γ Q Q I(π, Q) = α(h(π) E[H(π Q)]) Q P P π Q QP I(π, Q) I(π, QP ) I x ln y = 0 lim x ln y (x,y,z) (0,0,0) z = 0 lim (x,y) (0,0)

Q( ) D j {1,..., M } π i qi,j i {1,...,n} d,j (Q D Π) = M j=1 i {1,...,n} π i q i,j n j, j {1,..., M }, G j,j := π i qi,j π i qi,j i {1,...,n} i {1,...,n} Q Q Q Q j j q q,j + q,j q j π i qi,j = G j,j π i q i {1,...,n} i {1,...,n},j = i,j = π i qi,j + π i qi,j Q i {1,...,n} i {1,...,n} Q M j=1 i {1,...,n} π i q i,j = M j=1 i {1,...,n} π i q i,j Q Q j > n q,j j n q,j = { M } q k π i qi,j =j,k i {1,...,n} M j=1 i {1,...,n} π i q i,j = j=1 i {1,...,n} π i q i,j D D (Q D Π) = (Q D Π) (Q D Π) = M π i q i,j = n π i q i,i j=1 i {1,...,n} i=1 ( n C(π, Q ) = K Q Q r π i q i,i i=1 ) ( n n π i q i,i K π i q i,i i=1 i=1 ) π i q i,i = k π k q k,i i n q rq K(q) q i,i q i n q 1 q 1 n K q i,i r = K ( n i=1 π i q i,i )

K Q (q i,i) n i=1 Q q i,i = q i n q := (K ) 1 (r) q i,j = 1 q j i, i n n 1 P (r) = n π i q i,i = q n π i = q = (K ) 1 (r) i=1 r K i=1 β ζ δ Φ((ξ + β)ζδ) Φ(ξζδ) ξ ξ ξ ξ ζδ[ϕ((ξ + β)ζδ) ϕ(ξζδ)] ξ ξ ξ = 2k + 1 β = 2 k 1 ξ = 2k+3 θ n 1 θ n θ n 1 θ n Φ(ζδ) Φ(0) > Φ(2ζδ) Φ(ζδ) Φ(ζδ) Φ(0) > Φ(3ζδ) Φ(2ζδ) = 2[Φ(ζδ) Φ(0)] > Φ(3ζδ) Φ(ζδ) = Φ(ζδ) Φ( ζδ) > Φ(3ζδ) Φ(ζδ) Φ(ζδ) Φ( ζδ) ζ [0, ) r [2Φ (ζδ) + (n 2) (2Φ (ζδ) 1)] K(ζ) n

ζ [0, ) r [(2n 2)Φ (ζδ) (n 2)] K(ζ) n F (r, ζ) (2n 2)rδ ϕ(ζδ) K (ζ) = 0 n (2n 2)rδ3 ζϕ(ζδ) K (ζ) < 0, n ζ P (r) = 1 [(2n 2)Φ(ζ(r)δ) (n 2)] n P (r) d 2 P = dp [ (2n 2)δ ϕ(ζδ) dζ ] dr 2 dr n dr (2n 2)δ3 = ζϕ(ζδ) dζ (2n 2)δ + ϕ(ζδ) d2 ζ n dr n dr 2 dζ d2 ζ dr dr 2 dζ dr = F r F = > 0 ζ (2n 2)δ n (2n 2)rδ 3 n ϕ(ζδ) ζϕ(ζδ) + K (ζ)

r [ ] d 2 ζ (2n 2)rδ 3 2 dr = ζϕ(ζδ) + K (ζ) 2 n (2n 2)δ3 { ζϕ(ζδ) dζ ( ) (2n 2)rδ 3 ζϕ(ζδ) + K (ζ) n dr n [( (2n 2)δ 3 (2n 2)rδ3 dζ ζϕ(ζδ) + n n dr ϕ(ζδ) (2n 2)rδ5 ζ 2 dζ ) n dr ϕ(ζδ) + K (ζ) ( )]} (2n 2)δ ϕ(ζδ) n [ ] (2n 2)rδ 3 2 = ζϕ(ζδ) + K (2n 2)δ3 (ζ) { ζϕ(ζδ) dζ n n dr K (ζ) [( ) (2n 2)δ 3 (2n 2)rδ3 dζ ζϕ(ζδ) + n n dr ϕ(ζδ) + K (ζ) ( )]} (2n 2)δ ϕ(ζδ) n < 0 d 2 ζ dr 2 d2 P dr 2 < 0 < 0 dζ dr > 0 x r (n 1) exp ( r α ) + 1 r x x r r (x) r ) ( (n 1) exp r ( x r α α ( (n 1) exp ( r α α) 1 ) ) 2 + 1

( ) x r α ( (n 1) exp r ) > (<)1 α α ( ) x r α ( r (n 1) > (<) exp α α) r r r r r (n 1) ( ) ( x r α α = exp r ) α d 2 dr 2 [(x r)p (r)] = d dr [ P (r) + (x r) d dr P (r)] = 2 d dr P (r) + (x r) d2 dr 2 P (r) P (r) x > r r r x r (x) 35 40 45 50 55

2 10 5 < < <

n = 118 n = 117 n = 118 n = 118 p < 0.001 p = 0.003 p < 0.001 p = 0.009

j i E[(a i,j â i,j ) 2 ] + C(f i,j, g i,j ) g r i,j(r i,j )dr i,j = 1 a R i j [ ( ) ] 2 L = r + r a 2 a a g(r a)f(a) b g(r b)f(b)dbda a g(r a)f(a) b g(r b)f(b)dbda g(r b)f(b)db dr b ( ) g(r a)f(a) κg(r a)f(a) ln g(r b)f(b)db da dr + λ(a)g(r a)f(a)da dr b r a λ(a) = ν(a)/f(a) ν(a) η(r, a) lim r η(r, a) = lim r η(r, a) = lim a η(r, a) = lim a η(r, a) = 0 λ(a)

J(ε) := r [ + + r r a 2 (g(r a) + εη(r, a))f(a)da ( a a a a a((g(r a) + εη(r, a))f(a))2 g(r a)f(a)da a ( (g(r a) + εη(r, a))f(a) κ(g(r a) + εη(r, a))f(a) ln (g(r b) + εη(r, b))f(b)db b ] λ(a)(g(r a) + εη(r, a))f(a)da dr ) da J ε ε=0 [ ( ( ) g(r a)f(a) 0 = η(r, a)f(a) a 2 + κ ln r a g(r b)f(b)db b ) ( a + η(r, a)f(a)da (ag(r a)f(a)da)2 a a ( (g(r a)f(a)da)2 2 aη(r, a)f(a)da ag(r a)f(a)da ) a a g(r a)f(a)da a ] g(r a)f(a) + κη(r, b)f(b)db b g(r b)f(b)dbda dr a b ) + κ + λ(a) da η eta ln h(a r) = λ(a) κ (a µ h) 2 κ h µ h ( h(a r) = 1 σ 2π exp 1 ( ) ) 2 a µh 2 σ σ = κ λ(a) = κ ln( κπ) 2 µ h κ 2 h κ < 2 σ2 f σ2 f a i,j r i,j a i,j a i,j r i,j a N(µ, σ 2 f ) r N(a, σ2 g)

a r N ( σ 2 f r+σ 2 g µ σ 2 f +σ2 g ), σ2 f σ2 g σf 2+σ2 g σh 2 κ 2 κ = σ2 f σ2 g 2 σf 2+σ2 g σ 2 g = κσ2 f 2σ 2 f κ µ h := σ2 f r+σ2 gµ σ 2 f +σ2 g 2σf 2 κ r + 2σ 2 f κ 2σ 2 f µ x y z x > y > z x, y, z N(µ, σ 2 ) ϕ Φ x f(x x > y > z) f(x x > y > z) = Pr(x > y > z) x y f(x, y, z)dz dy = 1, x y z 6 x y ( ) ( ) ( ) 1 x µ y µ z µ = 6 σ ϕ ϕ ϕ dz dy 3 σ σ σ x ( ) ( ) ( ) 1 x µ y µ y µ = 6 σ ϕ ϕ Φ dy 2 σ σ σ = 6 1 ( ) x µ Φ ( x µ σ ϕ σ ) t dt, 2 σ = 3 ( ) ( ) x µ x µ σ ϕ Φ 2 σ σ

y f(x x > y > z) y ( ) ( ) ( ) 1 x µ y µ z µ = 6 y σ ϕ ϕ ϕ dz dx 3 σ σ σ = 6 1 ( ) ( ) [ ( )] y µ y µ y µ σ ϕ Φ 1 Φ σ σ σ = 6 ( ) [ ( ) ( )] y µ y µ y µ σ ϕ Φ Φ 2 σ σ σ z f(z x > y > z) ( ) ( ) ( ) 1 x µ y µ z µ = 6 x y σ ϕ ϕ ϕ dy dx 3 σ σ σ = 3 ( ) [ ( )] 2 z µ z µ σ ϕ 1 Φ σ σ µ = 0 x 3 = 3σ = 3σ [ ϕ = 3σ π x ( x ) ( x ) σ ϕ Φ 2 dx σ σ x ( x ) ( x ) σ ϕ Φ 2 dx 2 σ σ ( x ) ( x ) Φ 2 +2 σ σ ( ) 2 2x σ ϕ Φ σ ( x σ ) dx 1 ( x ) ( x ) ] σ ϕ2 Φ dx σ σ = 3σ 2, π z 3σ 2 π y

x x 2 ( x ) ( x ) 3 σ ϕ Φ 2 dx σ σ ( x = 3σ 4 2 ( x ) σ ϕ 1 ( x ) ) ( x ) 5 σ σ ϕ Φ 2 dx + 3σ 4 3 σ σ [ = 3σ 4 x ( x ) ( x ) σ ϕ Φ 2 2x ( x ) ( x ) + 3 σ σ σ 4 ϕ2 Φ dx σ σ ( ) = [ ϕ 3σ2 2x ( x ) Φ + 2π σ σ 3σ = σ 2 2 ( ) 3 2x + 2π σ ϕ dx σ ) 3 = σ (1 2 + 2π 1 σ ϕ ( ) 2x σ 1 ( x σ ϕ 3 σ ] + σ 2 ( x ) ϕ dx σ ] ) ( x ) Φ 2 dx σ + σ 2 z x z ) 3 V (x) = V (z) = σ (1 2 + 2π 9 4π y V (y) y 2 ( y = 6 σ ϕ σ ( y = 6σ 4 2 ) [ ( y ) ( y )] Φ Φ 2 dy σ σ σ 5 ϕ ( y σ 2σ 2 (1 + ) 3 2π 1 ( y ) ) ( y ) σ ϕ Φ dy + 6σ 4 3 σ σ ), [ = 6σ 4 y ( y ) ( y ) σ ϕ y ( y Φ + 3 σ σ σ 4 ϕ2 σ = 3σ ( ) ( 2 2y y π σ ϕ dy + 3σ 2 2σ 2 σ = σ 2 (1 ) 3 π 1 + 1 ( y ) ( y ) σ ϕ Φ dy 3 σ σ ) ] ( dy + 3σ 2 2σ 2 1 + ) 3 2π ) 3 2π V (y) < V (x) = V (z)

V (x) V (y) V (z)