The Modified Keifer-Weiss Problem, Revisited Bob Keener Department of Statistics University of Michigan IWSM, 2013 Bob Keener (University of Michigan) Keifer-Weiss Problem IWSM 2013 1 / 15
Outline 1 Symmetric Tests for Brownian Motion The SPRT Concerns and the Keifer-Weiss Problem Exact Solutions 2SPRTs or Triangular Tests Example 2 Discrete Time Formulation Excess Over the Boundary Refined Approximations Example To Do 3 References Bob Keener (University of Michigan) Keifer-Weiss Problem IWSM 2013 2 / 15
Symmetric tests for Brownian motion with drift: Process: X t = θt + W t, with W standard Brownian motion. Stopping time: τ = inf{t > 0 : X t b(t)}. Test: H 0 : θ 0 vs H 1 : θ > 0 based on observing X t for t [0, τ]. Power Function: β(θ) = P θ (X τ 0). SPRT: If we know that θ = ±, and an error probability α = 1 β( ) is desired, then the optimal boundary will be b(t) = 1 ( ) 1 α 2 log, t > 0, α minimizing both E (τ) and E (τ). Bob Keener (University of Michigan) Keifer-Weiss Problem IWSM 2013 3 / 15
Concerns with the SPRT: Poor performance if θ ±. In particular, E 0 (τ) may be large. Example. If α = 5% and = 1, then (by Wald s approximations) but E 1 (τ) = E 1 (τ) = 1.325, E 0 (τ) = 2.167. Bechhoffer (1960) suggested choosing a stopping rule τ to minimize sup θ E θ (τ) with error probabilities controlled at θ = ±. In this symmetric case, the best rule by this criteria will be the stopping rule minimizing E 0 (τ) with the same bounds for the error probabilities, called the modified Keifer-Weiss problem. Bob Keener (University of Michigan) Keifer-Weiss Problem IWSM 2013 4 / 15
Exact Solutions and Optimal Stopping: By a change of measure, β(θ) = P θ (X τ 0) = E 0 [I{X τ 0}e θxτ θ2 τ/2 ]. Introducing a Lagrange multiplier, for a suitable constant c the optimal stopping time for the modified Keifer-Weiss problem will minimize ] E 0 [exp( X τ τ 2 /2) + cτ. Bob Keener (University of Michigan) Keifer-Weiss Problem IWSM 2013 5 / 15
2SPRTs or Triangular Tests (for small α): Define τ + = inf{t : X t a t/2}, the stopping time for a one-sided SPRT of θ = versus θ = 0. Then P (τ + < ) = e a. Similarly, if τ = inf{t : X t a + t/2}, then P (τ < ) = e a. 2SPRTs, introduced by Lorden (1976), combine these by taking τ = min{τ +, τ }, and have error rate α e a. Lorden and other authors show that 2SPRTS are asymptotically optimal as α 0. Bob Keener (University of Michigan) Keifer-Weiss Problem IWSM 2013 6 / 15
Example with = 1, revisited: α E 1 (τ) E 0 (τ) Fixed Sample: 5% 2.70554 2.70554 SPRT: 5% 1.32400 2.16743 2SPRT: 2.5% 1.94403 2.80418 2SPRT: 5% 1.44602 1.93411 Optimal: 5% 1.39993 1.92173 Bob Keener (University of Michigan) Keifer-Weiss Problem IWSM 2013 7 / 15
Boundaries. Fixed, SPRT, 2SPRT, Optimal: 3 2 1 0 1 2 3 0 1 2 3 4 5 6 time Bob Keener (University of Michigan) Keifer-Weiss Problem IWSM 2013 8 / 15
Discrete Time: Adding the constraint that τ is a multiple of ɛ, let τ ɛ = inf{t > 0 : X t b(t), t ɛn}. If ɛ is small, the performance of a test using τ ɛ will be similar to the performance of a test using τ. With θ = θ def ɛ = ɛθ, the variables Y k, k 1, defined as Y k = [ X kɛ X (k 1)ɛ ] / ɛ, k = 1, 2,..., are i.i.d. from N( θ, 1). Taking N = τ ɛ /ɛ and S n = n k=1 Y k, we have N = inf { n > 0 : S n b(nɛ)/ ɛ }. Boundaries b( ) for Brownian motion should do a good job of minimizing E 0 N with error probabilities α when EY k = ± ɛ. Bob Keener (University of Michigan) Keifer-Weiss Problem IWSM 2013 9 / 15
Excess Over the Boundary: Since N = inf { n > 0 : S n b(nɛ)/ ɛ }, R ɛ def = S N b(nɛ)/ ɛ is the excess over the boundary. By Hogan (1984) or Keener (2013), if θ = 0, R ɛ R as ɛ 0. The distribution for R is related to fluctuation theory for random walks. If T is the ladder time T = inf{n > 0 : S n 0}, then R has density P 0 (S T > r)/e 0 (S T ), r > 0, and ρ def = E[R] = E[S2 T ] 2E[S T ] = 0.5826. Bob Keener (University of Michigan) Keifer-Weiss Problem IWSM 2013 10 / 15
Refined Approximations: Let C = {(t, x) : t > 0, x < b(t)} (the continuation region), and let u solve the heat equation u t + 1 2 u xx = 0, (t, x) C with boundary condition u ( t, ±b(t) ) = f ( t, b(t) ). [ ( Then u(0, 0) = E 0 f τ, Xτ )]. Defining ( ) ( ) H(t) = f x t, b(t) ux t, b(t), arguments similar to those in Keener (2013) give [ ( E 0 f τɛ, X τɛ )] = u(0, 0) + ρ ɛe 0 [H(τ)] + o ( ɛ ). Bob Keener (University of Michigan) Keifer-Weiss Problem IWSM 2013 11 / 15
Example in Discrete Time: With = 1 and ɛ = 0.04, α will be the error probability when E[Y k ] = ±0.2. The approximations from the previous slide give: α E 1 (τ) E 0 (τ) Fixed Sample: 4.955% 68 68 SPRT: 5%-1.107% 33.10+6.50 54.19+8.58 2SPRT: 5%-0.714% 36.15+2.36 48.35+4.21 Optimal: 5%-0.582% 35.00+2.12 48.04+3.51 Bob Keener (University of Michigan) Keifer-Weiss Problem IWSM 2013 12 / 15
Discrete Time Example with Adjusted Designs: Adjusting the design parameters for the stopping rules we can achieve an adjusted error rate α 5%. This gives the following approximate sample sizes: α E 1 (τ) E 0 (τ) Fixed Sample: 4.955% 68 68 SPRT: 5% 36.49 55.25 2SPRT: 5% 36.07 48.58 Optimal: 5% 35.18 48.27 Bob Keener (University of Michigan) Keifer-Weiss Problem IWSM 2013 13 / 15
To Do: Simulations. Numerical issues finding u x on the boundary. Exponential families. Dealing with asymmetry/skewness. Bob Keener (University of Michigan) Keifer-Weiss Problem IWSM 2013 14 / 15
References: Anderson, Ann. Math. Statist. 1960. Huang, IMS Lecture Notes on Math. Statist. 43, 2003. Lai, Ann. Statist. 1973. Lorden, Ann. Statist. 1976. Huffman, Ann. Statist. 1983. Bob Keener (University of Michigan) Keifer-Weiss Problem IWSM 2013 15 / 15