An Efficient Sequential Design for Sensitivity Experiments Yubin Tian School of Science, Beijing Institute of Technology

Sensitivity experiments: experimental settings in which each experimental unit has a critical stimulus level that cannot be observed directly. Introduction

Response Curve F(x) is the distribution of the critical stimulus levels over the test specimens. where, G is a known distribution function The general version of F(x) is

Observations stimulus settings x 1,…,x n, and corresponding response results y 1,…,y n. When x i is at or above the critical level of the i th unit, it responds and y i =1 ; otherwise, it does not respond and y i =0, i=1,…,n.

The interested parameter We are often interested in pth quantile of F(x),  p, Our goal Make inference for  p using small samples.

Historic Data In China, for explosives and materials, there often exists a data set from the documented method up-and-down procedure for analyzing the sensitivity. When considering the availability, interest would aim to estimate the extreme quantile  p by limited sample size n.

inappropriate for the existed methods Based on such kind of historic data set, we can use Wu’s method, Neyer’s method, or Joseph’s method to estimate  p. However, for this directly using, there will cast many units in the subsequent sequential procedure to modify the characteristic of centering on  0.5 of this historic data set before quickly converging to the interest parameter  p.

Proposed methods Our purpose is to develop a new sequential procedure that can make the estimate for  p more precisely by quickly and efficiently exploiting the information in the tested data and known knowledge.

Specification of the Prior Let  =  2 /  1,  =1/  1. Thus It is natural to assume that the prior distribution of  is normal, the prior distribution of  is lognormal, and they are independent.

For the historic data set, the experiment settings is a realization of the Markov chain with finite state space { a 0,…,a k-1 } and transition probability matrix where

Let be the MLE of ( ,  ) based on historic data set. Now, with as transition probability, generate m 1 Markov chain realizations, and obtain estimates of ( ,  ). Then with as transition probability, generate the m 2 second-level Markov chain realizations and obtain estimates

After the transformation from ( ,  ) to (  1,  2 ), we specify the prior distribution for (  1,  2 ). Let From we calculate their means and variances and then specify the prior distribution of  and .

Assume we have collected data (x 1, y 1 ),…, ( x t, y t ) (This includes the case of not having any data, for which t =0). The New Sequential Design Our objective: select a new level x* so that if we run there and obtain the result y x, the average posterior variance of  with respect to y x, but conditioned on (x 1, y 1 ),…,(x t, y t ), E n [Var(  |y 1,…,y t,y x )], is minimized. Set x t+1 =x*, make a run there and obtain the observation y t+1.

Estimation of  After a certain fixed number, n, of observations is obtained, we use the posterior expectation of  to estimate it. Let  be the parameter space for  =(  1,  2 ) and let x i (i  1) take values in A. Let the subset of x  1 –  2 for x  A,  be B. Some notations

High Order Approximation Theorem 1. Suppose the following conditions hold: (i) There exists an integer m such that the data set {(x i, y i ): i=1,…,m} has an overlap in responses and nonresponses. (ii) -lnG(u) and –ln[1-G(u)] are convex. G(u) is strictly increasing at every u satisfying 0<G(u)<1. (iii) The derivatives of five orders of G(u) are bounded over {u: 0<G(u)<1}. (iv) A is bounded and B  {u: 0<G(u)<1}. Also, B is compact and connected. Then for n  m Laplace formula produces the approximation

Consistency of the posterior estimator Theorem 2 Under conditions of Theorem 1 and following condition is consistent. B 1 ={(u,v): u=1, |v|<  1 <1}, B 2 ={(u,v): u  1, u>  2 >0,v=0}, B 3 ={(u,v): u  1, v  0, u>  31 >0, h(u,v)>  32 >0}

Comparisons in Simulation We now compare the performance of our method with other four methods through simulations. Under comparison are (a) the up and down method recommended by ASTM (1995); (b) Wu’s recursive design (1985); (c) Neyer’s method (1994); (d) the modified Robbins-Monro procedure proposed by Joseph (2004).

We use the logit model, as the true distribution, where  1 =1/2,  2 =5. We first simulate the historic data set with sample size 20 recommended by standards. Then simulate the data by using our method and other four methods with sample size n. We repeat above processes 500 times and obtain 500 estimates of .

Simulating Results The results of Monte Carlo mean and mean squared error of 500 estimates are given in following table.  0.1 (true value is 5.61)  0.2 (true value is 7.23) Design nn Mean MSE Proposed method Wu’s method Neyer’s method Roshan’s method Up-and–down method

The simulation study demonstrates the superior performance of our method over the other methods. Our method did a good job for commonly used extreme quantiles under the requested sample size. Conclution

Thank you for your attendance.

All involved integrals are two dimensional integrals, In this paper, we use the algorithm which is efficient and easy to implement given by James C. Fu and Liquan Wang (2002) to calculate this kind of integral. Here we don’t use MCMC algorithms because for the low dimensional integral MCMC does not perform very well and the functional forms of full conditionals of posterior distributions is not clear. are not very easy to be obtained.

Voelkel J. G.