An Efficient Sequential Design for Sensiti

vity 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

2121 ,0),();( xGxF

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 x1,…,xn, and corresponding response results y1,…,yn.

When xi is at or above the critical level of the ith unit, it responds and yi=1; otherwise, it does not respond and yi=0, i=1,…,n.

The interested parameter

We are often interested in pth quantile of F(x), p ,

)()/1()/( 1112 pGp

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 wo

uld 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

)()( 21 xGxF

Let =2/1, =1/1. Thus

)(

x

G

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 { a0,…,ak-1} and transition

probability matrix

11

22

22

11

00

10000000

10000000

000000000

00000100

00000010

00000001

),(

kk

kk

pp

pp

pp

pp

pp

P

1,...,0),(

kja

Gp jj

where

)ˆ,ˆ( Let be the MLE of (, ) based on

historic data set.

)ˆ,ˆ( P

1** ,...,1),ˆ,ˆ( miii

Now, with as transition probability, generate m1 Markov chain realizations, and obtain estimates of (, ).

)ˆ,ˆ( **iiP

2**,

**, ,...,1),ˆ,ˆ( mjjiji

Then with as transition probability, generate the m2 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.ˆ/ˆ1 2

1

***,

2

M

jijii M

)/ˆ,ˆ(),...,/ˆ,ˆ(111

**1

*1

*1 MMM

we calculate their means and variances and then specify the prior distribution of and .

Assume we have collected data (x1, y1),…, ( xt, yt) (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 yx, the average posterior variance of with respect to yx, but conditioned on (x1, y1),…,(xt, yt), En[Var(|y1,…,yt,yx)], is minimized

Set xt+1=x*, make a run there and obtain the observation yt+1.

Estimation of

After a certain fixed number, n, of observationsis obtained, we use the posterior expectation of

to estimate it.

),...,|( 1 nyyE

Let be the parameter space for =(1,2) and let xi(i1) take values in A. Let the subset of x1–2 for xA, be B.

Some notations

High Order ApproximationTheorem 1. Suppose the following conditions hold:

(i) There exists an integer m such that the data set

{(xi, yi): 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.

)}( ])}3[)(ˆ({6

])ˆ[(2

ˆ),...,|( 20,

10

0

10

1

nOkkhkn

khn

yyE uln

ijnllijun

ijnijijn

Then for nm Laplace formula produces the approximation

Consistency of the posterior estimator

),...,|( 1 nyyE

nasBBBxnxnPn

ii

n

ii ,1}),{( 321

1

1

1

210

Theorem 2 Under conditions of Theorem 1

and following condition

is consistent.B1={(u,v): u=1, |v|<1<1},

B2={(u,v): u1, u>2>0,v=0},

B3={(u,v): u1, v0, u>31>0, h(u,v)> 32>0}

}4

)4)1(|1(|

1

12

1

2{

)4)1(|1(|4

4),(

2

22222

2222

2

v

vuu

u

uv

u

vu

vuuv

vvuh

Comparisons in Simulation

We now compare the performance of ourmethod 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.

)52

1exp(1

)52

1exp(

)(

x

xxF

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 ResultsThe 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 n n

10 15 10 15

Mean MSE Mean MSE Mean MSE Mean MSE

Proposed method

5.68 1.32 5.62 1.28 7.31 0.95 7.24 0.90

Wu’s method 5.90 1.50 5.81 1.36 7.45 1.06 7.40 0.93

Neyer’s method 6.04 1.53 5.95 1.35 7.56 1.07 7.49 0.96

Roshan’s method

6.07 1.49 5.88 1.32 7.54 1.06 7.46 0.92

Up-and–down method

5.59 3.00 5.63 2.61 7.21 2.26 7.22 1.89

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.

Voelkel J. G.