# Synchrotron-based high-pressure research in materials science Page: 3 of 12

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extreme values are where most of the interesting features for damage detection are located, therefore it is important to

model them correctly. By re-formulating the SPRT to accurately model the extreme values of the data, accuracy and

quickness of the decisions may be increased.

Described within this report are the development of the sequential probability ratio test and the merger of sequential

testing with the field of extreme value statistics. Test data came from a three-story test structure with bolted

connections between floors and columns to simulate a building joint. The data are then analyzed using both a SPRT

formulated with normality assumptions and extreme value distributions. The two tests are then compared.

2. METHODOLOGY

2.1. Sequential test

A sequential statistical inference starts with observing a sequence of random variables {xi } (i =1, 2,...). This

accumulated data set at stage n is denoted as:

X" = (x,,... x") (1)

The goal of a statistical inference is to reveal the probability model of X,, which is assumed to be at least partially

unknown. When the statistical inference is cast as a parametric problem, the functional form of X, is assumed known

and the statistical inference poses some questions regarding the parameters of the probability model. For instance, if

{xi } are independent and identically distributed (i.i.d.) normal variables, one may pose some statistical test about the

mean and/or the variance of this normal distribution.

A sequential test is one of the simplest tests for such a statistical inference where the number of samples required

before reaching a decision is not determined in advance. An advantage of the sequential test is that on average a

smaller number of observations are needed to make a decision compared to the conventional fixed-sample size test.

First, a simple hypothsis test containing only two distinct distributions is considered. Here, the interest is in

discriminating two simple hypotheses:

Ho :9=90, Hi :9=01, 0 901 (2)

where 9 is a particular parameter value in question, and it is assumed that 9 can take either 9o or 9i only. A type I

error arises if Ho is rejected when in fact it is true. Type II errors arise if Ho is accepted when it is false. When a

sequence of observations {xi } are available, the purposes of any sequential test for the above hypotheses are (1) to

reach the correct decision about Ho with the least probability of type I and II errors, (2) to minimize the sampling

number before the correct decision is made, and (3) to eventually terminate with either the acceptance or rejection of

Ho as the sampling size n increases. When a sequential test satisfies the last condition, the test is defined closed.

Otherwise, an open test may continue infinitely observing data without reaching any terminal decision about Ho.

It turns out that the simultaneous achievment of all three goals is impossible by any test. Therefore, a reasonable

compromise among the these conflicting goals needs to be achieved. For the well-established fixed-sampling tests, the

sample size n is fixed, and an upper bound on the type I error is pre-specified. Then, an optimal fixed-sample test is

selected by minimizing the probability of type II error. On the other hand, a sequential test specifies upper bounds on

the probabilities of type I and II errors and minimizes the following average sample number, E(n 10):

E(n10)= n p(n 1) (3)

n=1

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Chen, Bin; Lin, Jung-Fu; Chen, Jiuhua; Zhang, Hengzhong & Zeng, Qiaoshi. Synchrotron-based high-pressure research in materials science, article, June 1, 2016; (digital.library.unt.edu/ark:/67531/metadc928997/m1/3/: accessed January 23, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.