Robustness Evaluation in Sequential Testing of Composite Hypotheses

The problem of sequential testing of composite hypotheses is considered. Asymptotic expansions are constructed for the conditional error probabilities and expected sample sizes under “contamination” of the probability distribution of observations. To obtain these results a new approach based on approximation of the generalized likelihood ratio statistic by a specially constructed Markov chain is proposed. The approach is illustrated numerically.


Introduction
Sequential testing of hypotheses (Wald, 1947) is used as an adequate statistical methodology in applications where not only accuracy (percentage of correct decisions), but also the number of observations used is important: medicine (Jennison and Turnbull, 2000), quality control, finance (Ghosh and Sen, 1991), etc.
In the applications sequential methods are often applied to "contaminated" data (see Huber, 1981), and this fact results in significant increasing of percentage of incorrect decisions (A.Kharin, 2002); the optimal properties of sequential procedures (see Aivazian, 1959) become to be broken.Some results on robustness analysis of sequential tests for simple hypotheses case are presented in Quang (1985); A. Kharin (2002); A. Y. Kharin and Kishylau (2005).The robust sequential tests are constructed in case of simple hypotheses for several models of data in A. Kharin (2002); A. Kharin and Kishylau (2005).
In this paper we evaluate robustness of the conditional error probabilities and expected sample sizes under "contamination" of the probability distribution of data for composite hypotheses.

Mathematical Model
Let on a measurable space (Ω, F) a random sequence x 1 , x 2 , . . .∈ R with an n-variate probability density function (pdf) p n (x 1 , . . ., x n | θ) be observed, where n ∈ N = {1, 2, . ..}, θ ∈ Θ ⊆ R k is an unknown value of the random vector of parameters.The pdf p(θ) of the parameter θ is supposed to be known.Consider two composite hypotheses: where Austrian Journal of Statistics, Vol. 37 (2008), No. 1, 51-60 To construct a sequential test for the hypotheses (1) in the considered Bayesian setting, let us use the method of weight functions proposed in Wald (1947).Introduce the notation: is the indicator function of a set S; The following parametric family of sequential tests is used in the notation (2) for the hypotheses (1): where N is the random number of observation (stopping time), at which the decision d is made according to (4).The decision d = i means that the hypothesis H i , i = 0, 1, is accepted; C − < 0, C + > 0 are two parameters of the test (3), (4).In Wald (1947) the following expressions are used for these parameters: where α 0 , β 0 are maximal admissible values for the error type I (accept H 1 provided H 0 is really true) and type II (accept H 0 provided H 1 is really true) probabilities, β 0 < 1 − α 0 .The true values α, β of the error type I and II probabilities differ from α 0 , β 0 (see e.g. A. Kharin, 2002), and calculation of these characteristics is an important problem (Lai, 2001) in the aspect of quantitative robustness analysis.It is also important to get expressions for calculation of conditional mathematical expectation of the sample size N for the sequential test (3), (4) with the fixed parameters (5).

Approximation of the Random Sequence Λ n by a Markov Chain
Let us split the state space R of the sequence Λ n on m + 2 "cells": where m ∈ N, Introduce the notation: S m 1 = {A 1 , . . ., A m } is the set of the intervals; p Λn (u) is the pdf of the generalized likelihood ratio statistic ( 2 the asymptotic expansion at h → 0 (m → ∞) holds: where Proof.Denote the left and right bounds of the interval A j , j ∈ N, by a j L , a j R , respectively.First, prove (7) for k = 1, A n+1 ∈ S m 1 : From the definition, (9) Transform the numerator of (9) using the Markov property of Λ n : From the property of a Markov sequence and the condition (6) it follows that ∃c > 0: Using the rectangular approximation formula (see, e.g.Bahvalov, 1973), we get For the denominator of (9) we get analogously: Using ( 11), ( 12) in ( 9) and then performing asymptotic expansion, we find By similar transformations for the conditional probability from the right side of ( 8) we find Comparing ( 13) and ( 14), we get (8).
For A n+1 ∈ {A 0 , A m+1 } the scheme of the proof is the same, except integration on z in (10), where the rectangular approximation formula is not applied.For this case we get: (15) Summarizing ( 8) and (15), we get (7) for k = 1.
The scheme of the proof of the result (7) for k = 2, 3, . . . is the same as for k = 1.
Note, that the main term in the asymptotic expansion (7) has the order O(h) for A n+1 ∈ S m 1 , and O(1) for A n+1 ∈ {A 0 , A m+1 }.Let [x] mean the integer part of x (maximal integer number, which is not greater than x).For the random sequence Λ n introduce the discrete random sequence Z m n , n ∈ N, with the finite state space V = {0, 1, . . ., m + 1}: 7) be equal to 0, the random sequence ( 16) should be a Markov chain.Theorem 1 states that the Markov property is approximately valid for the sequence (16), and gives the accuracy for this approximation.
Introduce the matrix of the size (m + 2) × (m + 2) of the conditional probabilities for the random sequence (16): A. Kharin 55 Note, that the theory of calculation of the characteristics for Markov chains is much better developed than for Markov sequences, so let us consider an approximation of Z m n by the Markov chain z m n ∈ V , n ∈ N, that has the initial probability distribution and the transition probabilities matrix P (n) (θ) the same as for the sequence (16).
Because {0}, {m + 1} are absorbing states, it is convenient to renumerate the states: V = {{0}, {m + 1}, {1}, . . ., {m}}.The matrix P (n) (θ) after renumeration can be represented in the form: where R (n) (θ) and Q (n) (θ) are blocks of the size m × 2 and m × m respectively, I k is the identity matrix of the size k, 0 2×m is the matrix of the size 2 × m with all elements equal to 0. Introduce the notation: π(θ) = (π i (θ)) is the vector of initial probabilities of the states 1, . . ., m for the sequence ( 16), π 0 (θ) and π m+1 (θ) are the initial probabilities of the absorbing states 0 and m + 1 respectively for ( 16); 1 m is the vector of the size m with all components equal to 1; γ H i (θ) = P{d = i | θ}, i = 0, 1, is the conditional probability to accept the hypothesis H i provided the parameter takes the value θ ∈ Θ; matrices S(θ) of the size m × m and B(θ) of the size m × 2 are is the j-th column of the matrix B(θ), j = 1, 2; t(θ) = E{N | θ}, t i = E{N | θ ∈ Θ i } (i = 0, 1) are the conditional sample sizes; t = E{N } is the unconditional sample size.
Theorem 2 Under the Theorem 1 conditions, if a sequential test from the family (3), ( 4) is used for the hypotheses (1), then ∀θ ∈ Θ at h → 0 the probabilistic characteristics of the test satisfy the following asymptotic expansions: Proof.The proof consists of two stages.1. Calculation of the main terms in the asymptotic expansions for E{N | θ} and γ H i (θ), i = 0, 1.These terms are calculated as the appropriate characteristics for the nonhomogeneous Markov chain z m n , n ∈ N (see, e.g.Kemeni and Snell, 1959) with the state space V (two of the states are absorbing: {0} and {m + 1}), initial probabilities vector of transient states π(θ), initial probabilities of absorbing states π 0 (θ), π m+1 (θ), and the transition probabilities matrix P (n) (θ) presented in the form (17).
2. Asymptotic analysis of the remainders at h → 0. The result of Theorem 1 is applied to analyze the differences between the characteristics of the random sequence ( 16) and the Markov chain z m n ; the relation h = (C + − C − )/m is used, where m is the size of the square matrix Q (n) (θ) and the number of rows in the matrix R (n) (θ).In practice the observed data usually do not follow the hypothetical model exactly, the hypothetical model is distorted (see Hampel, Ronchetti, Rousseeuw, and Stahel, 1986).The model of "contamination" of the hypothetical probability distributions (see Huber, 1981) is often used to analyze robustness of the statistical procedures.Suppose the hypothetical model considered in Section 2 is distorted: the data observed are really obtained from a "contaminated" probability distribution with a pdf ) where p(x 1 , . . ., x n | θ) is a "contaminating" pdf, ε ∈ [0, 1/2) is a probability of "contamination" presence ("contamination" level).
Theorem 4 For the distorted model ( 26), if the conditions of Theorem 2 are valid, then ∀θ ∈ Θ, the following asymptotic expansions hold at ε → 0, h → 0: Proof.The proof consists of two stages.1. Calculation of the characteristics for the Markov chain which has the state space V , the initial probabilities vector (π 0 (θ), π(θ), πm+1 (θ)) and the transition probabilities matrix 2. Asymptotic analysis at ε → 0, h → 0 of the correspondent differences between the calculated characteristics under "contamination" and their hypothetical values, using the results of Theorem 2.
Using the result of Theorem 4 one can approximate the characteristics of the sequential test (3), (4) under the distortion (26), and use these approximation in construction of the robust sequential test by the minimax criterion (see A. Kharin, 2002).

Conclusion
In this paper the approach to calculate with a given accuracy the characteristics of sequential tests is proposed for composite hypotheses.This approach is used for quantitative robustness analysis of sequential tests under "contamination" of the probability distribution of observations.The results can be used for minimax robust sequential test construction.

Table 1 :
Results of numerical experiments