PERFORMANCE ANALYSIS AND ROBUSTNESS EVALUATION OF A SEQUENTIAL PROBABILITY RATIO TEST FOR NON-IDENTICALLY DISTRIBUTED OBSERVATIONS

Abstract. In this article the problem of a sequential test for the model of independent non-identically distributed observations is considered. Based on recursive calculation a new numerical approach to approximate test characteristics for a sequential probability ratio test (SPRT) and a truncated SPRT (TSPRT) is constructed. The problem of robustness evaluation is also studied when the contamination is presented by the distortion of the distributions of all increments of the log-likelihood ratio statistics. The two-side truncated functions are proposed to be used for constructing the robustified SPRT. An algorithm to choose the thresholds of these truncated functions is indicated. The results are applied for a sequential test on parameters of time series with trend. Some kinds of the contaminated models of time series with trend are used to study the robustness of the truncated SPRT. Numerical examples confirming the theoretical results mentioned above are given.

the indicated test characteristics. Based on Wald's fundamental identity and likelihood ratio identity, some approximations for the average numbers of observations have been obtained [1][2][3]. An important improvement in computing these characteristics is that the operating characteristics (OC) and average sample number (ASN) functions were proved to satisfy the Fredholm integral equation of the second kind (FIESK) [3,4]. Neglecting the conditions on the existence of their solutions, we can resort numerical methods to get the approximations of these characteristics. Another approach to calculate is to use the properties of absorbing Markov chains [5][6][7]. This approach allows not only to get the approximate values of test characteristics but also to evaluate the robustness of statistical procedures [6,8,9]. For the TSPRT, the upper bounds for the error probabilities of type I and II were achieved by using normal approximation for the accumulated log-likelihood ratio statistic when the maximum number of observations is relatively large [1], or in more general case [2]. In the case of non-identical distributed observations, Liu Y. and Li X. R. [10] have shown numerical solutions in some special cases to the OC and ASN functions by constructing the sequence of the FIESK with respect to the sequence of new stopping times. In this paper, another method based on recursive calculations is constructed for approximating the test characteristics of the SPRT and TSPRT as well. Evaluation of robustness for the truncated sequential test is also studied and these results will be applied for sequentially testing the parameters of time series with trend.

Mathematical model and auxiliary results
Let { , 1} n X n ≥ be a sequence of independent random variables on the same probability space ( , , ) P Ω F with probability density functions 1 { ( , ), , 1} n p x x n θ ∈ ≥ R respectively, where θ is an unknown vector of parameters.
Consider two simple hypotheses: where 0 1 , m θ θ ∈R are known vectors, 0 1 . θ ≠ θ Denote the accumulated log-likelihood ratio statistic for n observations: where the thresholds C − and C + are the parameters of the test. According to Wald [12], C − and C + can be calculated as follows: where α 0 , β 0 are the given values for error probabilities of types I and II respectively. Denote , f x f y C x y x y a b − ≤ − ∀ ∈ then the following expansion holds: x satisfies the following recurrent relation: x is the cumulative distribution functions of λ n under hypothesis H k , and ( ) ∈ are continuous functions in R. Then, from the definitions of α, β and Lemma 1 the test characteristics can be expressed as follows: Assume that ( we obtain the following systems of linear equations: ,..., , and ( ), 1, .
was neglected, the following theorem has been proved.
T h e o r e m 3. Assume that ( ) p r o o f. Note that by the way of selecting ( ) 0 k n , we have . The result is directly derived from (6), (7), (10) and Theorem 2.
R e m a r k 1. In practice, it is not easy to determine ( ) 0 k n theoretically with respect to a given value ε 0 . However, if we know are much less than ε 0 and the test will terminate finitely with probability 1 as well.
R e m a r k 2. In general, there is still a problem of calculating the probability ( ) Λ ∈ because of the difficulty in getting theoretically the probability distribution for the sum of independent random variables , Therefore, if the way of finding index ( ) 0 k n in Remark 1 is not feasible, these indices can be possibly chosen from the following conditions: R e m a r k 3. In the case of independent identically distributed observations, due to Stein's lemma [3] < ∈ Next, we modify the method above to approximate error probabilities of type I and II for the TSPRT. Let M be the maximal number of observations that we may measure. The Wald's TSPRT is formulated as follows. If the sampling process has progressed to the n-th stage (n < M): and takes one more observation if From (11)- (12) and (13)-(14), we obtain From that we get: x k ∈ without any conclusion about the order of accuracy. For the TsPRT, due to the limited number of terms in the sum we can increase the number H to get better approximation.

Robustness evaluation.
In practice, there is often the case that the observed data do not follow the hypothetical model exactly, e. g. the hypothetical model is distorted [14]. This leads to the distortion in the dist ributions of increments λ n of log-likelihood statistic Λ n . In this section, we study the case where these influences can be described in the form of contaminated model of Huber type [15] for each increment λ n as follows: is a contaminating CDF, and [0,1 / 2) δ∈ is the level of contamination. Introduce the notation: are the elements also calculated analogously to ( ) and for n ≥ 2: .. .
n n n n n n n n n From that, we have:

Robustifying the TSPRT.
To reduce the influence of outliers in λ n , we can truncate the values of λ n by the following function ( Figure a): ( , ] ( , ) ( ) · ( ) · ( ) · ( ), g g g g g g f x g x x x g x so, N and N have the same probability distributions.
(ii) similarly, 0 1  R e m a r k 5. There are some remarks for choosing the thresholds gand g + : (i) If g -≥ 0, then β M = 0; if g + ≤ 0, then α M = 0. Therefore, the possible choice is that we should select If gincreases, β M will decrease, but α M will increase. If g + decreases, there is an opposite picture. so, the possible and reasonable criterion for choosing gand g + is to minimize the sum α M + β M for the TsPRT.
Using the truncated function (15), the distribution function of n λ is: which is generally a discontinuous function. Therefore, the numerical results in Theorem 3 and Theorem 4 cannot be applied for calculating the test characteristics. To make use of the proposed numerical approach, we can use a modified version of the function (15) in the following form (Figure b): g g g g x g In this case, when ( ) n F x λ is continuous, the distribution function of n λ is also continuous and has the following form: n n n n x g is small, we have to take more observations for the sequential test (e. g. the number of observations tends to the maximum number M). This means that we have more information for the test and this leads to the downward trend of both error probabilities. However, when | | g g is sufficiently small, the number of observations are mostly M and we have to make the final decision according to (15). In this case, both error probabilities can increase again.
The following algorithm can be used to choose thresholds gand g + : -choose a positive value K ∈ N and a small value ε > 0; using Theorem 4 and truncated function (16); Introduce the notation: Without loss of generality assume that hypothesis H 0 is true and we are interested in studying type I error probability α and the average number of observations (0) ( ).

E N 2.4.1. Calculation of the test characteristics.
A sufficient condition for the termination of the test can be found in [16].
Furthermore, in this case we know the exact probability distribution of ( )  is a sequence of independent random variables, { , 1} t t τ ≥ is a sequence of independent identically distributed random variables, From that we get: is the distribution function of random variable C a s e 2. Distortion in the basic function of trend ψ(t). We consider the following model: is a basic function of trend such that with a given positive δ, T h e o r e m 9. For the model (20) and the TSPRT (3), (11)-(12), the following expressions are valid: From the proof of Theorem 9, we knew The rest part of the proof is similar to the proof of Theorem 9.

Numerical examples
The probability model (17) was considered and the hypotheses (2) were tested with the following values of parameters: The thresholds Cand C + were calculated according to Wald [1]. Denote the sample estimate of a characteristic γ with Monte-Carlo method by ˆ.
γ The number of repetitions used in Monte-Carlo simulation was 100 000. The index (0) 0 n was chosen according to Remark 1 with   With very small value  For the TSPRT, there is no requirement of determining the index n 0 , and the maximum number of observations M is usually not too large. Due to these advantages we can possibly increase the number of partitions H to get better accuracy of approximation. In Tab. 2, with the same levels of α 0 , β 0 when the value M increases, the error probability α M decreases but the average number of observations (0) ( ) M E N increases. This can easily be understood because the more observations we have, the higher accuracy of the test is. In addition, the average number of observation has an upward trend with respect to M to reach its real expected values in Tab. 1. With H = 200, the approximate values M α and 0 ( ) t M are relatively close to their Monte-Carlo estimates. Furthermore, compared with Tab. 1, the limitation of maximum number of observations leads to so remarkable change in error probabilities of the test.