Robustness Analysis in Sequential Statistical Decisions

The sequential statistical decision making is considered. Performance characteristics (error probabilities and expected sample sizes) for the sequential statistical decision rules (tests) are analysed. Both cases of simple and composite hypotheses are considered. Asymptotic expansions under distortions are constructed for the performance characteristics enabling robust sequential test construction.


Introduction
In many fields of statistical methodology applications, especially in medicine, finance, quality control, problems of statistical decision making are topical (Ghosh and Sen 1991), (Jennison and Turnbull 2000). In statistical decision making, one of the most important problems is to construct decisions providing the requested accuracy on the basis of the minimal information (minimal number of observations). To solve this problem, the sequential approach (Wald 1947), (Lai 2001) is used. Within this approach, the number of observations required to provide the prescribed performance is not fixed a priori, but is considered as a random variable that depends on random observations. To find the optimal dependency is a complicated task, but this scheme of decision making requires a number of observations that is essentially less than what is required by the approach based on fixed sample sizes (see Aivazian (1959)).
In (Quang 1985) the robustness analysis is given for a special hypothetical model in case of simple hypotheses. An empirical study of robustness is performed in (Pandit and Gudaganavar 2009) for the scale parameter of gamma and exponential distributions. Here we systemize some of the results developed by the author for quantitative robustness analysis of sequential decision rules.
Concerning the parameter value θ of the probability distribution (1) the following two simple hypotheses are considered: Denote the statistic where λ t = log a (P (x t ; θ 1 )/P (x t ; θ 0 )) = J(x t ; θ 0 ) − J(x t ; θ 1 ) ∈ Z is the logarithm of the likelihood ratio, calculated by the observation x t .
In the sequential probability ratio test (SPRT, Wald test) for hypotheses (3) testing after n observations (n = 1, 2, . . .) the decision is made according to: The values d = 0 and d = 1 correspond to stopping of the observation process and acceptance of the hypotheis H 0 (if d = 0) or H 1 (if d = 1) by n observations. If d = 2, the observation number (n + 1) should be made. In (5) parameters C − , C + ∈ Z are calculated as follows: where α 0 , β 0 are the requested values of the error type I and II probabilities respectively; [·] means the integer part of an argument.
Theorem 1. If the hypothetical model (1), (4) is distorted according to (7), (8), and |S (k) | = 0, |Ŝ (k) | = 0, k ∈ {0, 1}, then for the SPRT the conditional mathematical expectation of the sample sizet (k) and the factual valuesᾱ,β of the error type I and II probabilities for the distorted model differ from the correspondent characteristics calculated for the hypothetical model, by the values of the order O(ε): Proof. The detailed proof is given in (Kharin 2013).
Using this theory, the performance characteristics can be calculated for tests from a family of sequential tests that makes possible to construct the robust sequential test by the minimax of the risk criterion.
In (Kharin and Kishylau 2015) the results are generalized to the case of arbitrary probability distribution of observations. The case of inhomogeneous data -the model of time series with a trend -is analyzed in (Kharin and Tu 2017).
The case of Markov chains is considered in details in (Kharin 2013).
To test the hypotheses (9), the following parametric family of Bayesian sequential tests is used: where N is the random number of the observation that determines the stopping time, after that observation the decision d is made according to the decision rule (12). The decision d = i means that the hypothesis H i is accepted, i = 0, 1; C − < 0, C + > 0 are parameters of the test (11), (12): where α 0 , β 0 ∈ (0, 1 2 ) are some values close to maximal admissible levels of error type I and II probabilities (Wald 1947). The actual values α, β of the error type I and II probabilities may deviate from α 0 , β 0 .
For calculation of α, β and mathematical expectations of the random variable N determined by (11), let us use a stochastic approximation of the statistic Λ n , n ∈ N. Let m ∈ N be a parameter of the approximation, h = (C + − C − )/m. Let p Λn (u) be the probability density function of the statistic (10); p Λ n+1 |Λn (u | y) be the conditional probability density function, n ∈ N; let R (n) (θ) and Q (n) (θ) be the blocks of the sizes m × 2 and m × m respectively for the approximating Markov chain, 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. Let π(θ) = (π i (θ)) be the vector of initial probabilities of the states 1, . . . , m for the approximating random sequence; π 0 (θ), π m+1 (θ) be the initial probabilities of the absorbing states 0 and m + 1; 1 m be the vecor of size m, with all components equal to 1. Denote:
Denote byt i , i = 0, 1, the conditional mathematical expectation of the random number of observations N provided the hypothesis H i is true, if the hypothetical model is distorted according to (13), (14).
Theorem 3. Under the conditions of Theorem 2, the conditional expected sample sizes satisfy the asymptotic expansions: Theorem 4. If the conditions of Theorem 2 are satisfied, then the following asymptotic expansion holds for the expected sample size: Proof. Proofs of Theorems 3, 4 are presented in Kharin (2013).
This theory is used to calculate the performance characteristics of the sequential tests under distortions and to construct the robust sequential test (Kharin 2017). The approach is presented in (Kharin 2016) with some numerical results.

Conclusion
The problem of robustness analysis for sequential statistical decision rules is considered in the paper. The cases of simple and composite hypotheses are analyzed. Asymptotic expansions are constructed for the performance characteristics of the sequential statistical decision rules under distortion. Analyzing the constructed expansions and constructing the similar for sequential test from a generalized families, robust sequential statistical decision rules can be constructed. The results are also applied for the decision making in case of many hypotheses (Ton and Kharin 2019).