Waveform design based multi-target hypothesis testing under unknown clutter parameters

A non-parametric sequential rank-sum probability ratio test method for binary hypothesis testing

The semiconductor manufacturing is one of the most sophisticated processes in modern industries. It contains hundreds of complex operations, and this makes semiconductor reliability a challenging task especially in recent years that integrated circuits (ICs) have been broadly used at much wider areas including the health cares, automotive, and aerospace applications, where safety is the major concern. To make things worse, due to the shorter product life cycles and severe competitions, time-consuming reliability tests become unacceptable. The industries are in urgent need of a fast and effective way to quickly assess reliability risks. Under all these stringent limitations and the high reliability targets, we propose a consolidated approach that consists of an efficient and effective reliability baseline (BL) management system and a test scheme developed based on sequential probability ratio test (SPRT). To ensure a trustworthy setting for the SPRT test scheme, we apply the Bayesian approach to continuously update the BL database. An integrated flow chart is introduced to elucidate our proposed methodology. It is the first time SPRT is applied on reliability risk assessments with two unknown parameters. Several actual cases from semiconductor arena are reported to demonstrate the merits and powers of our proposed methodology. From these real cases, as a side benefit, we show the wide applicability of our method, which does not limit to reliability but on yield and functionality checks as well. To facilitate the use of our method by field practitioners, we do the best to explain derivations in a plain manner and provide analysis codes in R (in Supplementary Material) to make our approach as portable as possible.

Because computers (except for parallel computers) generate simulation outputs sequentially, we recommend sequential probability ratio tests (SPRTs) for the statistical analysis of these outputs. However, until now simulation analysts have ignored SPRTs. To change this situation, we review SPRTs for the simplest case; namely, the case of choosing between two hypothesized values for the mean simulation output. For this case, the classic SPRT of Wald (Wald A. Sequential tests of statistical hypotheses. Ann Math Stat 1945; 16: 117–186) allows general types of distribution, including normal distributions with known variances. A modification permits unknown variances that are estimated. Hall (Hall WJ. Some sequential analogs of Stein’s two-stage test. Biometrika 1962; 49: 367–378) developed a SPRT that assumes normal distributions with unknown variances estimated from a pilot sample. A modification uses a fully sequential variance estimator. In this paper, we quantify the performance of the various SPRTs, using several Monte Carlo experiments. In experiment #1, simulation outputs are normal. Whereas Wald’s SPRT with estimated variance gives too high error rates, Hall’s original and modified SPRTs are “conservative”; that is, the actual error rates are smaller than those prespecified (nominal). Furthermore, our experiment shows that the most efficient SPRT is Hall’s modified SPRT. In experiment #2, we estimate the robustness of these SPRTs for non-normal output. For these two experiments, we provide details on their design and analysis; these details may also be useful for simulation experiments in general.

In medical, health, and sports sciences, researchers desire a device with high reliability and validity. This article focuses on reliability and validity studies with n subjects and m ≥2 repeated measurements per subject. High statistical power can be achieved by increasing n or m, and increasing m is often easier than increasing n in practice unless m is too high to result in systematic bias. The sequential probability ratio test (SPRT) is a useful statistical method which can conclude a null hypothesis H0 or an alternative hypothesis H1 with 50% of the required sample size of a non-sequential test on average. The traditional SPRT requires the likelihood function for each observed random variable, and it can be a practical burden for evaluating the likelihood ratio after each observation of a subject. Instead, m observed random variables per subject can be transformed into a test statistic which has a known sampling distribution under H0 and under H1. This allows us to formulate a SPRT based on a sequence of test statistics. In this article, three types of study are considered: reliabilityof a device, reliability of a device relative to a criterion device, and validity of a device relative to a  criterion device. Using SPRT for testing the reliability of a device, for small m, results in an average sample size of about 50% of the fixed sample size for a non-sequential test. For comparing a device to criterion, the average sample size approaches to 60% approximately as m increases. The SPRT tolerates violation of normality assumption for validity study, but it does not for reliability study.

In this article, we deal with problems of testing hypotheses in the framework of sequential statistical analysis. The main concern is the optimal design and performance evaluation of sampling plans in the Kiefer-Weiss problems. For the case of observations following a discrete exponential family, we provide algorithms for optimal design in the modified Kiefer-Weiss problem and obtain formulas for evaluating their performance, calculating operating characteristic function, average sample number, and some related characteristics. These formulas cover, as a particular case, sequential probability ratio tests (SPRTs) and their truncated versions, as well as optimal finite-horizon sequential tests. On the basis of the developed algorithms we propose a method of construction of optimal tests and their performance evaluation for the original Kiefer-Weiss problem. All algorithms are implemented as functions in the R programming language and can be downloaded from https://github.com/tosinabase/Kiefer-Weiss, where the code for the binomial, Poisson, and negative binomial distributions is readily available. Finally, we make numerical comparisons of the Kiefer-Weiss solution with the SPRT and the fixed-sample-size test having the same levels of the error probabilities.

The sequential probability ratio test (SPRT) is a fundamental tool for sequential analysis. It forms the basis of numerous sequential techniques for different applications; for example, the truncated SPRT and Page's cumulative sum test (CUSUM). The performance of SPRT is characterized by two important functions—operating characteristic (OC) and average sample number (ASN), and CUSUM's performance is revealed by the average run length (ARL) function. These functions have been studied extensively under the assumption of independent and identically distributed log-likelihood ratios (LLRs) with constant bounds, which is too stringent for many applications. In this article, inductive integral equations governing these functions are developed under very general settings—the bounds can be time-varying and the LLRs are assumed independent but nonstationary. These inductive equations provide a theoretical foundation for performance analysis. Unfortunately, they have nonunique solutions in the general case except for the truncated SPRT. Numerical solutions for some frequently encountered special cases are developed and are compared with the results of Monte Carlo simulations.

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 operating characteristic (OC) and average sample number (ASN) of the sequential probability ratio test (SPRT) and multi-hypothesis SPRT (MSPRT) are studied. We consider the case where the observation sequence is independent but not necessarily identically distributed. Also, the thresholds for the test can be time varying. Based on the governing equations for OC and ASN of the SPRT developed in our previous work, a solution for the general case is proposed. The governing equations for OC and ASN of the MSPRT are also obtained. Numerical solutions for MSPRT are developed. Basically, the solutions rely on approximating the original test by truncation, that is, truncating the test at some finite time $K$ . We show that under some mild conditions, the approximation error diminishes as $K$ increases, at the cost of increased computation. Numerical examples are provided to demonstrate our solutions by comparing with Monte Carlo simulations, Simon’s lower bound, and Dragalin’s method (if available) for ASN.

Lindley's concept of the information in an experiment is used to study alternate-hypothesis statistical tests. A test is considered to be a combination of a sampling rule and a decision rule which is based on the samples taken. Two information functions are formed: the information about the unknown parameter contained in the samples taken and the information contained in the decision to which these samples lead. The former quantity is at least as great as the latter, with equality if, and only if, the test is an SPRT (sequential probability ratio test) neglecting the excess over the boundary.

Conditional frequentist tests of a precise hypothesis versus a composite alternative have recently been developed, and have been shown to be equivalent to conventional Bayes tests in the very strong sense that the reported frequentist error probabilities equal the posterior probabilities of the hypotheses. These results are herein extended to sequential testing, and yield fully frequentist sequential tests that are considerably easier to use than are conventional sequential tests. Among the interesting properties of these new tests is the lack of dependence of the reported error probabilities on the stopping rule, seeming to lend frequentist support to the stopping rule principle. Keywords:Bayes factor; Composite hypothesis; Conditional test; Error probability; Likelihood ratio; Sequential probability ratio test; Sequential test.

A sequential test for correlation coefficients is presented. The method consists of reducing the test of a correlation coefficient to that of a variance ratio and making use of Helmert transformations of sequential observations. The resulting test is identified as the sequential probability ratio test regarding the parameter of the Cauchy distribution. The average sample number of the test is given and it is shown that the proposed sequential test yields an appreciable saving in the required average number of observations compared to the usual fixed sample size test.

We consider a single-hop, random-access, wireless sensor network (WSN) performing sequential distributed detection under a bandwidth constraint. At each time slot, the sensor nodes will make their decisions on whether the observed event is happening. The sensor censoring strategy is applied such that only the local decisions equal to one (i.e., the event happens) will be sent to the fusion center (FC). Because only one transmission channel between the FC and the sensor nodes is assumed, if two or more local decisions are sent at a time slot, a packet collision happens. We design a sequential probability ratio test (SPRT) at the FC that is aware of the collisions. The performance measures of the collision-aware SPRT are analyzed based on a Markov chain. In addition, we derive the asymptotic relative efficiency of the collision-aware SPRT relative to another SPRT. Numerical results show some interesting characteristics of the collision-aware SPRT. For example, the collision-aware SPRT prefers the scenario that many local decisions are likely to be sent per time slot.

The backstepping control design procedure has been used to develop stabilizing controllers for time invariant plants that are linear or belong to some class of nonlinear systems. The use of such a procedure to design stabilizing controllers for plants with time varying parameters has been an open problem. In this paper we consider the backstepping design procedure for linear time varying (LTV) plants with known and unknown parameters. We first show that a backstepping controller can be designed for an LTV plant by following the same steps as in the linear time-invariant (LTI) case and treating the plant parameters as constants at each time t. Its stability properties however cannot be established by using the same Lyapunov function and techniques as in the LTI case. We then introduce a new parametrization and filter structure that takes into account the plant parameter variations leading to a new backstepping controller. The new control design guarantees exponential convergence of the tracking error to zero if the plant parameters are exactly known. If the parameters are not precisely known but the time variations of the parameters associated with the system zeros are known, the appropriate choice of certain design parameters, without using any adaptive law, leads to closed-loop stability and perfect regulation. This new control design is modified and supplemented with an update law to be applicable to LTV plants with unknown parameters. In the adaptive control design, the notion of structured parameter variations is used in order to include possible a priori information about the plant parameter variations. With this formulation, only the unstructured plant parameters are estimated and are required to be slowly time varying, and the structured plant parameters are allowed to have any finite speed of variation. The adaptive controller is shown to be robust with respect to the unknown but slow parameter variations in the global sense. We derive performance bounds which can be used to select certain design parameters for performance improvement. The properties of the proposed control scheme are demonstrated using simulation results.

In this paper, we will discuss the sequential probability ratio test to test the zero inflation in counting data with the sample size being a random variable with specified Type I and Type II errors. We will introduce the zero-inflated non-central negative binomial distribution, and explain how its operating characteristic and average sample number functions are derived. Simulations are studied to indicate the performances of the proposed tests with different scenarios. A real application is also discussed to explain the process of the sequential probability ratio test.

We apply high-dimensional analysis of variance (HANOVA) and sequential probability ratio test (SPRT) to detect buried land mines from array ground penetrating radar (GPR) measurements. The GPR array surveys a region of interest in a progressive manner starting at a known position and moving step by step in a fixed direction. Our detection method consists of two stages. Because, at each stop of the array the path lengths are different from every transmitter/receiver pair to a mien target, there exists statistically significant difference among received signals when a mine target is presented. Thus, the first step in our processing consists of a HANOVA test to detect this statistical difference at each stop. HANOVA does not incorporate new data as the GPR array moves down-track. So secondly, we resort to a sequential probability ratio test to look for changes in the HANOVA statistics as the array proceeds down track. The SPRT allows for real-time processing as anew data are obtained by the GPR array. Finally, real sensor data are processed to verify the method.