Chapter 6 - Stopping Time of Invariant Sequential Probability Ratio Testst

Monte Carlo operating characteristic (OC) and average sample number (ASN) functions were compared with Wald's OC and ASN equations for sequential sampling plans based on Wald's sequential probability ratio test (SPRT) using the binomial, negative binomial, normal, and Poisson distributions. This comparison showed that the errors inherent in Wald's equations as a result of "overshooting" the decision boundaries of the SPRT can be large. Relative errors increased for the OC and ASN equations as the difference between the null (θ0)) and alternative (θ1) test parameter values increased. Relative errors also increased for the ASN equation as the probabilities of type I (α) and type II (β) errors increased. For discrete distributions, the relative errors also increased as θ0 increased with θ1/θ0 fixed. Wald's equations, in general, overestimate the true error probabilities and underestimate the true ASN. For the values of θ0, θ1, α, and β used in many sequential sampling plans in forestry, Wald's equations may not be adequate. For those cases where the errors in Wald's equations are important compared with the other errors associated with the sampling plan, two alternative Monte Carlo OC and ASN functions are proposed.

The virtual bumper is a safeguarding mechanism for autonomous and remotely operated robots to conservatively avoid collisions even in the face of sensor uncertainty. In this paper we take a new approach to the virtual bumper system by applying a powerful but rarely examined statistical test. By using a modified version of Wald's sequential probability ratio test, we demonstrate that we can reduce the number of false positives reported by the virtual bumper, thereby saving valuable mission time. We further use the concept of sequential probability ratio to control vehicle speed in the presence of possible obstacles in order to increase certainty about whether or not obstacles are present. With this principled obstacle certainty measure, our new algorithm reduces the chances of collision by approximately 98% relative to traditional virtual bumper safeguarding without speed control.

An algorithm is developed for determining exact values of the Operating Characteristic and Average Sample Number functions for Wald's Sequential Probability Ratio Test (SPRT) when observations are drawn from a discrete, Koopman‐Darmois density with positive probability on the nonnegative integers. The sample size distribution and percentiles of sample size are also considered. An example is given for the negative binomial distribution.

Sequential sampling plans are appealing because fewer observations are needed than those needed for fixed sampling plans to make a terminating decision when insect populations are sparse or abundant. Wald's Sequential Probability Ratio Test (SPRT) commonly has been used in monitoring insect populations and damages from these populations. At times, particularly for moderate infestations, the SPRT with its parallel boundaries takes longer than a comparable fixed-sample-size method to make a terminating decision. An alternative sequential testing procedure called the 2-SPRT for the negative-binomial parameter P is demonstrated. The value of the exponent k is assumed to be known. In this procedure two one-sided SPRTs are performed simultaneously, which results in the convergent boundary lines. The average number of observations needed for the terminating decision for medium infestations (i.e., for P 1 < P < P 2 ) is observed to be considerably smaller than that for the SPRT with a slight increase in sampling effort when infestations are light or heavy. An application of the 2-SPRT is demonstrated using cotton f1eahopper, Pseudatomoscelis seriatus (Reuter), data obtained in southwestern Oklahoma.

It is supposed that there is a multisensor system in which each sensor performs sequential detection of a target. Then the binary decisions on target presence and absence are transmitted to a fusion center, which combines them to improve the performance of the system. We assume that sensors represent multichannel systems with possibly each one having different number of channels. Sequential detection of a target in each sensor is done by implementing a generalized Wald's sequential probability ratio test which is based on the maximum likelihood ratio statistic and allows one to fix the false alarm rate and the mis-detection rate at specified levels. We first show that this sequential detection procedure is asymptotically optimal for general statistical models in the sense of minimizing the expected sample size when the probabilities of errors vanish. We then construct the optimal non-sequential fusion rule that waits until all the local decisions in all sensors are made and then fuses them. It is optimal in the sense of maximizing the probability of target detection for a fixed probability of a false alarm or minimizing the maximal probability of error (minimax criterion). An analysis shows that the final decision can be made substantially more reliable even for a small number of sensors (3-5). The performance of the system is illustrated by the example of detecting a deterministic signal in correlated (color) Gaussian noise. In this example, we provide both the results of theoretical analysis and the results of Monte Carlo experiment. These results allow us to conclude that the use of the sequential detection algorithm substantially reduces the required resources of the system compared to the best non-sequential algorithm.© (2001) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

Sequential sampling plans based on the negative binomial distribution were developed for egg mass density of spruce budworm, Choristoneura fumiferana (Clemens), in Michigan's Upper Peninsula. Plans developed were modifications of Wald's sequential probability ratio test (SPRT) based on Monte Carlo simulation. Parameters of the negative binomial distribution were estimated by the maximum likelihood method. Sampling models developed classify egg mass populations into low and high categories using the number of egg masses on whole branch samples of balsam fir, Abies balsamifera (L.) Voss. Plans are presented for three pairs of population density hypotheses to give forest pest managers the flexibility to select a plan that best meets management objectives. Monte Carlo estimates indicated that Wald's average sample number (ASN) and operating characteristic (OC) equations underestimated the actual ASN values, and the actual alpha and beta error probabilities were smaller than the values prescribed by Wald's equations. Errors were sufficiently large to have serious economic implications during implementation of plans developed from Wald's approximate procedure. The sequential plans developed by modifying the decision boundaries of Wald's SPRT using the Monte Carlo procedure had error probabilities, OC functions, and ASN values that were approximately equal to those desired.

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

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 &amp;ge;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&amp;nbsp; 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.

Striacosta albicosta (Smith) (Lepidoptera: Noctuidae), is a native pest of dry beans (Phaseolus vulgaris L.) and corn (Zea mays L.). As a result of larval feeding damage on corn ears, S. albicosta has a narrow treatment window; thus, early detection of the pest in the field is essential, and egg mass sampling has become a popular monitoring tool. Three action thresholds for field and sweet corn currently are used by crop consultants, including 4% of plants infested with egg masses on sweet corn in the silking-tasseling stage, 8% of plants infested with egg masses on field corn with approximately 95% tasseled, and 20% of plants infested with egg masses on field corn during mid-milk-stage corn. The current monitoring recommendation is to sample 20 plants at each of five locations per field (100 plants total). In an effort to develop a more cost-effective sampling plan for S. albicosta egg masses, several alternative binomial sampling plans were developed using Wald's sequential probability ratio test, and validated using Resampling for Validation of Sampling Plans (RVSP) software. The benefit-cost ratio also was calculated and used to determine the final selection of sampling plans. Based on final sampling plans selected for each action threshold, the average sample number required to reach a treat or no-treat decision ranged from 38 to 41 plants per field. This represents a significant savings in sampling cost over the current recommendation of 100 plants.

The article presents particulars of developing a plant disease detection system based on analysis of photo-graphic images by deep convolutional neural networks. A original lightweight neural network architecture is used (only 13 480 trained parameters) that is tens and hundreds of times more compact than typical solutions. Real-life field data is used for training and testing, with photographs taken in adverse conditions: variation in hardware quality, angles, lighting conditions, scales (from macro shots of individual fragments of leaf and stem to several rose bushes in one picture), and complex disorienting backgrounds. An adaptive decision-making rule is used, based on the Bayes' theorem and Wald's sequential probability ratio test, in order to improve reliability of the results. A following example is provided: detection of disease on leaves and stems of rose from images taken in the visible spectrum. The authors were able attain the quality of 90.6% on real-life data (F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> score, one input image, test dataset).

Item response theory (IRT) has most often been used in research on computerized adaptive testing (CAT). Depending on the model used, IRT requires between 200 and 1,000 examinees for estimating item parameters. Thus, it is not practical for instructional designers to develop their own CAT based on the IRT model. Frick improved Wald's sequential probability ratio test (SPRT) by combining it with normative expert systems reasoning, referred to as an EXSPRT-based CAT. While previous studies were based on re-enactments from historical test data, the present study is the first to examine how well these adaptive methods function in a real-time testing situation. Results indicate that the EXSPRT-I significantly reduced test lengths and was highly accurate in predicting mastery. EXSPRT is apparently a viable and practical alternative to IRT for assessing mastery of instructional objectives.

The use of the cumulative sum (CUSUM) charting procedure in process monitoring is first described. The relationship between the CUSUM charting procedure and Wald's sequential probability ratio test is then explained. This is followed by three main approaches of determining the average run length. The strengths and weaknesses of the CUSUM charting procedure, with respect to the Shewhart charting procedure, are illustrated with simulated data sets. Various other aspects of the CUSUM chart are also discussed. These include estimation of process mean using the CUSUM, weighting of samples for signaling, fast initial response for a newly start‐up process, simultaneous CUSUM schemes, design of CUSUM chart, multivariate CUSUM schemes, and risk‐adjusted CUSUM charts.

We study sequential collusion-resistant fingerprinting, where the fingerprinting code is generated in advance but accusations may be made between rounds, and show that in this setting both the dynamic Tardos scheme and schemes building upon Wald's sequential probability ratio test (SPRT) are asymptotically optimal. We further compare these two approaches to sequential fingerprinting, highlighting differences between the two schemes. Based on these differences, we argue that Wald's scheme should in general be preferred over the dynamic Tardos scheme, even though both schemes have their merits. As a side result, we derive an optimal sequential group testing method for the classical model, which can easily be generalized to different group testing models.

The banana root borer, Cosmopolites sordidus Germar (Coleoptera: Curculionidae), is a major pest in many banana (Musa spp.)‐producing regions of the world, causing yield losses of up to 100%. The aims of this study were to define the spatial distribution pattern of C. sordidus in banana plantations, to determine the probability distribution model that best describes the sampling data, and to develop a sequential sampling plan for this pest. Aggregation was investigated using aggregation indices and theoretical frequency distributions. To quantify the number of insects, 80 traps were distributed in a commercial banana plantation with an area of 4 ha and inspections were performed every 2 wk from 2005 to 2008. The sequential sampling plan was based on Wald's sequential probability ratio test. The calculated indices indicated moderate aggregation, with a common k parameter of the negative binomial distribution equal to 3.29. The best model for predicting the spatial distribution of the insects was the negative binomial distribution, which fit 93% of the inspections, followed by the Neyman Type A distribution, which fit 90% of the surveys, and Poisson's distribution, which fit only one survey. A sequential sampling plan was generated with Type I and II error rates of 5%, for which 19 was the maximum expected number of samples required to reach a decision for an infestation mean of approximately 3.7 insects per trap. The validation showed that the constructed plan is reliable and provides correct recommendations for decision‐making in pest control. The banana root borer has an aggregated spatial distribution in the field. Negative binomial distribution is the best model to explain species spatial pattern. A sequential sampling plan was developed and validated. The sequential sampling plan showed reliable decision rates. The sequential sampling plan showed a reasonable number of required sample units.

ABSTRACTWe develop truncated sequential probability ratio test (SPRT) procedures for multivariate normal data. The framework includes a general cost structure and arbitrary mean and covariance structures. The truncated SPRT solutions have a practical and easy-to-use decision boundary representation. In the homogeneous case, a very fast recursive algorithm is presented for calculating the decision boundaries. Misclassification rates and expected sample size are investigated and the results are compared with a nonsequential procedure. A real-life data set on kidney dysfunction following heart surgery is used to illustrate the truncated SPRT procedure.