Accuracy of sequential sampling plans based on Wald's sequential probability ratio test

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.

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.

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.

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.

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 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.

The expression for the Operating Characteristics (OC) and Average Sample Number (ASN) functions are derived under inspection error. Two types of error are defined i.e. (i) failing to find one or more of the nonconformities in the sample and (ii) declaring one or more nonconformities when none exist. It is seen by OC and ASN curves that inspector errors have considerable effects on the results of a sequential test for Poisson distribution.

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.

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.

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.

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.

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.