Xiaoou Li

Generalized Sequential Probability Ratio Test for Separate Families of Hypotheses

Abstract In this article, we consider the problem of testing two separate families of hypotheses via a generalization of the sequential probability ratio test. In particular, the generalized likelihood ratio statistic is considered and the stopping rule is the first boundary crossing of the generalized likelihood ratio statistic. We show that this sequential test is asymptotically optimal in the sense that it achieves asymptotically the shortest expected sample size as the maximal type I and type II error probabilities tend to zero.

Decentralized Sequential Composite Hypothesis Test Based on One-Bit Communication

This paper considers the sequential composite hypothesis test with multiple sensors. The sensors observe random samples in parallel and communicate with a fusion center, who makes the global decision based on the sensor inputs. On the one hand, in the centralized scenario, where local samples are precisely transmitted to the fusion center, the generalized sequential likelihood ratio test (GSPRT) is shown to be asymptotically optimal in terms of the expected sample size as error rates tend to zero. On the other hand, for systems with limited power and bandwidth resources, decentralized solutions that only send a summary of local samples (we particularly focus on a one-bit communication protocol) to the fusion center is of great importance. To this end, we first consider a decentralized scheme where sensors send their one-bit quantized statistics every fixed period of time to the fusion center. We show that such a uniform sampling and quantization scheme is strictly suboptimal and its suboptimality can be quantified by the KL divergence of the distributions of the quantized statistics under both the hypotheses. We then propose a decentralized GSPRT based on level-triggered sampling. That is, each sensor runs its own GSPRT repeatedly and reports its local decision to the fusion center asynchronously. We show that this scheme is asymptotically optimal as the local thresholds and global thresholds grow large at different rates. Finally, two particular models and their associated applications are studied to compare the centralized and decentralized approaches. Numerical results are provided to demonstrate that the proposed level-triggered sampling based decentralized scheme aligns closely with the centralized scheme with substantially lower communication overhead, and significantly outperforms the uniform sampling and quantization-based decentralized scheme.

Chernoff index for Cox test of separate parametric families

The asymptotic efficiency of a generalized likelihood ratio test proposed by Cox is studied under the large deviations framework for error probabilities developed by Chernoff. In particular, two separate parametric families of hypotheses are considered (Cox, 1961, 1962). The significance level is set such that the maximal type I and type II error probabilities for the generalized likelihood ratio test decay exponentially fast with the same rate. We derive the analytic form of such a rate that is also known as the Chernoff index (Chernoff, 1952), a relative efficiency measure when there is no preference between the null and the alternative hypotheses. We further extend the analysis to approximate error probabilities when the two families are not completely separated. Discussions are provided concerning the implications of the present result on model selection.

Rare-event simulation and efficient discretization for the supremum of Gaussian random fields

In this paper we consider a classic problem concerning the high excursion probabilities of a Gaussian random field f living on a compact set T. We develop efficient computational methods for the tail probabilities ℙ{sup T f(t) > b}. For each positive ε, we present Monte Carlo algorithms that run in constant time and compute the probabilities with relative error ε for arbitrarily large b. The efficiency results are applicable to a large class of Hölder continuous Gaussian random fields. Besides computations, the change of measure and its analysis techniques have several theoretical and practical indications in the asymptotic analysis of Gaussian random fields.

Recommendation System for Adaptive Learning

An adaptive learning system aims at providing instruction tailored to the current status of a learner, differing from the traditional classroom experience. The latest advances in technology make adaptive learning possible, which has the potential to provide students with high-quality learning benefit at a low cost. A key component of an adaptive learning system is a recommendation system, which recommends the next material (video lectures, practices, and so on, on different skills) to the learner, based on the psychometric assessment results and possibly other individual characteristics. An important question then follows: How should recommendations be made? To answer this question, a mathematical framework is proposed that characterizes the recommendation process as a Markov decision problem, for which decisions are made based on the current knowledge of the learner and that of the learning materials. In particular, two plain vanilla systems are introduced, for which the optimal recommendation at each stage can be obtained analytically.

Exploratory Item Classification Via Spectral Graph Clustering

Large-scale assessments are supported by a large item pool. An important task in test development is to assign items into scales that measure different characteristics of individuals, and a popular approach is cluster analysis of items. Classical methods in cluster analysis, such as the hierarchical clustering, K-means method, and latent-class analysis, often induce a high computational overhead and have difficulty handling missing data, especially in the presence of high-dimensional responses. In this article, the authors propose a spectral clustering algorithm for exploratory item cluster analysis. The method is computationally efficient, effective for data with missing or incomplete responses, easy to implement, and often outperforms traditional clustering algorithms in the context of high dimensionality. The spectral clustering algorithm is based on graph theory, a branch of mathematics that studies the properties of graphs. The algorithm first constructs a graph of items, characterizing the similarity structure among items. It then extracts item clusters based on the graphical structure, grouping similar items together. The proposed method is evaluated through simulations and an application to the revised Eysenck Personality Questionnaire.

On the Tail Probabilities of Aggregated Lognormal Random Fields with Small Noise

We develop asymptotic approximations for the tail probabilities of integrals of lognormal random fields. We consider the asymptotic regime that the variance of the random field converges to zero. Under this setting, the integral converges to its limiting value. This analysis is of interest in considering short-term portfolio risk analysis (such as daily performance), for which the variances of log-returns could be as small as a few percent.

Multi-sensor generalized sequential probability ratio test using level-triggered sampling

This paper investigates the generalized sequential probability ratio test (GSPRT) with multiple sensors. Focusing on the communication-constrained scenario, where sensors transmit one-bit messages to the fusion center, we propose a decentralized GSRPT based on level-triggered sampling scheme (LTS-GSPRT). The proposed LTS-GSPRT amounts to the algorithm where each sensor successively reports the decisions of local GSPRTs to the fusion center. Interestingly, with significantly lower communication overhead, LTS-GSPRT preserves the same asymptotic performance of the centralized GSPRT as the local thresholds and global thresholds grow large at different rates.

Robust Measurement via A Fused Latent and Graphical Item Response Theory Model

Item response theory (IRT) plays an important role in psychological and educational measurement. Unlike the classical testing theory, IRT models aggregate the item level information, yielding more accurate measurements. Most IRT models assume local independence, an assumption not likely to be satisfied in practice, especially when the number of items is large. Results in the literature and simulation studies in this paper reveal that misspecifying the local independence assumption may result in inaccurate measurements and differential item functioning. To provide more robust measurements, we propose an integrated approach by adding a graphical component to a multidimensional IRT model that can offset the effect of unknown local dependence. The new model contains a confirmatory latent variable component, which measures the targeted latent traits, and a graphical component, which captures the local dependence. An efficient proximal algorithm is proposed for the parameter estimation and structure learning of the local dependence. This approach can substantially improve the measurement, given no prior information on the local dependence structure. The model can be applied to measure both a unidimensional latent trait and multidimensional latent traits.

Uniformly efficient simulation for extremes of Gaussian random fields

This paper considers the problem of simultaneously estimating rare-event probabilities for a class of Gaussian random fields. A conventional rare-event simulation method is usually tailored to a specific rare event and consequently would lose estimation efficiency for different events of interest, which often results in additional computational cost in such simultaneous estimation problem. To overcome this issue, we propose a uniformly efficient estimator for a general family of Holder continuous Gaussian random fields. We establish the asymptotic and uniform efficiency of the proposed method and also conduct simulation studies to illustrate its effectiveness.