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Sparse clustering divides samples into distinct groups while simultaneously reducing dimensionality in ultra-high dimensional unsupervised learning. To achieve sparse clustering that can screening out noise variables, an adaptive sparse clustering framework based on sufficient variable screening, abbreviated as adaptive sufficient sparse clustering (ASSC), is developed by controlling false discovery. Without any specific model, ASSC employs a composite hypothesis testing procedure that leverages conditional marginal correlations of variables across distinct groups, aiming to pinpoint sufficient variables via sparse clustering and promoting the performance of sparse clustering in ultra-high dimensionality. To control false discoveries of the hypothesis testing procedure at a predetermined level, ASSC provides the adaptive threshold for identifying conditional marginal dependency, which ensures to accuracy of clustering with high probability. Under mild conditions, the sufficient screening properties and sufficient clustering properties of ASSC are established based on variable screening procedure by controlling false discovery. Numerical studies on synthetic data and real datasets corroborate the performance and flexibility of ASSC, and underscore its potent utility in unsupervised learning. [ABSTRACT FROM AUTHOR]

AbstractFor the hierarchical gamma and gamma model, we calculate the Bayes estimator of the rate parameter of the gamma distribution under Stein’s loss function which penalizes gross overestimation and gross underestimation equally and the corresponding Posterior Expected Stein’s Loss (PESL). We also obtain the Bayes estimator of the rate parameter under the squared error loss and the corresponding PESL. Moreover, we obtain the empirical Bayes estimators of the rate parameter of the gamma distribution with a conjugate gamma prior by the moment and MLE methods. In numerical simulations, we have illustrated five aspects: the two inequalities of the Bayes estimators and the PESLs for the oracle method; the moment estimators and the Maximum Likelihood Estimators (MLEs) are consistent estimators of the hyperparameters; the goodness-of-fit of the model to the simulated data; the comparisons of the Bayes estimators and the PESLs of the oracle, moment, and MLE methods; and the marginal densities of the model for various hyperparameters. The numerical results indicate that the MLEs are better than the moment estimators when estimating the hyperparameters. Finally, the model could potentially be used to fit right-skewed data, not left-skewed data. [ABSTRACT FROM AUTHOR]

For the hierarchical uniform and inverse gamma model, we calculate the Bayes posterior estimator of the parameter of the uniform distribution under Stein's loss function which penalizes gross overestimation and gross underestimation equally and the corresponding Posterior Expected Stein's Loss (PESL). We also obtain the Bayes posterior estimator of the parameter under the squared error loss function and the corresponding PESL. Moreover, we obtain empirical Bayes estimators of the parameter of the uniform distribution by two methods. Note that the estimators of the hyperparameters of the model by the Maximum Likelihood Estimation (MLE) method are summarized in a theorem, whose proof involves the upper incomplete gamma function and a special case of the Meijer G-function. In numerical simulations, we address from four perspectives. First, we exemplify the two inequalities of the Bayes posterior estimators and the PESLs. Second, we illustrate that the moment estimators and the Maximum Likelihood Estimators (MLEs) are consistent estimators of the hyperparameters. Third, we calculate the goodness-of-fit of the model for the simulated data. Fourth, we plot the marginal densities of the model for various hyperparameters. Finally, we utilize the current prices of the 300 component stocks of Shenzhen 300 Index to illustrate our theoretical studies. [ABSTRACT FROM AUTHOR]

In this paper, we consider aggregated functional data composed by a linear combination of component curves and the problem of estimating these component curves. We propose the application of a bayesian wavelet shrinkage rule based on a mixture of a point mass function at zero and the logistic distribution as prior to wavelet coefficients to estimate mean curves of components. This procedure has the advantage of estimating component functions with important local characteristics such as discontinuities, spikes and oscillations for example, due the features of wavelet basis expansion of functions. Simulation studies were done to evaluate the performance of the proposed method, and its results are compared with a spline-based method. An application on the so-called Tecator dataset is also provided. [ABSTRACT FROM AUTHOR]

In dose-finding (DF) trials, methods for discovering an optimal criterion that controls toxicity while demonstrating a potential for efficacy have been the subject of statistical research. Although continual reassessment methods (CRMs) have been utilized, the consensus among practitioners and clinical trial specialists is that there is always room for improvement with CRMs. Within the paradigm of a full-Bayesian method for CRMs, we examine the performance of dose selection algorithms based on a family of loss functions defined in Huber (The Annals of Mathematical Statistics, vol. 35, issue 1, pp. 73–101, 1964), namely, the Huber loss function, with a special focus on the modified Huber loss function (MHLF). Our exploration suggests that, compared to Bayesian optimal interval design (BOIN) and toxicity interval loss function (TILF), the approach based on the MHLF has been able to select correct doses with fewer average number of patients across the spectrum of dose escalation schemes such as those seen in clinical settings. [ABSTRACT FROM AUTHOR]

For the hierarchical normal and inverse gamma model, we calculate the Bayes posterior estimator of the variance parameter of the normal distribution under Stein's loss function which penalizes gross overestimation and gross underestimation equally and the corresponding Posterior Expected Stein's Loss (PESL). We also obtain the Bayes posterior estimator of the variance parameter under the squared error loss function and the corresponding PESL. Moreover, we obtain the empirical Bayes estimators of the variance parameter of the normal distribution with a conjugate inverse gamma prior by two methods. In numerical simulations, we have illustrated five aspects: The two inequalities of the Bayes posterior estimators and the PESLs; the moment estimators and the Maximum Likelihood Estimators (MLEs) are consistent estimators of the hyperparameters; the goodness-of-fit of the model to the simulated data; the numerical comparisons of the Bayes posterior estimators and the PESLs of the oracle, moment, and MLE methods; and the plots of the marginal densities for various hyperparameters. The numerical results indicate that the MLEs are better than the moment estimators when estimating the hyperparameters. Finally, we utilize the %bodyfat data of 250 men of various ages to illustrate our theoretical studies. [ABSTRACT FROM AUTHOR]

Statistical inference can be seen as information processing involving input information and output information that updates belief about some unknown parameters. We consider the Bayesian framework for making inferences about dynamical systems from ergodic observations, where the Bayesian procedure is based on the Gibbs posterior inference, a decision process generalization of standard Bayesian inference (see [7, 37]) where the likelihood is replaced by the exponential of a loss function. In the case of direct observation and almost-additive loss functions, we prove an exponential convergence of the a posteriori measures to a limit measure. Our estimates on the Bayes posterior convergence for direct observation are related and extend those in [47] to a context where loss functions are almost-additive. Our approach makes use of non-additive thermodynamic formalism and large deviation properties [39-40, 57] instead of joinings. [ABSTRACT FROM AUTHOR]

Scenario optimization is a broad methodology to perform optimization based on empirical knowledge. One collects previous cases, called "scenarios", for the set-up in which optimization is being performed, and makes a decision that is optimal for the cases that have been collected. For convex optimization, a solid theory has been developed that provides guarantees of performance, and constraint satisfaction, of the scenario solution. In this paper, we open a new direction of investigation: the risk that a performance is not achieved, or that constraints are violated, is studied jointly with the complexity (as precisely defined in the paper) of the solution. It is shown that the joint probability distribution of risk and complexity is concentrated in such a way that the complexity carries fundamental information to tightly judge the risk. This result is obtained without requiring extra knowledge on the underlying optimization problem than that carried by the scenarios; in particular, no extra knowledge on the distribution by which scenarios are generated is assumed, so that the result is broadly applicable. This deep-seated result unveils a fundamental and general structure of data-driven optimization and suggests practical approaches for risk assessment. [ABSTRACT FROM AUTHOR]

The article presents the discussion on reef monitoring programs playing an important role in managing reef activities. Topics include providing vital services and products such as maintaining the carbon dioxide level in the water; and coral reefs being an example of such ecosystems supporting enormous biodiversity and being essential to the coastal populations surrounding them.

A latent class model is proposed to examine couples’ breadwinning typologies and explain the wage differentials according to the socio-demographic characteristics of the society with data collected through surveys. We derive an ordinal variable indicating the couple’s income provision-role type and suppose the existence of an underlying discrete latent variable to model the effect of covariates. We use a two-step maximum likelihood inference conducted to account for concomitant variables, informative sampling scheme and missing responses. The weighted log-likelihood is maximised through the Expectation-Maximization algorithm and information criteria are used to develop the model selection. Predictions are made on the basis of the maximum posterior probabilities. Disposing of data collected in Japan over thirty years we compare couples’ breadwinning patterns across time. We provide some evidence of the gender wage-gap and we show that it can be attributed to the fact that, especially in Japan, duties and responsibilities for the child care are supported exclusively by women.

The present study aims to explore the effects of an adapted classical dance intervention on the psychological and functional status of institutionalized elder people using a Bayesian network. All participants were assessed at baseline and after the 9 weeks period of the intervention. Measures included balance and gait, psychological well-being, depression, and emotional distress.

For the hierarchical inverse gamma and inverse gamma model, we calculate the Bayes posterior estimator of the rate parameter of the inverse gamma distribution under Stein's loss function which penalizes gross overestimation and gross underestimation equally and the corresponding Posterior Expected Stein's Loss (PESL). We also obtain the Bayes posterior estimator of the rate parameter under the squared error loss and the corresponding PESL. Moreover, we obtain the empirical Bayes estimators of the rate parameter of the inverse gamma distribution with a conjugate inverse gamma prior by the moment and MLE methods. In numerical simulations, we have illustrated four aspects: Consistency, goodness-of-fit, comparison, and marginal densities. The numerical results indicate that the MLEs are better than the moment estimators when estimating the hyperparameters. Finally, the model could potentially be used to fit right skewed data, not left-skewed data. [ABSTRACT FROM AUTHOR]

Two-stage hierarchical models have been widely used in small area estimation to produce indirect estimates of areal means. When the areas are treated exchangeably and the model parameters are assumed to be the same over all areas, we might lose the efficiency in the presence of spatial heterogeneity. To overcome this problem, we consider a two-stage area-level model based on natural exponential family with spatially varying model parameters. We employ geographically weighted regression approach to estimating the varying parameters and suggest a new empirical Bayes estimator of the areal mean. We also discuss some related problems, including the mean squared error estimation, benchmarked estimation and estimation in non-sampled areas. The performance of the proposed method is evaluated through simulations and applications to two data sets. [ABSTRACT FROM AUTHOR]

Movements of equity indices are very important information for an investment decision. Empirical studies illustrate that the movements switch among different regimes. The Markov regime-switching model has important applications to such analysis. However, parameters estimated under normality assumption might not be stable and the corresponding change-point detection algorithm might face some challenges when either the error distribution is heavy-tailed or observed data contain outliers. In this paper, we relax the normality assumption and propose a generalized Markov regime-switching (GMRS) model. We propose a GMRS model based change-point detection algorithm, which is tested on both simulation data and Hang Seng monthly index. Simulation studies show that this algorithm can improve the accuracy of identifying change-points when either the error distribution is heavy-tailed or observed data contain outliers. It is also evident that the identified change-points on Hang Seng monthly index data match the observed market behaviours. [ABSTRACT FROM AUTHOR]

For the hierarchical Poisson and gamma model, we calculate the Bayes posterior estimator of the parameter of the Poisson distribution under Stein's loss function which penalizes gross overestimation and gross underestimation equally and the corresponding Posterior Expected Stein's Loss (PESL). We also obtain the Bayes posterior estimator of the parameter under the squared error loss and the corresponding PESL. Moreover, we obtain the empirical Bayes estimators of the parameter of the Poisson distribution with a conjugate gamma prior by two methods. In numerical simulations, we have illustrated: The two inequalities of the Bayes posterior estimators and the PESLs; the moment estimators and the Maximum Likelihood Estimators (MLEs) are consistent estimators of the hyperparameters; the goodness-of-fit of the model to the simulated data. The numerical results indicate that the MLEs are better than the moment estimators when estimating the hyperparameters. Finally, we exploit the attendance data on 314 high school juniors from two urban high schools to illustrate our theoretical studies. [ABSTRACT FROM AUTHOR]

Most of the samples in the real world are from the normal distributions with unknown mean and variance, for which it is common to assume a conjugate normal-inverse-gamma prior. We calculate the empirical Bayes estimators of the mean and variance parameters of the normal distribution with a conjugate normal-inverse-gamma prior by the moment method and the Maximum Likelihood Estimation (MLE) method in two theorems. After that, we illustrate the two theorems for the monthly simple returns of the Shanghai Stock Exchange Composite Index. [ABSTRACT FROM AUTHOR]

Empirical Bayes (EB) methods are very useful for post selection inference. Following Datta et al. (2002), we construct EB confidence intervals for the selected population mean. The EB intervals are adjusted to achieve the target coverage probabilities asymptotically up to the second order. Both unconditional coverage probabilities of EB intervals and corresponding probabilities conditional on ancillary statistics are found. [ABSTRACT FROM AUTHOR]

In this paper, we devote to constructing the one-sided empirical Bayes (EB) test for the location parameter in the Gamma distribution by nonparametric method. Under some mild conditions, we prove that the EB test is asymptotically optimal with the rate of the order O(n−δs2s+1), where 1/2 ≤ δ < 1 and s > 1 is a given natural number. An example is also given to illustrate that the conditions of the main theorems are easily satisfied. [ABSTRACT FROM AUTHOR]

We consider parametric exponential families of dimension K on the real line. We study a variant of boundary crossing probabilities coming from the multi-armed bandit literature, in the case when the real-valued distributions form an exponential family of dimension K. Formally, our result is a concentration inequality that bounds the probability that Bψ(θ̂n, θ*) ≥ f(t/n)/n, where θ* is the parameter of an unknown target distribution, θ̂n is the empirical parameter estimate built from n observations, ψ is the log-partition function of the exponential family and Bψ is the corresponding Bregman divergence. From the perspective of stochastic multi-armed bandits, we pay special attention to the case when the boundary function f is logarithmic, as it is enables to analyze the regret of the state-of-the-art KL-ucb and KL-ucb+ strategies, whose analysis was left open in such generality. Indeed, previous results only hold for the case when K = 1, while we provide results for arbitrary finite dimension K, thus considerably extending the existing results. Perhaps surprisingly, we highlight that the proof techniques to achieve these strong results already existed three decades ago in the work of T. L. Lai, and were apparently forgotten in the bandit community. We provide a modern rewriting of these beautiful techniques that we believe are useful beyond the application to stochastic multi-armed bandits. [ABSTRACT FROM AUTHOR]

In a functional setting, we elaborate and study two test statistics to highlight the Poisson nature of a Cox process when n copies of the process are available. Our approach involves a comparison of the empirical mean and the empirical variance of the functional data and can be seen as an extended version of a classical overdispersion test for count data. The limiting distributions of our statistics are derived using a functional central limit theorem for càdlàg martingales. Our procedures are easily implementable and do not require any knowledge on the covariate. We address a theoretical comparison of the asymptotic power of our tests under some local alternatives. A numerical study reveals the good performances of the method. We also present two applications of our tests to real datasets. [ABSTRACT FROM AUTHOR]

We construct a two-sided empirical Bayes (EB) test for the location parameter in the gamma distribution by non parametric method. Under some mild conditions, we proved that the EB test is asymptotically optimal with the rate of the order, where 1/2 ⩽ δ<; 1 ands>; 1 is a given natural number. [ABSTRACT FROM AUTHOR]

For decision purpose, one of the commonly used statistical applications is the comparison of two or more objects or characteristics. Sometimes, it is not possible to compare the objects at a time or when the number of objects under study is large and the differences between the objects become small, then a useful way is to compare them in pairwise manner. Because of its practical nature, the fields in which paired comparison techniques are being used are numerous. Many Bayesian statisticians have focused their attention on the practical and usable paired comparison technique and have successfully performed the Bayesian study of many of the paired comparison models. In the current study, analysis of the amended Davidson model (ADM) which has been extended after incorporating the order effect parameter is narrated. For this intention, both the informative and non informative priors are used. The said model is studied for the case of four treatments which are compared pairwise. [ABSTRACT FROM PUBLISHER]

This paper develops Bayesian analysis in the context of progressively Type II censored data from the compound Rayleigh distribution. The maximum likelihood and Bayes estimates along with associated posterior risks are derived for reliability performances under balanced loss functions by assuming continuous priors for parameters of the distribution. A practical example is used to illustrate the estimation methods. A simulation study has been carried out to compare the performance of estimates. The study indicates that Bayesian estimation should be preferred over maximum likelihood estimation. In Bayesian estimation, the balance general entropy loss function can be effectively employed for optimal decision-making. [ABSTRACT FROM PUBLISHER]

This article considers an empirical Bayes testing problem for the guarantee lifetime in the two-parameter exponential distributions with non identical components. We study a method of constructing empirical Bayes tests under a class of unknown prior distributions for the sequence of the component testing problems. The asymptotic optimality of the sequence of empirical Bayes tests is studied. Under certain regularity conditions on the prior distributions, it is shown that the sequence of the constructed empirical Bayes tests is asymptotically optimal, and the associated sequence of regrets converges to zero at a rateO(n− 1 + 1/[2(r + α) + 1]) for some integerr⩾ 0 and 0 ⩽ α ⩽ 1 depending on the unknown prior distributions, wherenis the number of past data available when the (n+ 1)st component testing problem is considered. [ABSTRACT FROM PUBLISHER]

The problem of selecting a population according to "selection and ranking" is an important statistical problem. The ideas in selecting the best populations with some demands having optimal criterion have been suggested originally by Bechhofer (1954) and Gupta (1956, 1965). In the area of ranking and selection, the large part of literature is connected with a single criterion. However, this may not satisfy the experimenter's demand. We followmethodology of Huang and Lai (1999) and themain focus of this article is to select a best population under Type-II progressively censored data for the case of right tail exponential distributions with a bounded and unbounded supports forμi. We formulate the problem and develop a Bayesian setup with two kinds of bounded and unbounded prior forμi. We introduce an empirical Bayes procedure and study the large sample behavior of the proposed rule. It is shown that the proposed empirical Bayes selection rule is asymptotically optimal. [ABSTRACT FROM AUTHOR]

This paper explores properties of the E-Bayesian and hierarchical Bayesian estimations of the system reliability parameter. E-Bayesian estimation and hierarchical Bayesian estimation of Pascal distribution's parameter under two loss function, LINEX loss function and entropy loss function can be found. We obtained limits of that the E-Bayesian estimation and hierarchical Bayesian estimation are equal. A Monte Carlo simulation is used to compare performances of the two methods. [ABSTRACT FROM PUBLISHER]

The combined universal probability M ( D ) of strings x in sets D is close to max x ∈ D ⁡ M ( { x } ) : their ∼logs differ by at most D 's information j = I ( D : H ) about the halting sequence H . Thus if all x have complexity K ( x ) ≥ k , D carries ≥ i bits of information on each x where i + j ∼ k . Note, there are no ways (whether natural or artificial) to generate D with significant I ( D : H ) . [ABSTRACT FROM AUTHOR]

This article considers the monotone empirical Bayes (MEB) test problem for the parameter of Pareto distribution with censored data. We construct the MEB test rule for the parameter of Pareto distribution under random censorship. It is shown that the proposed MEB test have asymptotically optimal property. The convergence rate of the MEB test is obtained and the rate can approach toO(n– ½) arbitrarily under suitable conditions. [ABSTRACT FROM PUBLISHER]

In this paper, the Bayes estimator and the parametric empirical Bayes estimator (PEBE) of mean vector in multivariate normal distribution are obtained. The superiority of the PEBE over the minimum variance unbiased estimator (MVUE) and a revised James-Stein estimators (RJSE) are investigated respectively under mean square error (MSE) criterion. Extensive simulations are conducted to show that performance of the PEBE is optimal among these three estimators under the MSE criterion. [ABSTRACT FROM AUTHOR]

We study empirical and hierarchical Bayes approaches to the problem of estimating an infinite-dimensional parameter in mildly ill-posed inverse problems. We consider a class of prior distributions indexed by a hyperparameter that quantifies regularity. We prove that both methods we consider succeed in automatically selecting this parameter optimally, resulting in optimal convergence rates for truths with Sobolev or analytic 'smoothness', without using knowledge about this regularity. Both methods are illustrated by simulation examples. [ABSTRACT FROM AUTHOR]

An algorithm for sampling from non-log-concave multivariate distributions is proposed, which improves the adaptive rejection Metropolis sampling (ARMS) algorithm by incorporating the hit and run sampling. It is not rare that the ARMS is trapped away from some subspace with significant probability in the support of the multivariate distribution. While the ARMS updates samples only in the directions that are parallel to dimensions, our proposed method, the hit and run ARMS (HARARMS), updates samples in arbitrary directions determined by the hit and run algorithm, which makes it almost not possible to be trapped in any isolated subspaces. The HARARMS performs the same as ARMS in a single dimension while more reliable in multidimensional spaces. Its performance is illustrated by a Bayesian free-knot spline regression example. We showed that it overcomes the well-known ‘lethargy’ property and decisively find the global optimal number and locations of the knots of the spline function. [ABSTRACT FROM AUTHOR]

In this article, we aim at assessing hierarchical Bayesian modeling for the analysis of multiple exposures and highly correlated effects in a multilevel setting. We exploit an artificial data set to apply our method and show the gains in the final estimates of the crucial parameters. As a motivating example to simulate data, we consider a real prospective cohort study designed to investigate the association of dietary exposures with the occurrence of colon-rectum cancer in a multilevel framework, where, e.g., individuals have been enrolled from different countries or cities. We rely on the presence of some additional information suitable to mediate the final effects of the exposures and to be arranged in a level-2 regression to model similarities among the parameters of interest (e.g., data on the nutrient compositions for each dietary item). [ABSTRACT FROM AUTHOR]

In classical Bayesian inference the prior is treated as fixed and its effects are ignored asymptotically, and useful information, if any, is wasted. However, in practice often an informative prior is summarized from previous similar or the same kind of studies, which contains useful cumulative information for the current study. We treat such prior to be non-fixed, i.e., we give the data sizes in the prior studies similar status as the that of the current dataset. Under this formulation, the prior is asymptotically non-negligible, and its original information is transferred to the new study. We explore some basic properties of Bayesian estimators under such prior formulation, and illustrate the method via simulation. [ABSTRACT FROM AUTHOR]

Various studies have been performed in relation to the influence that a number of characteristics of drinking water distribution systems (DWDSs) have on biofilm development. Nevertheless, their joint influence, apart from a few exceptions, has scarcely been studied due to the complexity of the community and the environment. In this paper, we apply various machine learning algorithms based on naïve Bayesian networks. Alternatives for the base naïve Bayesian model to outperform individual performances while maintaining simplicity are suggested. These alternatives include augmentation of the arcs in the graph, and initial bagging approaches. Finally, a combination of different naïve approaches in a bagging process that produces explanatory hybrid decision trees is proposed. As a result, it is possible to achieve a deeper understanding of the consequences that the interaction of the relevant hydraulic and physical factors of DWDSs has on biofilm development. [ABSTRACT FROM PUBLISHER]

Many algorithms originated from decision trees have been developed for classification problems. Although they are regarded as good algorithms, most of them suffer from loss of prediction accuracy, namely high misclassification rates when there are many irrelevant variables. We propose multi-step classification trees with adaptive variable selection (the multi-step GUIDE classification tree (MG) and the multi-step CRUISE classification tree (MC) to handle this problem. The variable selection step and the fitting step comprise the multi-step method. We compare the performance of classification trees in the presence of irrelevant variables. MG and MC perform better than Random Forest and C4.5 with an extremely noisy dataset. Furthermore, the prediction accuracy of our proposed algorithm is relatively stable even when the number of irrelevant variables increases, while that of other algorithms worsens [ABSTRACT FROM AUTHOR]

We derive the minimum risk estimates of the scalar means for Normal, Exponential, and Gamma distributions, under the convex combination of SEL and LIN EX loss functions. The functional forms of the proposed estimates for the three examples are general in nature, and for the boundary conditions provide us with the corresponding estimates under SEL and LINEX loss, respectively. We authenticate our proposed models using different iterative as well as meta-heurislic techniques, and through extensive simulation as well as application of live data sets, validate the efficacy of our proposed results. [ABSTRACT FROM AUTHOR]

The focus of this article is on the choice of suitable prior distributions for item parameters within item response theory (IRT) models. In particular, the use of empirical prior distributions for item parameters is proposed. Firstly, regression trees are implemented in order to build informative empirical prior distributions. Secondly, model estimation is conducted within a fully Bayesian approach through the Gibbs sampler, which makes estimation feasible also with increasingly complex models. The main results show that item parameter recovery is improved with the introduction of empirical prior information about item parameters, also when only a small sample is available. [ABSTRACT FROM AUTHOR]

This article deals with non identical components empirical Bayes testing for uniform distributions. First, we derive the Bayes rule. Then, by mimicking the behavior of the preceding Bayes rule, we construct a sequence of empirical Bayes tests [image omitted] for the sequence of component testing problem. The asymptotic optimality of [image omitted] is studied. It has been shown that [image omitted] possesses the asymptotic optimality, and the associated sequence of regrets converge to zero at a rate O(n-2(r+α)/[2(r+α)+1]), where n is the number of past data available when the present testing problem is considered, and r is a positive integer, 0 ≤ α ≤1, r and α depending on conditions pertaining to the unknown prior distribution. [ABSTRACT FROM AUTHOR]