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That is, the data is generated by or .

Spiders make silk of various types, numbers, properties and purposes that vary with the species, environment, dietary composition and metabolic status of the organism (7). Not surprisingly, early research has focused primarily on the mechanical properties of spider silk with applications in light and strong composite materials in mind.

Call  the ``null'' hypothesis and  the ``alternative''.

Replace the stresses in the force and moment resultants with strains via the constitutive equations, we haveBy applying the summation and integration operations to their respective components, the force and moment resultants can be further simplified toCombine the above equations we can write:where is called the extensional stiffness, is called the coupling stiffness, and is called the bending stiffness of the laminate.

The alternative hypothesis indicates a disturbance is present.

If we decide , we signal an ``alarm'' for the disturbance.

During the process of synthesis of the crystals on the silk fiber, it is possible to make it a porous composite. This method would entail altering the calcium carbonate crystal structures, the quantity and the packing orientation of the calcium carbonate layers by changing the conditions during crystal formation (like pH, temperature, calcium ion concentration), alteration of the silk fiber surface morphology and the rate of bubbling of carbon dioxide.

Previous research showed that sealing restorations increases their longevity and clinical performance (14, 15). Sealants are already currently used in elementary school-based programs to prevent caries (cavities) formation in the pit and fissure surfaces of posterior teeth. Given that these sealants are responsible for reducing the occurrence of new decay by 60%, they will likely play a similar role with restorations (16). Therefore, we anticipate a layer of sealing composite resin will not only reduce mercury release, but also protect a restoration from physical forces, extend its lifespan, and reduce the risk of new caries formation.

Then, as , and for large , we get:

We define, where  and  are densities:The following facts about Kullback-Leibler distance hold:

Abstract: In this paper, we consider the detection problem with intermittent observations, due to the unreliable shared communication link between local sensors and the fusion center. Detection performance is analyzed using Neyman-Pearson criterion of maximizing the probability of detection, for a given probability of false alarm. The detector performance is compared, with and without intermittent observations, and a formal approach is presented to restore the original detector performance. Keywords: NP Detection; Wireless sensor network; Passive sensor; Hypothesis testing; Z-test. 1.

For the universal hypothesis testing problem, where the goal is to decide between the known null hypothesis distribution and some other unknown distribution, Hoeffding proposed a universal test in the nineteen sixties. Hoeffding's universal test statistic can be written in terms of KullbackLeibler (K-L) divergence between the empirical distribution of the observations and the null hypothesis distribution. In this paper a modification of Hoeffding's test is considered based on a relaxation of the K-L divergence, referred to as the mismatched divergence. The resulting mismatched test is shown to be a generalized likelihood-ratio test (GLRT) for the case where the alternate distribution lies in a parametric family of distributions characterized by a finite-dimensional parameter, i.e., it is a solution to the corresponding composite hypothesis testing problem. For certain choices of the alternate distribution, it is shown that both the Hoeffding test and the mismatched test have the same asymptotic performance in terms of error exponents. A consequence of this result is that the GLRT is optimal in differentiating a particular distribution from others in an exponential family. It is also shown that the mismatched test has a significant advantage over the Hoeffding test in terms of finite sample size performance for applications involving large alphabet distributions. This advantage is due to the difference in the asymptotic variances of the two test statistics under the null hypothesis.

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JO - IEEE Transactions on Information Theory

A novel approach is presented for the long-standing problem of composite hypothesis testing. In composite hypothesis testing, unlike in simple hypothesis testing, the probability function of the observed data given the hypothesis, is uncertain as it depends on the unknown value of some parameter. The proposed approach is to minimize the worst-case ratio between the probability of error of a decision rule that is independent of the unknown parameters and the minimum probability of error attainable given the parameters. The principal solution to this minimax problem is presented and the resulting decision rule is discussed. Since the exact solution is, in general, hard to find, and a-fortiori hard to implement, an approximation method that yields an asymptotically minimax decision rule is proposed. Finally, a variety of potential application areas are provided in signal processing and communications with special emphasis on universal decoding.

T2 - IEEE Transactions on Information Theory

Abstract—This paper is concerned with error exponents in testing problems raised by autoregressive (AR) modeling. The tests to be considered are variants of generalized likelihood ratio testing corresponding to traditional approaches to autoregressive moving-average (ARMA) modeling estimation. In several related problems, such as Markov order or hidden Markov model order estimation, optimal error exponents have been determined thanks to large deviations theory. AR order testing is specially chal-lenging since the natural tests rely on quadratic forms of Gaussian processes. In sharp contrast with empirical measures of Markov chains, the large deviation principles (LDPs) satisfied by Gaussian quadratic forms do not always admit an information-theoretic representation. Despite this impediment, we prove the existence of nontrivial error exponents for Gaussian AR order testing. And furthermore, we exhibit situations where the exponents are optimal. These results are obtained by showing that the log-like-lihood process indexed by AR models of a given order satisfy an LDP upper bound with a weakened information-theoretic representation. Index Terms—Error exponents, Gaussian processes, large devi-ations, Levinson–Durbin, order, test, time series.

JF - IEEE Transactions on Information Theory

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So the K-L distance is not a metric.

N2 - For the universal hypothesis testing problem, where the goal is to decide between the known null hypothesis distribution and some other unknown distribution, Hoeffding proposed a universal test in the nineteen sixties. Hoeffding's universal test statistic can be written in terms of KullbackLeibler (K-L) divergence between the empirical distribution of the observations and the null hypothesis distribution. In this paper a modification of Hoeffding's test is considered based on a relaxation of the K-L divergence, referred to as the mismatched divergence. The resulting mismatched test is shown to be a generalized likelihood-ratio test (GLRT) for the case where the alternate distribution lies in a parametric family of distributions characterized by a finite-dimensional parameter, i.e., it is a solution to the corresponding composite hypothesis testing problem. For certain choices of the alternate distribution, it is shown that both the Hoeffding test and the mismatched test have the same asymptotic performance in terms of error exponents. A consequence of this result is that the GLRT is optimal in differentiating a particular distribution from others in an exponential family. It is also shown that the mismatched test has a significant advantage over the Hoeffding test in terms of finite sample size performance for applications involving large alphabet distributions. This advantage is due to the difference in the asymptotic variances of the two test statistics under the null hypothesis.

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