Determination of uniformly most powerful tests in discrete sample spaces

Uniformly most powerful tests are statistical hypothesis tests that provide the greatest power against a fixed null hypothesis among all tests of a given size. In this article, the notion of uniformly most powerful tests is extended to the Bayesian setting by defining uniformly most powerful Bayesian tests to be tests that maximize the probability that the Bayes factor, in favor of the alternative hypothesis, exceeds a specified threshold. Like their classical counterpart, uniformly most powerful Bayesian tests are most easily defined in one-parameter exponential family models, although extensions outside of this class are possible. The connection between uniformly most powerful tests and uniformly most powerful Bayesian tests can be used to provide an approximate calibration between p-values and Bayes factors. Finally, issues regarding the strong dependence of resulting Bayes factors and p-values on sample size are discussed.

This letter is focused on the classic problem of testing samples drawn from independent Bernoulli probability mass functions, when the success probability under the alternative hypothesis is not known. The goal is to provide a systematic taxonomy of the viable detectors (designed according to theoretically-founded criteria) which can be used for the specific instance of the problem. Both One-Sided (OS) and Two-Sided (TS) tests are considered, with reference to: ( $i$ ) identical success probability (a homogeneous scenario) or ( $ii$ ) different success probabilities (a non-homogeneous scenario) for the observed samples. As a result of the study, a complete summary (in tabular form) of the relevant statistics for the problem is provided, along with a discussion on the existence of the Uniformly Most Powerful (UMP) test. Finally, when the Likelihood Ratio Test (LRT) is not UMP, existence of the UMP detector after reduction by invariance is investigated.

Uniformly Most Powerful (UMP) centralized detection for the composite binary hypothesis problem has been well researched. This paper extends the UMP methodology to the parallel distributed detection problem, when the local observations are independently distributed. A collection of general theorems and corollaries define sufficient conditions for the existence of a UMP parallel Distributed Detection (UMP-DD) under one set of fusion rules. These same conditions under another set of general fusion rules result in at least a Locally Most Powerful Distributed Detection (LMP-DD) rule. The subtleties of these conclusions are explored using informative examples that highlight the strengths of this approach and introduce new groups of UMP-DD tests.

In this paper, we study some relationships between the detection and estimation theories for a binary composite hypothesis test <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> against <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> and a related estimation problem. We start with a one-dimensional (1D) space for the unknown parameter space and one-sided hypothesis problems and then extend out results into more general cases. For one-sided tests, we show that the uniformly most powerful (UMP) test is achieved by comparing the minimum variance and unbiased estimator (MVUE) of the unknown parameter with a threshold. Thus for the case where the UMP test does not exist, the MVUE of the unknown parameter does not exist either. Therefore for such cases, a good estimator of the unknown parameter is deemed as a good decision statistic for the test. For a more general class of composite testing with multiple unknown parameters, we prove that the MVUE of a separating function (SF) can serve as the optimal decision statistic for the UMP unbiased test where the SF is continuous, differentiable, positive for all parameters under <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> and is negative for the parameters under <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> . We then prove that the UMP unbiased statistic is equal to the MVUE of an SF. In many problems with multiple unknown parameters, the UMP test does not exist. For such cases, we show that if one detector between two detectors has a better receiver operating characteristic (ROC) curve, then using its decision statistic we can estimate the SF more ε-accurately, in probability. For example, the SF is the signal-to-noise ratio (SNR) in some problems. These results motivate us to introduce new suboptimal SF-estimator tests (SFETs) which are easy to derive for many problems. Finally, we provide some practical examples to study the relationship between the decision statistic of a test and the estimator of its corresponding SF.

The Uniformly Most Powerful (UMP) test, which is one with the highest probability of detection over all unknown parameters in a composite hypothesis test, does not exist in the most practical problems. It is common to derive the suboptimal Generalize Likelihood Ratio (GLR) detector, that is shown to perform close to optimal invariant tests in some problems. The Uniformly Most Powerful Invariant (UMPI) test is the one that is used for the performance evaluation of the GLR detector. However, the GLR or UMPI detectors are not simply derivable in many problems. In this paper, we suggest that the problem be transformed to a new space. We derive the conditions of the transformation, such that the detectors for the original problem and the transformed ones become equivalent. In addition, using this method, we discuss two practical problems (one in radar and other in communication) and derive the GLR detectors. We also find the UMPI detectors in these problems to verify the performance of the derived GLR detector.

SUMMARY The use of major genes is a component of livestock improvement. This requires identification of the genotypes of reproducers, in particular progeny-tested sires after mating with homozygous females. Two test statistics derived from the nonnested test of Cox (1961, Proceedings of the Fouirth Berkeley Svmposium on Mathematical Statistics and Probability 1, 105-123; 1962, Journal of the Royal Statistical Society, Series B 24, 406-424) are proposed. For both tests, asymptotic normality is proved algebraically and the speed of convergence is studied through simulations. In the cases where a uniformly most powerful (UMP) test exists, the power of one of the tests is close to the power of that UMP test. Finally, the tests are applied to a population of poultry carrying the dwarf gene.

CoopMAC has been recently proposed as a possible implementation of cooperation protocols in the medium access control (MAC) layer of a wireless network. However, some nodes may refrain from cooperation for selfish purposes, e.g. in order to save energy, in what is called selfish behavior or misbehavior. This protocol violation worsens other nodes' performance and can be avoided if other nodes detect and punish (e.g. banning from the network) misbehaving nodes. However, fading and interference may prevent nodes from cooperating even if they are willing, therefore it is not trivial to identify misbehaving nodes. In a fading scenario where an automatic repeat request (ARQ) protocol is used, we propose a mechanism that allows to detect misbehaving nodes. Two approaches, either based on the uniformly most powerful (UMP) test or on the sequential probability ratio test (SPRT) are considered. The two techniques are characterized and compared in terms of their average detection delay and resulting network performance.

In this study, the authors study the modulation classification of linearly modulated signals including amplitude shift keying (ASK), phase shift keying (PSK) and quadrature amplitude modulation (QAM) signals. The authors consider an unknown frequency non-selective slowly fading channel with an unknown variance additive white Gaussian noise. The authors treat this classification problem as a multi-hypotheses test which is invariant under the complex scale. In such a case, the authors objective is to find uniformly most powerful (UMP) test in the class of invariant decisions. However, the authors find out that the UMPI test does not exist; instead, they provide a most powerful invariant (MPI) PSK signal classifier for known signal to noise ratio and use it as the upper performance bound for any invariant classifier. The authros also propose a hybrid likelihood ratio test (HLRT) solution which can be employed for the classification of linearly modulated signals, inter-family and intra-family. The authors also explain the efficient implementation of these algorithms in some steps. In order to reduce the computational cost, the authors propose quasi-HLRT classifiers for PSK signals. Some simulation examples are provided that show the power of the proposed algorithms in the classification of linearly modulated signals.

The concept of invariance in hypothesis testing is brought to bear on the problem of detecting signals of known form and unknown energy in Gaussian noise of unknown level. The noise covariance function is assumed to be <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K(t,u) = \sigma^2 \pho(t,u)</tex> where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\rho(t,u)</tex> is the known form of the covariance function and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\sigma^2</tex> is the unknown level. Classical approaches to signal detection depend on the assumption that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K(t,u)</tex> is known completely. Then, a correlation-type receiver that is the uniformly most powerful (UMP) test of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H_o</tex> (signal absent) versus <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H_1</tex> (signal present) can be derived. When <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\sigma^2</tex> is unknown, there exists no UMP test. However, it is shown in this paper that there exists a test of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H_o</tex> versus <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H_1</tex> that is UMP-invariant for a very natural group of transformations on the space of observations. The derived test is found to be independent of knowledge about the noise level <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\sigma^2</tex> , since the derived test (receiver) contains an error-free estimate of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\sigma^2</tex> . This utopian conclusion is reconciled by noting that the derived receiver can never be physically realized. It is shown that any physically realizable version of the receiver has a <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</tex> -distributed test statistic. This permits choice of operating receiver thresholds and evaluation of performance characteristics.

Recently, a design was proposed for the Simultaneous Global Drug Development Program (SGDDP) to assess the impact of ethnic factors on the effect of a new treatment for a targeted ethnic (TE) population. It used weighted Z tests to combine the information collected from the TE and non-TE (NTE) subgroups in the SGDDP based on the fundamental assumption on their shared biological commonality. In this article, we mathematically formulated this assumption as the quantitative interaction between treatment effect and subgroup. We used it to more rigorously describe the hypotheses, and showed the unbiasedness of the weighted Z test. Moreover, to study the loss of efficiency from down weighting the NTE information in this SGDDP design, we compared the power of their test with that of the uniformly most powerful (UMP) test, which we showed was also a weighted Z test. We discussed that the choice of weight should balance the maximization of power when the assumption holds and the minimization of bias otherwise.

Graph-based composite local Bregman divergences on discrete sample spaces

Up to now, we have been working with discrete sample spaces and discrete random variables (see Chapters 1 and 7). There are problems, however, for which a discrete sample space is not appropriate because there are just too many possible outcomes. Suppose, for instance, that I want to choose a number on the interval between 0 and 1 “at random.” Ignoring for a moment exactly what I mean by the term “at random” in this context, we note that the number chosen can be any value on the interval, so there is a continuum of possible values. This continuum has so many numbers in it that they cannot all be counted off using the positive integers; for this reason the sample space of this interval is not discrete but is what is called a continuous sample space.

Many statistical models over a discrete sample space often face the computational difficulty of the normalization constant. Because of that, the maximum likelihood estimator does not work. In order to circumvent the computation difficulty, alternative estimators such as pseudo-likelihood and composite likelihood that require only a local computation over the sample space have been proposed. In this paper, we present a theoretical analysis of such localized estimators. The asymptotic variance of localized estimators depends on the neighborhood system on the sample space. We investigate the relation between the neighborhood system and estimation accuracy of localized estimators. Moreover, we derive the efficiency bound. The theoretical results are applied to investigate the statistical properties of existing estimators and some extended ones.

We study Markov bases for sampling from a discrete sample space equipped with a convenient metric. Starting from any two states in the sample space, we ask whether we can always move closer by an element of a Markov basis. We call a Markov basis distance-reducing if this is the case. The particular metric we consider in this paper is the L1-norm on the sample space. Some characterizations of L1-norm-reducing Markov bases are derived.

Many types of data are often incompletely observed. How incompletely is typically randomly determined. Heitjan and Rubin (Annals of Statistics, 1991) proposed a condition, “coarsened at random” or CAR, ensuring ignorability of this randomness in discrete sample spaces. In general sample spaces CAR comes in two flavors according to whether it is defined in terms of probabilities or densities. In this paper, CAR defined in terms of densities, called relative CAR, is discussed as a condition for ignorability in a statistical model allowing for partial observation of random elements determining the degree of incompleteness in the observation of the data.