We investigate the problem of jointly testing multiple hypotheses and estimating a random parameter of the underlying distribution in a sequential setup. The aim is to jointly infer the true hypothesis and the true parameter while using on average as few samples as possible and keeping the detection and estimation errors below predefined levels. Based on mild assumptions on the underlying model, we propose an asymptotically optimal procedure, i.e., a procedure that becomes optimal when the tolerated detection and estimation error levels tend to zero. The implementation of the resulting asymptotically optimal stopping rule is computationally cheap and, hence, applicable for high-dimensional data. We further propose a projected quasi-Newton method to optimally choose the coefficients that parameterize the instantaneous cost function such that the constraints are fulfilled with equality. The proposed theory is validated by numerical examples.

A general class of Bayesian lower bounds when the underlying loss function is a Bregman divergence is demonstrated. This class can be considered as an extension of the Weinstein–Weiss family of bounds for the mean squared error and relies on finding a variational characterization of Bayesian risk. This approach allows for the derivation of a version of the Cramér–Rao bound that is specific to a given Bregman divergence. This new generalization of the Cramér–Rao bound reduces to the classical one when the loss function is taken to be the Euclidean norm. In order to evaluate the effectiveness of the new lower bounds, the paper also develops upper bounds on Bayesian risk, which are based on optimal linear estimators. The effectiveness of the new bound is evaluated in the Poisson noise setting.

We investigate the problem of jointly testing two hypotheses and estimating a random parameter based on sequentially observed data whose distribution belongs to the exponential family. The aim is to design a scheme which minimizes the expected number of used samples while limiting the detection and estimation errors to pre-set lev-els. This constrained problem is first converted to an unconstrained problem which is then reduced to an optimal stopping problem. To solve the optimal stopping problem, we propose an asymptotically pointwise optimal (APO) stopping rule, i.e., a stopping rule that is optimal when the tolerated detection and estimation errors tend to zero. The policy parameterizing coefficients are then chosen such that the constraints on the detection and estimation errors are fulfilled. The proposed theory is illustrated with a numerical example.

We address the problem of robust network localization in realistic mixed LOS/NLOS environments. We make use of the fact that the bias of range measurement errors is not only non-negative but also sparse when LOS dominates, which has been long overlooked in the existing literature. To exploit these two properties, we introduce a sparsity-promoting regularization term and relax the resulting optimization problem to a semi-definite programming (SDP) problem. The proposed method admits a neat mathematical formulation and is computationally cheap. Moreover, its global convergence is guaranteed and it achieves good robustness against NLOS measurements. In numerical results, the proposed method outperforms representative state-of-the-art SDP approaches, in terms of both localization accuracy and computational efficiency.

Jointly testing multiple hypotheses and estimating a random parameter of the underlying model is investigated in a sequential setup. The optimal scheme is designed such that it minimizes the expected number of used samples while keeping the probabilities of falsely rejecting a hypothesis and the mean-squared estimation errors below a pre-set level. The underlying constrained problem is first converted to an unconstrained problem and then reduced to an optimal stopping problem, whose solution is characterized by a non-linear Bellman equation. The optimal cost coefficients are obtained by exploiting a connection between the derivatives of the cost function and the detection/estimation errors. The paper concludes with a numerical example, namely solving the problem of sequential joint amplitude-shift keying symbol decoding and noise power estimation.

This paper considers a problem of estimation of the Fisher information for location from a random sample of size n. First, an estimator proposed by Bhattacharya is revisited and improved convergence rates are derived. Second, a new estimator, termed clipped estimator, is proposed. The new estimator is shown to have superior rates of convergence as compared to the Bhattacharya estimator, albeit with different regularity conditions. Third, both of the estimators are evaluated for the practically relevant case of a random variable contaminated by Gaussian noise. Moreover, using Brown's identity, which relates the Fisher information to the minimum mean squared error (MMSE) in Gaussian noise, a consistent estimator for the MMSE is proposed.

We propose a robust likelihood-ratio test (LRT) to detect landmines and unexploded ordnance using forward-looking ground-penetrating radar. Instead of modeling the distributions of the target and clutter returns with parametric families, we construct a band of feasible probability densities under each hypothesis. The LRT is then devised based on the least favorable densities within the bands. This detector is designed to maximize the worst case performance over all feasible density pairs and, hence, does not require strong assumptions about the clutter and noise distributions. The proposed technique is evaluated using electromagnetic field simulation data of shallow-buried targets. We show that, compared to detectors based on parametric models, robust detectors can lead to significantly reduced false alarm rates, particularly in cases where there is a mismatch between the assumed model and the true distributions.

In both wired and wireless communication networks, power allocation is an important technique to improve system performance. This paper investigates the power allocation problem for parallel Gaussian channels from an information-theoretic perspective with the aim of maximizing the sum of mutual informations (i.e., an achievable data rate). If all the inputs are Gaussian, it is well-known that the waterfilling policy provides an optimal solution. For arbitrary input distributions, a generalization of waterfilling, so-called mercury/waterfilling, provides an optimal solution in terms of the minimum mean square errors (MMSEs). However, the difficulty of obtaining closed-form analytical expressions of the MMSE often makes computing the mercury/waterfilling solution challenging. This paper proposes a robust waterfilling power allocation (RPA) policy for parallel Gaussian channels when the input distributions are close to Gaussian distributions in the Kullback-Leibler (KL) divergence (relative entropy). First, it is shown that the proposed policy results in water-levels that are close to the optimal ones in a well-defined sense. Second, tight bounds for the loss in mutual information (data rate) are given. This bounded loss property makes the proposed power allocation policy robust and approximately optimal, which is illustrated by means of various simulation setups. Moreover, the RPA policy provides a general framework for solving the power allocation problem for parallel channels, with the classical waterfilling being included as a special case. Finally, the RPA policy is argued to be scalable with the number of users since it inherently uses the classical low complexity waterfilling.

This paper investigates the power allocation problem for parallel Gaussian channels from an information-theoretic perspective with the aim of maximizing the sum of mutual informations (i.e., an achievable data rate). If all the inputs are Gaussian, it is well-known that the waterfilling policy provides an optimal solution. For arbitrary input distributions, a generalization of waterfilling, so-called mercury/waterfilling, provides an optimal power allocation in terms of the minimum mean square errors (MMSEs). However, the difficulty of obtaining closed-form analytical expression of the MMSE often makes the computation of mercury/waterfilling solution challenging. This paper proposes a robust waterfilling power allocation (RPA) policy for parallel Gaussian channels when the input distributions are close to Gaussian distributions in the Kullback-Leibler divergence (relative entropy). First, it is shown that the proposed policy results in water levels that are close to the optimum in a well-defined sense. Second, tight bounds for the loss in achievable rate are given. This bounded loss property makes the proposed power allocation policy robust and approximately optimal. Both aspects are illustrated by means of different simulation setups. Finally, the RPA is argued to be scalable with the number of users on the account of the fact that it inherently uses the classical low complexity waterfilling.

The problem of deciding whether a given set of data points forms one cluster or two clusters is investigated from a robust hypothesis testing perspective. It is assumed that a clustering algorithm exists that for both cases calculates cluster assignments and estimates of the corresponding probability density functions. Based on the latter, a statistical hypothesis test for the true number of clusters is formulated. In order to take falsely labeled data points into account, the clusters are then modeled as being contaminated with outliers. This leads to an uncertainty model for the cluster densities of the ε-contamination type, whose corresponding minimax optimal robust detector is well-known and can be implemented using least favorable densities. The performance of this detector under cluster overlap, cluster imbalance, and for different contamination ratios is evaluated numerically and is compared to that of a Bayesian cluster enumeration criterion. Significant performance improvements are shown in all cases.

We consider the problem of making a quick decision in favor of one of two possible physical signal models while the numerical measurements are acquired by sensing devices featuring minimal digitization complexity. Therefore, the digital data streams available for statistical processing are binary and exhibit temporal and spatial dependencies. To handle the intractable multivariate binary data model, we first consider sequential tests for exponential family distributions. Within this generic probabilistic framework, we identify adaptive approximations for the log-likelihood ratio and the Kullback-Leibler divergence. The results allow designing sequential detectors for binary radio systems and analyzing their average run-time along classical arguments of Wald. In particular, the derived tests exploit the spatio-temporal correlation structure of the analog sensor signals engraved into the binary measurements. As an application, we consider the specification of binary sensing architectures for cognitive radio and GNSS spectrum monitoring where our results characterize the sequential detection latency as a function of the temporal oversampling and the number of antennas. Finally, we evaluate the efficiency of the proposed algorithms and illustrate the accuracy of our analysis via Monte-Carlo simulations.

The paper deals with minimax optimal statistical tests for two composite hypotheses, where each hypothesis is defined by a nonparametric uncertainty set of feasible distributions. It is shown that for every pair of uncertainty sets of the$f$-divergence-ball type, a pair of uncertainty sets of the density-band type can be constructed, which is equivalent in the sense that it admits the same pair of least favorable distributions. This result implies that robust tests under$f$-divergence-ball uncertainty, which are typically only minimax optimal for the single sample case, are also fixed sample size minimax optimal with respect to the equivalent density-band uncertainty sets.

The problem of sequential multiple hypothesis testing in a distributed sensor network is considered and two algorithms are proposed: the Consensus + Innovations Matrix Sequential Probability Ratio Test (CIMSPRT for multiple simple hypotheses and the robust Least-Favorable-Density- CIMSPRT for hypotheses with uncertainties in the corresponding distributions. Simulations are performed to verify and evaluate the performance of both algorithms under different network conditions and noise contaminations.

The sequential analysis of the problem of joint signal detection and signal-to-noise ratio (SNR) estimation for a linear Gaussian observation model is considered. The problem is posed as an optimization setup where the goal is to minimize the number of samples required to achieve the desired (i) type I and type II error probabilities and (ii) mean squared error performance. This optimization problem is reduced to a more tractable formulation by transforming the observed signal and noise sequences to a single sequence of Bernoulli random variables; joint detection and estimation is then performed on the Bernoulli sequence. This transformation renders the problem easily solvable, and results in a computationally simpler sufficient statistic compared to the one based on the (untransformed) observation sequences. Experimental results demonstrate the advantages of the proposed method, making it feasible for applications having strict constraints on data storage and computation.

A carrier sense multiple access network is considered and two distributions for random slot selection that minimize the expected delay of the first or the first successful transmission under a constraint on the collision probability are derived. The distribution that minimizes the delay of the first successful transmission is of finite support and can be calculated in a recursive manner. The distribution that minimizes the unconditional delay of the first transmission is a standard geometric distribution with an appropriately chosen parameter. The results are proved by means of dynamic programming.

We show that the optimal design of non-randomized discrete sequential tests, i.e., tests whose test statistics take on only a countable number of states, can be modeled as a mixed integer linear problem. This is done by reformulating the difference equations describing the random walk on the integer lattice in terms of linear mixed integer constraints. We outline the general procedure and give a simple example to show how the proposed method can be used in practice.

Many of today's wireless sensor networks operate under the strict requirement that only a single sensor transmits data at a time. One way to guarantee this is to use protocols that detect and prevent package collisions on the MAC layer. These, however, come at the cost of increased transmission delays, reduced throughput and higher energy consumption. We propose a PHY layer approach to collision avoidance that is based on sequential detection and significantly reduces the risk of collisions while simultaneously minimizing the transmission delay. For this approach, a performance analysis is given whose results are shown to closely match numerical simulations.