Simple robust parameter estimation for the Birnbaum-Saunders distribution

Examining the robustness properties of maximum likelihood (ML) estimators of parameters in exponential power and generalized t distributions has been considered together. The well-known asymptotic properties of ML estimators of location, scale and added skewness parameters in these distributions are studied. The ML estimators for location, scale and scale variant (skewness) parameters are represented as an iterative reweighting algorithm (IRA) to compute the estimates of these parameters simultaneously. The artificial data are generated to examine performance of IRA for ML estimators of parameters simultaneously. We make a comparison between these two distributions to test the fitting performance on real data sets. The goodness of fit test and information criteria approve that robustness and fitting performance should be considered together as a key for modeling issue to have the best information from real data sets.

The Maximum Likelihood (ML) and Best Linear Unbiased (BLU) estimators of the location and scale parameters of an extreme value distribution (Lawless [1982]) are compared under conditions of small sample sizes and Type I censorship. The comparisons were made in terms of the mean square error criterion. According to this criterion, the ML estimator of σ in the case of very small sample sizes (n < 10) and heavy censorship (low censoring time) proved to be more efficient than the corresponding BLU estimator. However, the BLU estimator for σ attains parity with the corresponding ML estimator when the censoring time increases even for sample sizes as low as 10. The BLU estimator of σ attains equivalence with the ML estimator when the sample size increases above 10, particularly when the censoring time is also increased. The situation is reversed when it came to estimating the location parameter μ, as the BLU estimator was found to be consistently more efficient than the ML estimator despite the improved performance of the ML estimator when the sample size increases. However, computational ease and convenience favor the ML estimators. Keywords: maximum likelihood estimatorsbest linear unbiased estimatorslife-testingreliability

For a balanced two‐way mixed model, the maximum likelihood (ML) and restricted ML (REML) estimators of the variance components were obtained and compared under the non‐negativity requirements of the variance components by Lee and Kapadia (1984). In this note, for a mixed (random blocks) incomplete block model, explicit forms for the REML estimators of variance components are obtained. They are always non‐negative and have smaller mean squared error (MSE) than the analysis of variance (AOV) estimators. The asymptotic sampling variances of the maximum likelihood (ML) estimators and the REML estimators are compared and the balanced incomplete block design (BIBD) is considered as a special case. The ML estimators are shown to have smaller asymptotic variances than the REML estimators, but a numerical result in the randomized complete block design (RCBD) demonstrated that the performances of the REML and ML estimators are not much different in the MSE sense.

The Multiple Comparison Procedures with Modeling Techniques (MCP-Mod) framework has been recently approved by the U.S. Food, Administration, and European Medicines Agency as fit-for-purpose for phase II studies. Nonetheless, this approach relies on the asymptotic properties of Maximum Likelihood (ML) estimators, which might not be reasonable for small sample sizes. In this paper, we derived improved ML estimators and correction for their covariance matrices in the censored Weibull regression model based on the corrective and preventive approaches. We performed two simulation studies to evaluate ML and improved ML estimators with their covariance matrices in (i) a regression framework (ii) the Multiple Comparison Procedures with Modeling Techniques framework. We have shown that improved ML estimators are less biased than ML estimators yielding Wald-type statistics that controls type I error without loss of power in both frameworks. Therefore, we recommend the use of improved ML estimators in the MCP-Mod approach to control type I error at nominal value for sample sizes ranging from 5 to 25 subjects per dose.

Over-the-air computation (OAC) is a promising technique to realize fast model aggregation in the uplink of federated edge learning (FEEL). OAC, however, hinges on accurate channel-gain precoding and strict synchronization among edge devices, which are challenging in practice. As such, how to design the maximum likelihood (ML) estimator in the presence of residual channel-gain mismatch and asynchronies is an open problem. To fill this gap, this paper formulates the problem of misaligned OAC for FEEL and puts forth a whitened matched filtering and sampling scheme to obtain oversampled, but independent samples from the misaligned and overlapped signals. Given the whitened samples, a sum-product ML (SP-ML) estimator and an aligned-sample estimator are devised to estimate the arithmetic sum of the transmitted symbols. In particular, the computational complexity of our SP-ML estimator is linear in the packet length, and hence is significantly lower than the conventional ML estimator. Extensive simulations on the test accuracy versus the average received energy per symbol to noise power spectral density ratio (EsN0) yield two main results: 1) In the low EsN0 regime, the aligned-sample estimator can achieve superior test accuracy provided that the phase misalignment is not severe. In contrast, the ML estimator does not work well due to the error propagation and noise enhancement in the estimation process. 2) In the high EsN0 regime, the ML estimator attains the optimal learning performance regardless of the severity of phase misalignment. On the other hand, the aligned-sample estimator suffers from a test-accuracy loss caused by phase misalignment.

Maximum likelihood estimation of time delays in multipath acoustic channel

A maximum likelihood (ML) estimator is developed for determining time delay between signals received at two spatially separated sensors in the presence of uncorrelated noise. This ML estimator can be realized as a pair of receiver prefilters followed by a cross correlator. The time argument at which the correlator achieves a maximum is the delay estimate. The ML estimator is compared with several other proposed processors of similar form. Under certain conditions the ML estimator is shown to be identical to one proposed by Hannan and Thomson [10] and MacDonald and Schultheiss [21]. Qualitatively, the role of the prefilters is to accentuate the signal passed to the correlator at frequencies for which the signal-to-noise (S/N) ratio is highest and, simultaneously, to suppress the noise power. The same type of prefiltering is provided by the generalized Eckart filter, which maximizes the S/N ratio of the correlator output. For low S/N ratio, the ML estimator is shown to be equivalent to Eckart prefiltering.

In estimation theory, the asymptotic (in the number of samples) efficiency of the Maximum Likelihood (ML) estimator is a well known result [1]. Nevertheless, in some scenarios, the number of snapshots may be small. We recently investigated the asymptotic behavior of the Stochastic ML (SML) estimator at high Signal to Noise Ratio (SNR) and finite number of samples [2] in the array processing framework: we proved the non-Gaussiannity of the SML estimator and we obtained the analytical expression of the variance for the single source case. In this paper, we generalize these results to multiple sources, and we obtain variance expressions which demonstrate the non-efficiency of SML estimates.

The problem of distributed parameter estimation from binary quantized observations is studied when the unquantized observations are corrupted by combined multiplicative and additive Gaussian noise. These results are applicable to sensor networks where the sensors observe a parameter in combined additive and nonadditive noise and to a case where dispersed receivers are employed with analog communication over fading channels where the receivers employ binary quantization before noise-free digital communications to a fusion center. We first discuss the case in which all the quantizers use an identical threshold. The parameter identifiability condition is given, and, surprisingly, it is shown that unless the common threshold is chosen properly and the parameter lies in an open interval, the parameter will not generally be identifiable, in contrast to the additive noise case. The best achievable mean square error (MSE) performance is characterized by deriving the corresponding Cramer-Rao Lower Bound (CRLB). A closed-form expression describing the corresponding maximum likelihood (ML) estimator is presented. The stability of the performance of the ML estimators is improved when a nonidentical threshold strategy is utilized to estimate the unknown parameter. The thresholds are designed by maximizing the minimum asymptotic relative efficiency (ARE) between quantized and unquantized ML estimators. Although the ML estimation problem is nonconvex, it is shown that one can relax the optimization to make it convex. The solution to the relaxed problem is used as an initial solution in a gradient algorithm to solve the original problem. Next, the case where both the variances of the additive noise and multiplicative noise are unknown is studied. The corresponding CRLB is obtained, and the ML estimation problem is transformed to a convex optimization, which can be solved efficiently. Finally, numerical simulations are performed to substantiate the theoretical analysis.

A generalization of the exponential-Poisson distribution

Finite mixtures of multivariate skew t (MST) distributions have proven to be useful in modelling heterogeneous data with asymmetric and heavy tail behaviour. Recently, they have been exploited as an effective tool for modelling flow cytometric data. A number of algorithms for the computation of the maximum likelihood (ML) estimates for the model parameters of mixtures of MST distributions have been put forward in recent years. These implementations use various characterizations of the MST distribution, which are similar but not identical. While exact implementation of the expectation-maximization (EM) algorithm can be achieved for `restricted' characterizations of the component skew t-distributions, Monte Carlo (MC) methods have been used to fit the `unrestricted' models. In this paper, we review several recent fitting algorithms for finite mixtures of multivariate skew t-distributions, at the same time clarifying some of the connections between the various existing proposals. In particular, recent results have shown that the EM algorithm can be implemented exactly for faster computation of ML estimates for mixtures with unrestricted MST components. The gain in computational time is effected by noting that the semi-infinite integrals on the E-step of the EM algorithm can be put in the form of moments of the truncated multivariate non-central t-distribution, similar to the restricted case, which subsequently can be expressed in terms of the non-truncated form of the central t-distribution function for which fast algorithms are available. We present comparisons to illustrate the relative performance of the restricted and unrestricted models, and demonstrate the usefulness of the recently proposed methodology for the unrestricted MST mixture, by some applications to three real datasets.

The reverberation power spectral density (PSD) is often required for dereverberation and noise reduction algorithms. In this work, we compare two maximum likelihood (ML) estimators of the reverberation PSD in a noisy environment. In the first estimator, the direct path is first blocked. Then, the ML criterion for estimating the reverberation PSD is stated according to the probability density function of the blocking matrix (BM) outputs. In the second estimator, the speech component is not blocked. Instead, the ML criterion for estimating the speech and reverberation PSD is stated according to the probability density function of the microphone signals. To compare the expected mean square error (MSE) between the two ML estimators of the reverberation PSD, the Cramer–Rao Bounds (CRBs) for the two ML estimators are derived. We show that the CRB for the joint reverberation and speech PSD estimator is lower than the CRB for estimating the reverberation PSD from the BM outputs. Experimental results show that the MSE of the two estimators indeed obeys the CRB curves. Experimental results of multimicrophone dereverberation and noise reduction algorithm show the benefits of using the ML estimators in comparison with another baseline estimators.

We pose the estimation of the parameters of multiple superimposed exponential signals in additive Gaussian noise problem as a maximum likelihood (ML) estimation problem. The ML problem is non linear and hard to solve. Some previous works focused on finding alternative estimation procedures, for example by denoising. In contrast, we tackle the ML estimation problem directly. First, we use the same transformation as the first step of iterative quadratic maximum likelihood (IQML) and transform the ML problem into another optimization problem that gets rid of the amplitude coefficients. Second, we solve the remaining optimization problem with a gradient descent approach ("pseudo-quadratic maximum likelihood"). We also use this algorithm for ultra-wideband channel estimation and estimate ranging in non-line of sight environment.

Cooperative communication combats the effects offadingwithoutthe need for multiple antennas.The cooperative networks use multiple distributed nodes that virtually behave like multiple antennas each having their own local oscillators. But the oscillators cause multiple Carrier Frequency Offsets (CFO), unknown channel gains and Multiple Timing Offsets (MTO). For a good performance of the system perfect synchronization is needed which is achieved by estimation and compensation of multiple offsets. Also, the coverage and capacity can be improved by employing intermediate nodes called relays within cooperative networks. These cooperative relay networks can perform in a better manner even when considering MCFOs, MTOs and unknown channel gains by having more computationally efficient decoders that can use Maximum Likelihood (ML) estimation subject to the Cramer-Rao Lower Bound (CRLB) on the variance of any impartial estimate. The simulation results show that frequency offset still exist after Amplify and Forward (AF) relaying. The performance of a conventional cooperative network seems to have improved with respect to Bit Error Rate of the system when an AF relay is hosted. The estimate of the frequency offset is chosen such that it gives the signal, a high probability of occurrence using ML estimator. In ML estimate, since the bias and variance is difficult to compute, a lower bound called the CRLB on the variance of any unbiased estimate is derived which is satisfied by the ML estimator.Keywords: AF, Cooperative Communication, CFO, ML Estimation and Compensation, OFDM

The oscillator phase hit (PH) results in a temporary link loss in the communication system and bears an expensive cost on the operator companies. In this work, we propose a double-stage solution to mitigate the PH. In the first stage, we use Neyman-Pearson binary hypothesis testing to develop a low-cost PH detection algorithm, in which a likelihood ratio test is designed, and the optimal detection threshold is analytically calculated. The proposed PH detection scheme entails small real-time computations while using the existing pilot symbols in the system. Hence, no extra pilot overhead is required. In the second stage, the maximum likelihood (ML) estimation is used to develop a PH correction scheme. In particular, the phase noise and the PH are jointly mitigated by solving an ML estimation problem. By applying the proposed correction scheme, the number of remaining erroneous symbols due to PH (if any) is within the error correction capability of modern forward error-correcting codes. Numerical results verify the effectiveness of the proposed scheme in detecting and correcting the PH.