Fitting Second-Order Parametric Distributions to Data Using Maximum Likelihood Estimation

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

The goal of the present work is to estimate the nonlinear correlation between two random variables when the sample is drawn from a Farlie-Gumbel-Morgenstern (FGM)-type bivariate gamma distribution.In the context of estimating the dependence parameter, a maximum likelihood (ML) methodology is used.Thus, the present work offers ML estimators based on simple random sampling (SRS) and ranked set sampling (RSS).Additionally, we consider generalized modified RSS (GMRSS), which only requires a single rank to obtain a sample.Using GMRSS, we aim to observe the effect of the rth order statistic and its concomitant on the ML estimator.According to the Monte Carlo simulation, it is clearly seen that RSS provides an ML estimator as efficient as the ML estimator based on SRS.On the other hand, it appears that the ML estimator based on GMRSS (with minimum or maximum ranked pairs) is the best option among the studied ML estimators.Moreover, these findings are made even more meaningful by the fact that GMRSS is easier to obtain than SRS and RSS.

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.

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.

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.

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.

The assessment of earthquake triggered landslide hazard may be undertaken using both deterministic and probabilistic techniques. Probabilistic methods have been developed because much of the data can be considered as random variables where parameters such as the angle of internal friction and moisture content do not have a single fixed value but may assume any number of values across a range. This random variability can be modelled by a probability density function (PDF) which describes the relative likeli-hood that a random variable will assume a particular value. Instead of using just the average or expected value of an input parameter, the complete range of possible values can be used to estimate a range of possible outcomes. Thus the probability of a slope being unstable can be obtained rather than a single indicator of stability. Such proba-bilistic analyses allow for the incorporation of the likely variability of each parameter and therefore allow a more intimate assessment of slope stability to be derived. Utilising empirical relationships for calculating earthquake ground motions and associated slope displacement, an investigation was undertaken to identify the contribution that modern simulation techniques could make to the assessment of earthquake-triggered landslides. To achieve this, geotechnical and earthquake data obtained from a deep-seated landslide triggered during the M w 7.0 Loma Prieta earthquake was used. By incorporating the variability of the geotechnical parameters and the uncertainty in earthquake location the model derived the probabilities associated with increasing amounts of slope displacement during future probable earthquakes. Analysis was undertaken for four of the principal fault segments in the San Francisco Bay area. These estimates were then combined with the occurrence probabilities of the earthquakes to provide temporal estimates of dis-placement for a 30 year period. Results indicated that a M w 7.0 earthquake located on the Peninsula Segment of the San Andreas fault was most hazardous with a 11% chance of minor slope displacement (≥0.10 m) and a 6% chance of moderate slope displacement (≥0.30 m) within the next 30 years.

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.

Decomposition of Kullback–Leibler risk and unbiasedness for parameter-free estimators

A unified response surface framework for the interval and stochastic finite element analysis of structures with uncertain parameters

We consider transient flow in unsaturated heterogeneous porous media with uncertain hydraulic parameters. Our aim is to provide unbiased predictions (estimates) of system states, such as pressure head, water content, and fluxes, and to quantify the uncertainty associated with such predictions. We achieve this goal by treating hydraulic parameters as random fields and the corresponding flow equations as stochastic. Current stochastic analyses of transient flow in partially saturated soils require linearization of the constitutive relations, which may lead to significant inaccuracies when these relations are highly nonlinear. If relative conductivity and saturation vary exponentially with pressure and the corresponding scaling parameters are random variables, the transient Richards equation is mapped onto a linear equation by means of the Kirchhoff transformation. This allows us to develop deterministic differential equations for the first and second ensemble moments of pressure and saturation. We solve these equations analytically, for vertical infiltration, and compare them with direct Monte Carlo simulations.

We consider transient flow in unsaturated heterogeneous porous media with uncertain hydraulic parameters. Our aim is to provide unbiased predictions (estimates) of system states, such as pressure head, water content, and fluxes, and to quantify the uncertainty associated with such predictions. We achieve this goal by treating hydraulic parameters as random fields and the corresponding flow equations as stochastic. Current stochastic analyses of transient flow in partially saturated soils require linearization of the constitutive relations, which may lead to significant inaccuracies when these relations are highly nonlinear. If relative conductivity and saturation vary exponentially with pressure and the corresponding scaling parameters are random variables, the transient Richards equation is mapped onto a linear equation by means of the Kirchhoff transformation. This allows us to develop deterministic differential equations for the first and second ensemble moments of pressure and saturation. We solve these equations analytically, for vertical infiltration, and compare them with direct Monte Carlo simulations.

In this paper, the influence of uncertain model parameters on vehicle lateral dynamics steady states is studied. These steady states are widely used in driver assistant systems and enable highly automated driving functions. A methology for propagating random variables through the single-track vehicle model is presented. The random variables are described using continuous probability density functions. It is shown, that the influence of parameter uncertanties scales differently to different steady states. The effect on the path curvature and the lateral acceleration is highlighted. An example for a fault detection scheme is given and analysed in context of the given uncertainty. The faults that can be detected without violating a given true-positive rate and a given false-positive rate are calculated. The method can be used to quickly identify possible problems arising from uncertain parameters.

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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.