Approximate Maximum Likelihood Estimator Research Articles

Statistical inference of the stress-strength reliability for inverse Weibull distribution under an adaptive progressive type-Ⅱ censored sample

<abstract><p>In this paper, we investigate classical and Bayesian estimation of stress-strength reliability $\delta = P(X > Y)$ under an adaptive progressive type-Ⅱ censored sample. Assume that $X$ and $Y$ are independent random variables that follow inverse Weibull distribution with the same shape but different scale parameters. In classical estimation, the maximum likelihood estimator and asymptotic confidence interval are deduced. An approximate maximum likelihood estimator approach is used to obtain the explicit form. In Bayesian estimation, the Bayesian estimators are derived based on symmetric entropy loss function and LINEX loss function. Due to the complexity of integrals, we proposed Lindley's approximation to get the approximate Bayesian estimates. To compare the different estimators, we performed Monte Carlo simulations. Under gamma prior, the approximate maximum likelihood estimator performs better than Bayesian estimators. Under non-informative prior, the approximate maximum likelihood estimator has the same behavior as Bayesian estimators. In the end, two data sets are used to prove the effectiveness of the proposed methods.</p></abstract>

Computing Pseudolikelihood Estimators for Exponential-Family Random Graph Models

The reputation of the maximum pseudolikelihood estimator (MPLE) for Exponential Random Graph Models (ERGM) has undergone a drastic change over the past 30 years. While first receiving broad support, mainly due to its computational feasibility and the lack of alternatives, general opinions started to change with the introduction of approximate maximum likelihood estimator (MLE) methods that became practicable due to increasing computing power and the introduction of MCMC methods. Previous comparison studies appear to yield contradicting results regarding the preference of these two point estimators; however, there is consensus that the prevailing method to obtain an MPLE’s standard error by the inverse Hessian matrix generally underestimates standard errors. We propose replacing the inverse Hessian matrix by an approximation of the Godambe matrix that results in confidence intervals with appropriate coverage rates and that, in addition, enables examining for model degeneracy. Our results also provide empirical evidence for the asymptotic normality of the MPLE under certain conditions.

Venc Design and Velocity Estimation for Phase Contrast MRI.

In phase-contrast magnetic resonance imaging (PC-MRI), spin velocity contributes to the phase measured at each voxel. Therefore, estimating velocity from potentially wrapped phase measurements is the task of solving a system of noisy congruence equations. We propose Phase Recovery from Multiple Wrapped Measurements (PRoM) as a fast, approximate maximum likelihood estimator of velocity from multi-coil data with possible amplitude attenuation due to dephasing. The estimator can recover the fullest possible extent of unambiguous velocities, which can greatly exceed twice the highest venc. The estimator uses all pairwise phase differences and the inherent correlations among them to minimize the estimation error. Correlations are directly estimated from multi-coil data without requiring knowledge of coil sensitivity maps, dephasing factors, or the actual per-voxel signal-to-noise ratio. Derivation of the estimator yields explicit probabilities of unwrapping errors and the probability distribution for the velocity estimate; this, in turn, allows for optimized design of the phase-encoded acquisition. These probabilities are also incorporated into spatial post-processing to further mitigate wrapping errors. Simulation, phantom, and in vivo results for three-point PC-MRI acquisitions validate the benefits of reduced estimation error, increased recovered velocity range, optimized acquisition, and fast computation. A phantom study at 1.5T demonstrates 48.5% decrease in root mean squared error using PRoM with post-processing versus a conventional "dual-venc" technique. Simulation and 3T in vivo results likewise demonstrate the proposed benefits.

Regression Analysis of Multivariate Interval-Censored Failure Time Data under Transformation Model with Informative Censoring

We consider a regression analysis of multivariate interval-censored failure time data where the censoring may be informative. To address this, an approximated maximum likelihood estimation approach is proposed under a general class of semiparametric transformation models, and in the method, the frailty approach is employed to characterize the informative interval censoring. For the implementation of the proposed method, we develop a novel EM algorithm and show that the resulting estimators of the regression parameters are consistent and asymptotically normal. To evaluate the empirical performance of the proposed estimation procedure, we conduct a simulation study, and the results indicate that it performs well for the situations considered. In addition, we apply the proposed approach to a set of real data arising from an AIDS study.

On the Estimation of Stress Strength Reliability Parameter of Weibull-Rayleigh Distribution

The reliability of a component depends on the stress conditions of the operating environment, which are uncertainty and should be modeled as random. This article deals with the estimation of the stress-strength reliability parameter of Weibull-Rayleigh distribution. Let X and Y be two independent random variables, where X and Y follow Weibull-Rayleigh distribution. The maximum likelihood estimator and the approximate maximum likelihood estimator of the stress-strength reliability are obtained. Other properties of the Weibull-Rayleigh distributions are derived. Two real data applications are given for showing the flexibility of the Weibull-Rayleigh stress-strength reliability.

Measuring liquidity with return volatility: An analytical approach based on heavy-tailed Censored-GARCH model

The measurement of liquidity based on low-frequency data is a crucial issue in the financial market microstructure literature. This paper extends the commonly used LOT liquidity model by incorporating the characterization of the volatility dynamics and distributional properties of return series, thereby significantly improving both the goodness-of-fit of liquidity model and the estimation performance of the existing low-frequency liquidity measures. The new models, which have a special form of heavy-tailed Censored-GARCH model, have challenges in estimation. We then provide an approximate maximum likelihood estimation method to circumvent this problem. A real data analysis is conducted to evaluate the performance of the new liquidity measures. The results show overwhelming evidence that our new measures have advantages over the existing measures in both estimation error and correlation coefficient with high-frequency bid-ask spread. The robustness check analysis further illustrates that the advantages of our new measures are stable across different stock industries and different turnover levels.

Approximate Selective Inference via Maximum Likelihood

Several strategies have been developed recently to ensure valid inference after model selection; some of these are easy to compute, while others fare better in terms of inferential power. In this article, we consider a selective inference framework for Gaussian data. We propose a new method for inference through approximate maximum likelihood estimation. Our goal is to: (a) achieve better inferential power with the aid of randomization, (b) bypass expensive MCMC sampling from exact conditional distributions that are hard to evaluate in closed forms. We construct approximate inference, for example, p-values, confidence intervals etc., by solving a fairly simple, convex optimization problem. We illustrate the potential of our method across wide-ranging values of signal-to-noise ratio in simulations. On a cancer gene expression dataset we find that our method improves upon the inferential power of some commonly used strategies for selective inference. Supplementary materials for this article are available online.

Fast Likelihood Calculations for Automatic Identification of Macroevolutionary Rate Heterogeneity in Continuous and Discrete Traits.

Understanding phenotypic disparity across the tree of life requires identifying where and when evolutionary rates change on phylogeny. A primary methodological challenge in macroevolution is therefore to develop methods for accurate inference of among-lineage variation in rates of phenotypic evolution. Here, we describe a method for inferring among-lineage evolutionary rate heterogeneity in both continuous and discrete traits. The method assumes that the present-day distribution of a trait is shaped by a variable-rate process arising from a mixture of constant-rate processes and uses a single-pass tree traversal algorithm to estimate branch-specific evolutionary rates. By employing dynamic programming optimization techniques and approximate maximum likelihood estimators where appropriate, our method permits rapid exploration of the tempo and mode of phenotypic evolution. Simulations indicate that the method reconstructs rates of trait evolution with high accuracy. Application of the method to data sets on squamate reptile reproduction and turtle body size recovers patterns of rate heterogeneity identified by previous studies but with computational costs reduced by many orders of magnitude. Our results expand the set of tools available for detecting macroevolutionary rate heterogeneity and point to the utility of fast, approximate methods for studying large-scale biodiversity dynamics. [Brownian motion; continuous characters; discrete characters; macroevolution; Markov process; rate heterogeneity.].

Mixed-type multivariate response regression with covariance estimation.

We propose a new method for multivariate response regression and covariance estimation when elements of the response vector are of mixed types, for example some continuous and some discrete. Our method is based on a model which assumes the observable mixed‐type response vector is connected to a latent multivariate normal response linear regression through a link function. We explore the properties of this model and show its parameters are identifiable under reasonable conditions. We impose no parametric restrictions on the covariance of the latent normal other than positive definiteness, thereby avoiding assumptions about unobservable variables which can be difficult to verify in practice. To accommodate this generality, we propose a novel algorithm for approximate maximum likelihood estimation that works “off‐the‐shelf” with many different combinations of response types, and which scales well in the dimension of the response vector. Our method typically gives better predictions and parameter estimates than fitting separate models for the different response types and allows for approximate likelihood ratio testing of relevant hypotheses such as independence of responses. The usefulness of the proposed method is illustrated in simulations; and one biomedical and one genomic data example.

On the Stratonovich Estimator for the Itô Diffusion

For the parameter appearing non-linearly in the drift coefficient of homogeneous Itô stochastic differential equation having a stationary ergodic solution, the paper obtains the strong consistency of an approximate maximum likelihood estimator based on Stratonovich type approximation of the continuous Girsanov likelihood, under some regularity conditions, when the corresponding diffusion is observed at equally spaced dense time points over a long time interval in the high frequency regime. Pathwise convergence of stochastic integral approximations and their connection to discrete drift estimators is studied. Often it is shown that discrete drift estimators converge in probability. We obtain convergence of the estimator with probability one. Ornstein-Uhlenbeck process is considered as an example.

An Approximate Maximum Likelihood Estimator for Instantaneous Frequency Estimation of Multicomponent Nonstationary Signals

Nonstationary signals are widely observed in various applications. The instantaneous frequency (IF) is an important feature to capture the time-frequency (TF) characteristics of nonstationary signals, so the IF estimation problem has attracted increasing interests in the past three decades. In practice, the signal is usually composed of multiple components, and their IFs may intersect or are closely spaced with each other, which cannot be easily identified from the TF representation (TFR). In this case, the maximum likelihood estimator (MLE), which does not rely on the TFR, is commonly used. However, the MLE cannot deal with signals with time-varying amplitudes and suffers from a large computing load due to the multidimensional search within an indefinite parameter space. To address these problems, we propose an approximate MLE (AMLE). First, the IF is approximated by the Chebyshev interpolation polynomial and represented by several Chebyshev interpolation frequencies which are well bounded between zero and half of the sampling frequency, so that the multidimensional search can be performed within a bounded search space. Then, an approximate maximum likelihood function, which considers both the statistical characteristics of noise and the time-varying characteristics of amplitudes, is derived for amplitude-modulated and frequency-modulated signals. It is proven that the AMLE outperforms the MLE in the computing efficiency and estimation accuracy for nonstationary signals with more complicated IFs and amplitudes. Experimental hydroelectric rotor vibration and radar signals are tested, where the AMLE can accurately find those intersected and closely spaced IFs for their constituent components.

DNN-Based Peak Sequence Classification CFAR Detection Algorithm for High-Resolution FMCW Radar

Multitarget detection is very challenging, especially when targets are densely distributed. In conventional constant false alarm rate (CFAR) detection algorithms, the detection threshold is determined based on a pre-estimated background level (BL). However, interfering targets inevitably lead to inaccurate BL estimation, resulting in degraded detection performance. In our previous work, a new solution was developed and verified to be effective; in that solution, compressed sensing (CS)-based detector performs target detection without relying on BL estimation, and a subsequent CFAR regulation processor achieves the desired false alarm rate based on the statistical information of the reduced samples obtained by removing the detected targets and adjacent guard cells from the original samples. However, for scenes with a high target density, the CS-based detector suffers from performance degradation while acquiring the correlation between linear measurements of the signal and the sensing matrix due to the high local signal sparsity. To address this shortcoming, this article proposes a deep neural network (DNN)-based detector that further improves the detection performance by converting the target detection into a problem of peak sequence classification of frequency intensity (FI) measurements from radar. A DNN detector trained on augmented simulated data shows excellent generalization ability for deployment in real scenes. In addition, to achieve better computational efficiency, a Taylor-series-based approximate maximum likelihood estimator with an explicit expression is applied in the CFAR regulation process for the first time. Both simulation and field tests are performed to verify the superiority of the proposed algorithm over conventional algorithms.

Adaptive Detection of Radar Targets in Heavy-Tailed Sea Clutter With Lognormal Texture

This article deals with the problem of detecting a marine target with coherent radars in a correlated heavy-tailed sea clutter background. The heavy-tailed sea clutter is modeled by a compound-Gaussian model, and the clutter texture is characterized by the lognormal distribution with a new parameterization form. We develop an adaptive coherent detector on the basis of the two-step generalized likelihood ratio test. The proposed detector can achieve adaptation to sea clutter characteristics by using the maximum a posterior estimate of the clutter texture, the constrained approximate maximum likelihood estimator of the speckle covariance matrix, and the proposed negative- and positive-fractional moment estimate of amplitude parameters of sea clutter. Remarkably, the proposed detector inherently ensures a constant false alarm rate with respect to the clutter power mean and the speckle covariance matrix. Finally, numerical experiments using simulated data and real radar data demonstrate that the proposed estimator and adaptive coherent detector outperform their respective competitors.

Branching Process Models to Identify Risk Factors for Infectious Disease Transmission

ABSTRACT Simple branching processes for infectious disease transmission assume all individuals are homogeneous, which means that risk factors that may inhibit or increase transmission are unable to be identified. In this work, we develop a branching process model that allows for identification of risk factors by assuming the probability of onward transmission is determined by the individual’s covariates. Because enumerating the transmission trees is infeasible for large clusters, we develop an algorithm to sample transmission trees to compute approximate maximum likelihood estimates. We then discuss how our model can be extended to account for cases that are undetected but are part of the true transmission tree. We use our method to investigate individual characteristics that are associated with transmission of Tuberculosis using clusters of detected cases in Maryland from 2003 to 2009. We find that later detection within a cluster is associated with an increased probability of onward transmission (OR = 1.41 [95% CI: 1.31, 1.52]). We show some of most likely transmission trees from our model, and results can be reproduced via our R package InfectionTrees. Supplementary files for this article are available online.

Surgeon effects on cataract refractive outcomes are minimal compared with patient comorbidity and gender: an analysis of 490 987 cases

AimTo investigate effect of patient age, gender, comorbidities and surgeon on refractive outcomes following cataract surgery.MethodsStudy population: patients on UK national ophthalmic cataract database on cataract operations undertaken between 1...

Estimated reproduction ratios in the SIR model.

The aim of this article is to understand the extreme variability in estimates of the reproduction ratio R 0 observed in practice. For expository purposes, we consider a discrete‐time, stochastic version of the susceptible‐infected‐recovered model and introduce different approximate maximum likelihood estimators of R 0. We carefully discuss the properties of these estimators and illustrate, by a Monte Carlo study, the widths of confidence intervals for R 0.

Approximate maximum likelihood estimation for one‐dimensional diffusions observed on a fine grid

AbstractWe consider a one‐dimensional stochastic differential equation that is observed on a fine grid of equally spaced time points. A novel approach for approximating the transition density of the stochastic differential equation is presented, which is based on an Itô‐Taylor expansion of the sample path, combined with an application of the so‐called ‐expansion. The resulting approximation is economical with respect to the number of terms needed to achieve a given level of accuracy in a high‐frequency sampling framework. This method of density approximation leads to a closed‐form approximate likelihood function from which an approximate maximum likelihood estimator may be calculated numerically. A detailed theoretical analysis of the proposed estimator is provided and it is shown that it compares favorably to the Gaussian likelihood‐based estimator and does an excellent job of approximating the exact, but usually intractable, maximum likelihood estimator. Numerical simulations indicate that the exact and our approximate maximum likelihood estimator tend to be close, and the latter performs very well relative to other approximate methods in the literature in terms of speed, accuracy, and ease of implementation.

Single-Port Homodyne Detection in a Squeezed-State Interferometry with Optimal Data Processing

Performing homodyne detection at a single output port of a squeezed-state light interferometer and then separating the measurement quadrature into several bins can realize superresolving and supersensitive phase measurements. However, the phase resolution and the achievable phase sensitivity depend on the bin size that is adopted in the data processing. By maximizing classical Fisher information, we analytically derive an optimal value of the bin size and the associated best sensitivity for the case of three bins, which can be regarded as a three-outcome measurement. Our results indicate that both the resolution and the achievable sensitivity are better than that of the previous binary–outcome case. Finally, we present an approximate maximum Likelihood estimator to asymptotically saturate the ultimate lower bound of the phase sensitivity.

Fast covariance parameter estimation of spatial Gaussian process models using neural networks

Gaussian processes (GPs) are a popular model for spatially referenced data and allow descriptive statements, predictions at new locations, and simulation of new fields. Often, a few parameters are sufficient to parameterize the covariance function, and maximum likelihood (ML) methods can be used to estimate these parameters from data. ML methods, however, are computationally demanding. For example, in the case of local likelihood estimation, even fitting covariance models on modest size windows can overwhelm typical computational resources for data analysis. This limitation motivates the idea of using neural network (NN) methods to approximate ML estimates. We train NNs to take moderate size spatial fields or variograms as input and return the range and noise‐to‐signal covariance parameters. Once trained, the NNs provide estimates with a similar accuracy compared to ML estimation and at a speedup by a factor of 100 or more. Although we focus on a specific covariance estimation problem motivated by a climate science application, this work can be easily extended to other, more complex, spatial problems and provides a proof‐of‐concept for this use of machine learning in computational statistics.

A New Algorithm for Approximate Maximum Likelihood Estimation in Sub-fractional Chan-Karolyi-Longstaff-Sanders Model

The paper introduces several approximate maximum likelihood estimators of the parameters of the sub-fractional Chan-Karolyi-Longstaff-Sanders (CKLS) interest rate model and obtains their rates of convergence. A new algorithm inspired by Newton-Cotes formula is presented to improve the accuracy of estimation. The estimators are useful for simulation of interest rates. The proposed new algorithm could be useful for other stochastic computation. It also proposes a generalization of the CKLS interest rate model with sub-fractional Brownian motion drivers which preserves medium range memory.