Standard Maximum Likelihood Estimation Research Articles

Using a Response Time-Based Expected A Posteriori Estimator to Control for Differential Speededness in Computerized Adaptive Test.

This study investigates using response times (RTs) with item responses in a computerized adaptive test (CAT) setting to enhance item selection and ability estimation and control for differential speededness. Using van der Linden's hierarchical framework, an extended procedure for joint estimation of ability and speed parameters for use in CAT is developed following van der Linden; this is called the joint expected a posteriori estimator (J-EAP). It is shown that the J-EAP estimate of ability and speededness outperforms the standard maximum likelihood estimator (MLE) of ability and speededness in terms of correlation, root mean square error, and bias. It is further shown that under the maximum information per time unit item selection method (MICT)-a method which uses estimates for ability and speededness directly-using the J-EAP further reduces average examinee time spent and variability in test times between examinees above the resulting gains of this selection algorithm with the MLE while maintaining estimation efficiency. Simulated test results are further corroborated with test parameters derived from a real data example.

Estimation of the binomial probabilities in a two-stage phase II clinical trial with two co-primary endpoints

In cancer research, two-stage designs are usually used to assess the effect of a new agent in phase II clinical trials. Optimal two-stage designs with two co-primary endpoints have been proposed to assess the effects of new cancer treatments, such as cytostatic or molecularly targeted agents (MTAs), based on both response rate and early progression rate. Accurate estimation of response and early progression rates based on the data from the phase II trials conducted according to the optimal two-stage designs would be very useful for further testing of the agents in phase II trials. In this paper, we derive some estimation procedures, which include both standard and bias-corrected maximum likelihood estimates (MLE) and uniformly minimum variance unbiased estimate (UMVUE), for two binomial probabilities which are used to define the hypotheses for two co-primary endpoints tested in a two-stage phase II clinical trial. Simulation studies were performed to evaluate the performance of these procedures. These procedures are also applied to analyze the data from a phase II trial conducted by the Canadian Cancer Trials Group.

Transforming variables to central normality

Many real data sets contain numerical features (variables) whose distribution is far from normal (Gaussian). Instead, their distribution is often skewed. In order to handle such data it is customary to preprocess the variables to make them more normal. The Box–Cox and Yeo–Johnson transformations are well-known tools for this. However, the standard maximum likelihood estimator of their transformation parameter is highly sensitive to outliers, and will often try to move outliers inward at the expense of the normality of the central part of the data. We propose a modification of these transformations as well as an estimator of the transformation parameter that is robust to outliers, so the transformed data can be approximately normal in the center and a few outliers may deviate from it. It compares favorably to existing techniques in an extensive simulation study and on real data.

On some beta ridge regression estimators: method, simulation and application

The classic statistical method for modelling the rates and proportions is the beta regression model (BRM). The standard maximum likelihood estimator (MLE) is used to estimate the coefficients of the BRM. However, this MLE is very sensitive when the regressors are linearly correlated. Therefore, this study introduces a new beta ridge regression (BRR) estimator as a remedy to the problem of instability of the MLE. We study the mean squared error properties of the BRR estimator analytically and then based on the derived MSE, we suggest some new estimators of the shrinkage parameter. We also suggest a median squared error (SE) performance criterion, which can be used to achieve strong evidence in favour of the proposed method for the Monte Carlo simulation study. The performance of BRR and MLE is appraised through Monte Carlo simulation. Finally, an empirical application is used to show the advantages of the proposed estimator.

Financial Factors, Openness and the Natural Interest Rate in China

AbstractChina's financial market has undergone significant changes since financial deleveraging commenced and regulatory supervision was tightened in 2017. Intensifying China–US trade tensions have further increased the uncertainties of external environments. In this article, we use a Bayesian approach instead of the standard maximum likelihood estimation in the Laubach–Williams model to estimate the natural interest rate by considering financial factors and open conditions, and analyze the relationships among the natural interest rate, economic activities and monetary policies. We find that technological and demographic factors are the main drivers of natural interest rates, while financial factors and open conditions also play important roles. In particular, shocks in the financial markets and the external economic environment in recent years are important reasons for the decline of China's natural interest rate. Therefore, it is necessary to strengthen research on the estimation of the natural interest rate to ensure China's transformation into more price‐based monetary policy and high‐quality development.

Combining a self-exciting point process with the truncated generalized Pareto distribution: An extreme risk analysis under price limits

Abstract In this paper, we introduce a general framework of the self-exciting point process with the truncated generalized Pareto distribution to measure the extreme risks in the stock markets under price limits. We incorporate the predictable marks, defined as the variance of mark distribution depending on the previous events via the intensity, into the model setting. The proposed process can well accommodate many important empirical characteristics, such as the thick-tailness, extreme risk clustering and price limits. We derive a closed-form solution for the objective likelihood, based on which the proposed model can be estimated via the standard maximum likelihood estimation algorithm. Furthermore, the closed-form measures of the Value-at-Risk and Expected Shortfall are also derived. For empirical illustration, we use the China Securities Index 300 (with ± 10 % price restriction) in the analysis. In general, the results from both in-sample fitting and out-of-sample forecasting measures show that the proposed process can explain the empirical data well. We also investigate the cascade effect of the China stock market by introducing the branching process to distinguish the endogenous risks from the exogenous risks.

A note on the applicability of the standard nonparametric maximum likelihood estimator for combined incident and prevalent cohort data

Nonparametric estimation of the survival function for either incident or prevalent cohort failure time data, exclusively, has been well studied in the literature; the Kaplan‐Meier (KM) estimator is routinely used for right‐censored incident cohort failure time data, whereas a modified form of the KM estimator, sometimes referred to as the Tsai–Jewell–Wang (TJW) estimator, is the default estimator used for prevalent cohort data with follow‐up. Often, failure time data comprise observations from a combination of incident and prevalent cohorts. In this note, we justify the use of the TJW estimator for a combined sample of incident and prevalent cohort data with follow‐up. We suggest how the TJW estimator forms the basis for density estimation and hypothesis testing problems, when incident and prevalent cohorts are combined.

Within-cluster resampling for multilevel models under informative cluster size

Summary A within-cluster resampling method is proposed for fitting a multilevel model in the presence of informative cluster size. Our method is based on the idea of removing the information in the cluster sizes by drawing bootstrap samples which contain a fixed number of observations from each cluster. We then estimate the parameters by maximizing an average, over the bootstrap samples, of a suitable composite loglikelihood. The consistency of the proposed estimator is shown and does not require that the correct model for cluster size is specified. We give an estimator of the covariance matrix of the proposed estimator, and a test for the noninformativeness of the cluster sizes. A simulation study shows, as in Neuhaus & McCulloch (2011), that the standard maximum likelihood estimator exhibits little bias for some regression coefficients. However, for those parameters which exhibit nonnegligible bias, the proposed method is successful in correcting for this bias.

Neural Machine Translation with Adequacy-Oriented Learning

Although Neural Machine Translation (NMT) models have advanced state-of-the-art performance in machine translation, they face problems like the inadequate translation. We attribute this to that the standard Maximum Likelihood Estimation (MLE) cannot judge the real translation quality due to its several limitations. In this work, we propose an adequacyoriented learning mechanism for NMT by casting translation as a stochastic policy in Reinforcement Learning (RL), where the reward is estimated by explicitly measuring translation adequacy. Benefiting from the sequence-level training of RL strategy and a more accurate reward designed specifically for translation, our model outperforms multiple strong baselines, including (1) standard and coverage-augmented attention models with MLE-based training, and (2) advanced reinforcement and adversarial training strategies with rewards based on both word-level BLEU and character-level CHRF3. Quantitative and qualitative analyses on different language pairs and NMT architectures demonstrate the effectiveness and universality of the proposed approach.

On Robust Estimation for Slope in Linear Functional Relationship Model

In this paper, we propose a robust parameter estimation method for the linear functional relationship model. We improved the maximum likelihood estimation using robust estimators and robust correlation coefficients to estimate the slope parameter. The performance of the propose method, MMLE, is compared with the standard maximum likelihood estimation (MLE) and the nonparametric method in terms of mean square error. The results for simulation studies suggested the performance of the MMLE and nonparametric methods gives better estimate than the standard MLE in the presence of outliers. The novelty of the proposed method is that it is not affected by the presence of outliers and is simple to use. To illustrate practical application of the methods, we obtain the estimate of the slope parameter in a study of body-composition techniques for children.

A flexible likelihood approach for predicting neural spiking activity from oscillatory phase

BackgroundThe synchronous ionic currents that give rise to neural oscillations have complex influences on neuronal spiking activity that are challenging to characterize. New methodHere we present a method to estimate probabilistic relationships between neural spiking activity and the phase of field oscillations using a generalized linear model (GLM) with an overcomplete basis of circular functions. We first use an L1-regularized maximum likelihood procedure to select an active set of regressors from the overcomplete set and perform model fitting using standard maximum likelihood estimation. An information theoretic model selection procedure is then used to identify an optimal subset of regressors and associated coefficients that minimize overfitting. To assess goodness of fit, we apply the time-rescaling theorem and compare model predictions to original data using quantile-quantile plots. ResultsSpike-phase relationships in synthetic data were robustly characterized. When applied to in vivo hippocampal data from an awake behaving rat, our method captured a multimodal relationship between the spiking activity of a CA1 interneuron, a theta (5–10 Hz) rhythm, and a nested high gamma (65–135 Hz) rhythm. Comparison with existing methodsPrevious methods for characterizing spike-phase relationships are often only suitable for unimodal relationships, impose specific relationship shapes, or have limited ability to assess the accuracy or fit of their characterizations. ConclusionsThis method advances the way spike-phase relationships are visualized and quantified, and captures multimodal spike-phase relationships, including relationships with multiple nested rhythms. Overall, our method is a powerful tool for revealing a wide range of neural circuit interactions.

Validation of Improved Significant Wave Heights from the Brown-Peaky (BP) Retracker along the East Coast of Australia

Improved significant wave heights (SWHs) along the east coast of Australia have been estimated using the parameters resolved by a Brown-peaky (BP) retracker through reprocessing of three years of Jason-1 altimetric waveforms. The BP-estimated SWHs are validated against eight waverider buoys along the coast, and compared with the SWHs estimated by the standard four-parameter maximum likelihood estimator (MLE4). When assessing 1 Hz coastal SWHs for distances from 12 km to the coast, mean standard deviations (STDs) of BP SWHs vary from ca. 0.5 m to 0.9 m, while those of MLE4 SWHs increase from ca. 0.6 m to ca. 2.3 m, indicating a dramatically drop in the quality of MLE4-derived SWHs at the coast. The BP retracker has retrieved ca. 80% of 1 Hz coastal SWHs, which are more than those (ca. 50%) by the standard MLE4. The validation of 1 Hz SWHs is performed by calculating the along-altimeter-track pointwise bias, STD and correlation coefficient between altimetry and buoys. The results show that within 30 km off the coast the BP dataset has better agreement with buoy’s wave heights than the SGDR MLE4 dataset in terms of the BP’s small absolute biases and STDs, as well as high correlation coefficients.

Higher-order asymptotic theory of shrinkage estimation for general statistical models

In this study, we develop a higher-order asymptotic theory of shrinkage estimation for general statistical models, which includes dependent processes, multivariate models, and regression models (i.e., non-independent and identically distributed models). We introduce a shrinkage estimator of the maximum likelihood estimator (MLE) and compare it with the standard MLE by using the third-order mean squared error. A sufficient condition for the shrinkage estimator to improve the MLE is given in a general setting. Our model is described as a curved statistical model p(⋅;θ(u)), where θ is a parameter of the larger model and u is a parameter of interest with dimu<dimθ. This setting is especially suitable for estimating portfolio coefficients u based on the mean and variance parameters θ. We finally provide the results of our numerical study and discuss an interesting feature of the shrinkage estimator.

Covariate balancing propensity score for a continuous treatment: Application to the efficacy of political advertisements

Propensity score matching and weighting are popular methods when estimating causal effects in observational studies. Beyond the assumption of unconfoundedness, however, these methods also require the model for the propensity score to be correctly specified. The recently proposed covariate balancing propensity score (CBPS) methodology increases the robustness to model misspecification by directly optimizing sample covariate balance between the treatment and control groups. In this paper, we extend the CBPS to a continuous treatment. We propose the covariate balancing generalized propensity score (CBGPS) methodology, which minimizes the association between covariates and the treatment. We develop both parametric and nonparametric approaches and show their superior performance over the standard maximum likelihood estimation in a simulation study. The CBGPS methodology is applied to an observational study, whose goal is to estimate the causal effects of political advertisements on campaign contributions. We also provide open-source software that implements the proposed methods.

Classification of matrix-variate Fisher–Bingham distribution via Maximum Likelihood Estimation using manifold valued data

Abstract In the existing literature, the numerical estimation of multivariate normalizing constant is considered highly non-trivial, intractable or a big barrier to Bayesian approach, and hence non-parametric and many other techniques are adopted as an alternative for inferential problems, i.e., without the use of normalizing constant. This problem is a substantial one, and is mainly due to the difficult normalizing constant requiring accurate and efficient calculation for parametric models. The core objective of this paper is to estimate the numerical value of multivariate normalizing constant in Fisher–Bingham parametric model of distributions in the most general context. The ultimate goal is then the practicability of multivariate Fisher–Bingham model with the inclusion of normalizing constant for Bayesian inference via the standard Maximum Likelihood Estimation (MLE) algorithm. The normalizing constant is evaluated with the method of Saddle-Point Approximation (SPA) via Taylor Series expansion, where the first four terms of the series are considered the most crucial for obtaining the threshold value of the multivariate normalizing constant. Furthermore, it plays the most important role for the practicability of multivariate parametric models with manifold valued data being more useful for statistical inferences. We evaluate the numerical value of multivariate normalizing constant with best accuracy estimates. The numerical value is then used for statistical inferences with a simple platform of MLE for experiments on one synthetic dataset on Stiefel manifolds and three publicly available real world datasets with consideration of Grassmannian points on the Fisher–Bingham model. The results obtained via numerical experiments are compared with results of existing related techniques, where our algorithm outperforms or at least is best comparable to the results of existing techniques currently available in literature.

A practical method to test the validity of the standard Gumbel distribution in logit-based multinomial choice models of travel behavior

Most multinomial choice models (e.g., the multinomial logit model) adopted in practice assume an extreme-value Gumbel distribution for the random components (error terms) of utility functions. This distributional assumption offers a closed-form likelihood expression when the utility maximization principle is applied to model choice behaviors. As a result, model coefficients can be easily estimated using the standard maximum likelihood estimation method. However, maximum likelihood estimators are consistent and efficient only if distributional assumptions on the random error terms are valid. It is therefore critical to test the validity of underlying distributional assumptions on the error terms that form the basis of parameter estimation and policy evaluation. In this paper, a practical yet statistically rigorous method is proposed to test the validity of the distributional assumption on the random components of utility functions in both the multinomial logit (MNL) model and multiple discrete-continuous extreme value (MDCEV) model. Based on a semi-nonparametric approach, a closed-form likelihood function that nests the MNL or MDCEV model being tested is derived. The proposed method allows traditional likelihood ratio tests to be used to test violations of the standard Gumbel distribution assumption. Simulation experiments are conducted to demonstrate that the proposed test yields acceptable Type-I and Type-II error probabilities at commonly available sample sizes. The test is then applied to three real-world discrete and discrete-continuous choice models. For all three models, the proposed test rejects the validity of the standard Gumbel distribution in most utility functions, calling for the development of robust choice models that overcome adverse effects of violations of distributional assumptions on the error terms in random utility functions.

Mean targeting estimator for the integer-valued GARCH(1, 1) model

The integer-valued GARCH model is commonly used in modeling time series of counts. Maximum likelihood estimation (MLE) is used to estimate unknown parameters, but numerical results for MLE are sensitive to the choice of initial values, which also occurs in estimating the GARCH model. To alleviate this numerical difficulty, we propose an alternative to MLE and name it as mean targeting estimation (MTE), which is an analogue to variance targeting estimation used in the GARCH model. Consistency and asymptotic normality for MTE are established. Comparisons with the standard MLE are provided and the merits of the mean targeting method are discussed. In particular, it is shown that MTE can be superior to MLE for estimating parameters or prediction when the model is well specified and misspecified. We conduct numerical studies to confirm our theoretical findings and illustrate the practical utility of our proposals.

Identification and estimation of semi‐parametric censored dynamic panel data models of short time periods

Summary In this paper, we present a semi-parametric identification and estimation method for censored dynamic panel data models of short time periods and their average partial effects with only two periods of data. The proposed method transforms the semi-parametric specification of censored dynamic panel data models into a parametric family of distribution functions of observables without specifying the distribution of the initial condition. Then the censored dynamic panel data models are globally identified under a standard maximum likelihood estimation framework. The identifying assumptions are related to the completeness of the families of known parametric distribution functions corresponding to censored dynamic panel data models. Dynamic tobit models and two-part dynamic regression models satisfy the key assumptions. We propose a sieve maximum likelihood estimator and we investigate the finite sample properties of these sieve-based estimators using Monte Carlo analysis. Our empirical application using the Medical Expenditure Panel Survey shows that individuals consume more health care when their incomes increase, after controlling for past health expenditures.

Segmented targeted least squares estimator for material decomposition in multibin photon-counting detectors.

We present a fast, noise-efficient, and accurate estimator for material separation using photon-counting x-ray detectors (PCXDs) with multiple energy bin capability. The proposed targeted least squares estimator (TLSE) is an improvement of a previously described A-table method by incorporating dynamic weighting that allows the variance to be closer to the Cramér-Rao lower bound (CRLB) throughout the operating range. We explore Cartesian and average-energy segmentation of the basis material space for TLSE and show that, compared with Cartesian segmentation, the average-energy method requires fewer segments to achieve similar performance. We compare the average-energy TLSE to other proposed estimators-including the gold standard maximum likelihood estimator (MLE) and the A-table-in terms of variance, bias, and computational efficiency. The variance and bias were simulated in the range of 0 to 6cm of aluminum and 0 to 50cm of water with Monte Carlo methods. The Average-energy TLSE achieves an average variance within 2% of the CRLB and mean absolute error of [Formula: see text]. Using the same protocol, the MLE showed variance within 1.9% of the CRLB ratio and average absolute error of [Formula: see text] but was 50 times slower in our implementations. Compared with the A-table method, TLSE gives a more homogenously optimal variance-to-CRLB ratio in the operating region. We show that variance in basis material estimates for TLSE is lower than that of the A-table method by as much as [Formula: see text] in the peripheral region of operating range (thin or thick objects). The TLSE is a computationally efficient and fast method for material separation with PCXDs, with accuracy and precision comparable to the MLE.

Performance of Firth-and logF-type penalized methods in risk prediction for small or sparse binary data

BackgroundWhen developing risk models for binary data with small or sparse data sets, the standard maximum likelihood estimation (MLE) based logistic regression faces several problems including biased or infinite estimate of the regression coefficient and frequent convergence failure of the likelihood due to separation. The problem of separation occurs commonly even if sample size is large but there is sufficient number of strong predictors. In the presence of separation, even if one develops the model, it produces overfitted model with poor predictive performance. Firth-and logF-type penalized regression methods are popular alternative to MLE, particularly for solving separation-problem. Despite the attractive advantages, their use in risk prediction is very limited. This paper evaluated these methods in risk prediction in comparison with MLE and other commonly used penalized methods such as ridge.MethodsThe predictive performance of the methods was evaluated through assessing calibration, discrimination and overall predictive performance using an extensive simulation study. Further an illustration of the methods were provided using a real data example with low prevalence of outcome.ResultsThe MLE showed poor performance in risk prediction in small or sparse data sets. All penalized methods offered some improvements in calibration, discrimination and overall predictive performance. Although the Firth-and logF-type methods showed almost equal amount of improvement, Firth-type penalization produces some bias in the average predicted probability, and the amount of bias is even larger than that produced by MLE. Of the logF(1,1) and logF(2,2) penalization, logF(2,2) provides slight bias in the estimate of regression coefficient of binary predictor and logF(1,1) performed better in all aspects. Similarly, ridge performed well in discrimination and overall predictive performance but it often produces underfitted model and has high rate of convergence failure (even the rate is higher than that for MLE), probably due to the separation problem.ConclusionsThe logF-type penalized method, particularly logF(1,1) could be used in practice when developing risk model for small or sparse data sets.