Properties Of Maximum Likelihood Estimators Research Articles

Large sample properties of maximum likelihood estimator using moving extremes ranked set sampling

Large sample properties of maximum likelihood estimator using moving extremes ranked set sampling

Hypothesis testing for Panels of Semi-Markov Processes with parametric sojourn time distributions

This work deals with the asymptotic properties of maximum likelihood estimators for semi-Markov processes with parametric sojourn time distributions. It is motivated by the comparison, via a two-sample test procedure, of the distribution of two panels of qualitative trajectories modeled by semi-Markov processes and observed over a random number of transitions. Considering first one panel of growing size, we derive, under classical conditions, the convergence in probability of the estimators of the transition probabilities and the parameters of the sojourn time distributions as well as their asymptotic normality. We then consider panels of semi-Markov processes drawn from two different populations and study two-sample tests based on likelihood ratio. We also introduce a two-sample Wald type test. The finite sample performances of the proposed two-sample tests are evaluated with a brief simulation study.

Modeling biomarker variability in joint analysis of longitudinal and time-to-event data.

The role of visit-to-visit variability of a biomarker in predicting related disease has been recognized in medical science. Existing measures of biological variability are criticized for being entangled with random variability resulted from measurement error or being unreliable due to limited measurements per individual. In this article, we propose a new measure to quantify the biological variability of a biomarker by evaluating the fluctuation of each individual-specific trajectory behind longitudinal measurements. Given a mixed-effects model for longitudinal data with the mean function over time specified by cubic splines, our proposed variability measure can be mathematically expressed as a quadratic form of random effects. A Cox model is assumed for time-to-event data by incorporating the defined variability as well as the current level of the underlying longitudinal trajectory as covariates, which, together with the longitudinal model, constitutes the joint modeling framework in this article. Asymptotic properties of maximum likelihood estimators are established for the present joint model. Estimation is implemented via an Expectation-Maximization (EM) algorithm with fully exponential Laplace approximation used in E-step to reduce the computation burden due to the increase of the random effects dimension. Simulation studies are conducted to reveal the advantage of the proposed method over the two-stage method, as well as a simpler joint modeling approach which does not take into account biomarker variability. Finally, we apply our model to investigate the effect of systolic blood pressure variability on cardiovascular events in the Medical Research Council elderly trial, which is also the motivating example for this article.

Information-Geometric Approach for a One-Sided Truncated Exponential Family.

In information geometry, there has been extensive research on the deep connections between differential geometric structures, such as the Fisher metric and the α-connection, and the statistical theory for statistical models satisfying regularity conditions. However, the study of information geometry for non-regular statistical models is insufficient, and a one-sided truncated exponential family (oTEF) is one example of these models. In this paper, based on the asymptotic properties of maximum likelihood estimators, we provide a Riemannian metric for the oTEF. Furthermore, we demonstrate that the oTEF has an α = 1 parallel prior distribution and that the scalar curvature of a certain submodel, including the Pareto family, is a negative constant.

Statistical Inference of Spatial Econometric Models Based on Likelihood Functions

This paper is devoted to studying the probabilistic and statistical properties of maximum likelihood estimation of parameters in spatially mixed autoregressive models. When the response variable in the mixed autoregressive model obeys a continuous distribution, this paper verifies the monotonicity of the likelihood function of the mixed autoregressive model with respect to the autoregressive parameter P, which proves the existence and uniqueness of the maximum likelihood estimation of the autoregressive parameter. The results show that when the condition n > rank(x+1) is satisfied, the quasi-likelihood function of the mixed autoregressive model has a unique maximum value with probability 1 in the parameter space; when n > rank(x+1) and the regression coefficient When the matrix column is full rank, the maximum likelihood estimates of all parameters in the spatial mixed autoregressive model exist with probability 1 and are unique in the parameter space. In order to detect the strong influence points and abnormal points in the spatial mixed autoregressive model, this paper uses the first derivative method in the local influence analysis to obtain the statistics of the strong influence points and abnormal points in the spatial mixed autoregressive model in the form of variance perturbation. Simulation studies have shown that the first derivative of the maximum likelihood estimate of variance in the spatial mixed autoregressive model should be chosen to obtain the test statistic, which can effectively avoid the mearing and masking effects that often occur and are difficult to handle in local influence analysis. . As application and verification, this paper analyzes a real data to show that the conclusions obtained in this paper are reliable and practical.

On partially observed competing risk model under generalized progressive hybrid censoring for Lomax distribution

ABSTRACT We study a competing risks model under the assumption that latent failure times follow family of Lomax distributions. We obtain various inferences for model parameters when causes of failure are partially known and lifetime data are observed using a generalized progressive hybrid censoring scheme. The existence and uniqueness properties of maximum likelihood estimators of unknown parameters are established. Bayes estimators and associated credible intervals are also obtained. In addition, various inferences for unknown parameters are derived under order-restricted shape parameters of Lomax distributions. Finally, a simulation study is conducted to evaluate the performance of the proposed estimates. A real data set is also analysed for illustration purposes.

Inference in mixed causal and noncausal models with generalized Student’s t-distributions

The properties of Maximum Likelihood estimator in mixed causal and noncausal models with a generalized Student’s t error process are reviewed. Several known existing methods are typically not applicable in the heavy-tailed framework. To this end, a new approach to make inference on causal and noncausal parameters in finite sample sizes is proposed. It exploits the empirical variance of the generalized Student’s t, without the existence of population variance. Monte Carlo simulations show a good performance of the new variance construction for fat tail series. Finally, different existing approaches are compared using three empirical applications: the variation of daily COVID-19 deaths in Belgium, the monthly wheat prices, and the monthly inflation rate in Brazil.

Application of the Information Matrix Test on the National Productivity Analysis as the Cause of International Trade

Purpose In order to analyze the national competitiveness of the Korean economy, it is necessary to estimate the total factor productivity correctly, because incorrect assumptions in the model can lead to serious estimation errors. Therefore, in order to estimate the suitability of the model, an information matrix test was proposed as an applicable method to the actual stochastic frontier production function. Design/Methodology/Approach We first examine the large sample distribution properties of maximum likelihood estimation (MLE) and quasi-maximum likelihood estimation (QMLE), and introduce several versions of the stochastic frontier product function models proposed by different scholars. Finally, we derive the Information Matrix Test (Battese and Coelli, 1992) for the stochastic frontier production function model. Also, a Monte Carlo simulation was applied to verify the correctness of the model. Findings From the results, we derive an analytical form of the Information Matrix Test (Battese and Coelli, 1992) on the stochastic frontier production function model using the simplified form proposed by Lancaster (1984). and as a result of empirical analysis by applying this process to each of the seven countries (Korea, China, Japan, the United States, United Kingdom, Germany, France), each  value was below the 95% significance level in all countries, indicating that the translog production function model was specified correctly. Research Implications The Information Matrix Test by White (1982) is an important solution for misspecified regression models. To solve this issue, Lancaster (1984) suggested the simpler form of the Information Matrix Test of White (1982). However, this method is also complex to apply in a real model. Therefore, until now, the Information Matrix Test has not been properly implemented in a regression model. This study presents an analytical form of the Information Matrix Test for the stochastic frontier production function, which is very important for analyzing national, industrial, and corporate competitiveness.

Estimation of Cumulative Incidence Function in the Presence of Middle Censoring Using Improper Gompertz Distribution

In this paper we deal with the modelling of cumulative incidence function using improper Gompertz distribution based on middle censored competing risks survival data. Together with the unknown parameters, cumulative incidence function also estimated. In classical set up, we derive the point estimates using maximum likelihood estimator and midpoint approximation methods. The asymptotic confidence interval are obtained based on asymptotic normality properties of maximum likelihood estimator. We also derive the Bayes estimates with associated credible intervals based on informative and non-informative types of priors under two loss functions such as squared error and LINEX loss functions. A simulation study is conducted for comprehensive comparison between various estimators proposed in this paper. A real life data set is also used for illustration.

Maximum likelihood estimation for sub-fractional Vasicek model

Abstract We investigate the asymptotic properties of maximum likelihood estimators of the drift parameters for the fractional Vasicek model driven by a sub-fractional Brownian motion.

Stress-Strength Parameter Estimation under Small Sample Size: A Testing Hypothesis Approach.

In this paper, uniformly most powerful unbiased test for testing the stress-strength model has been presented for the first time. The end of the paper is recommending a method which is appropriate for no large data where a normal asymptotic distribution is not applicable. The previous methods for inference on stress-strength models use almost all the asymptotic properties of maximum likelihood estimators. The distribution of components is considered exponential and generalized logistic. A corresponding unbiased confidence interval is constructed, too. We compare presented methodology with previous methods and show the method of this paper is logically better than other methods. Interesting result is that our recommended method not only uses from small sample size but also has better result than other ones.

Application of PZT Ceramic Sensors for Composite Structure Monitoring Using Harmonic Excitation Signals and Bayesian Classification Approach.

The capabilities of ceramic PZT transducers, allowing for elastic wave excitation in a broad frequency spectrum, made them particularly suitable for the Structural Health Monitoring field. In this paper, the approach to detecting impact damage in composite structures based on harmonic excitation of PZT sensor in the so-called pitch–catch PZT network setup is studied. In particular, the repeatability of damage indication for similar configuration of two independent PZT networks is analyzed, and the possibility of damage indication for different localization of sensing paths between pairs of PZT sensors with respect to damage locations is investigated. The approach allowed for differentiation between paths sensitive to the transmission mode of elastic wave interaction and sensitive reflection mode. In addition, a new universal Bayesian approach to SHM data classification is provided in the paper. The defined Bayesian classifier is based on asymptotic properties of Maximum Likelihood estimators and Principal Component Analysis for orthogonal data transformation. Properties of the defined algorithm are compared to the standard nearest-neighbor classifier based on the acquired experimental data. It was shown in the paper that the proposed approach is characterized by lower false-positive indications in comparison with the nearest-neighbor algorithm.

Pareto Distribution under Hybrid Censoring: Some Estimation

In the present study, the Pareto model is considered as the model from which observations are to be estimated using a Bayesian approach. Properties of the Bayes estimators for the unknown parameters have studied by using different asymmetric loss functions on hybrid censoring pattern and their risks have compared. The properties of maximum likelihood estimation and approximate confidence length have also been investigated under hybrid censoring. The performances of the procedures are illustrated based on simulated data obtained under the Metropolis-Hastings algorithm and a real data set.

A new linearized ridge Poisson estimator in the presence of multicollinearity

Poisson regression is a very commonly used technique for modeling the count data in applied sciences, in which the model parameters are usually estimated by the maximum likelihood method. However, the presence of multicollinearity inflates the variance of maximum likelihood (ML) estimator and the estimated parameters give unstable results. In this article, a new linearized ridge Poisson estimator is introduced to deal with the problem of multicollinearity. Based on the asymptotic properties of ML estimator, the bias, covariance and mean squared error of the proposed estimator are obtained and the optimal choice of shrinkage parameter is derived. The performance of the existing estimators and proposed estimator is evaluated through Monte Carlo simulations and two real data applications. The results clearly reveal that the proposed estimator outperforms the existing estimators in the mean squared error sense.

Statistical inference for the reliability of Burr-XII distribution under improved adaptive Type-II progressive censoring

We propose an improved adaptive Type-II progressive censoring scheme, which can effectively guarantee that the test time will not exceed a certain threshold time. And the adaptive Type-II progressive censoring scheme is its special case. Based on this censoring scheme, the parameters, reliability and hazard functions of Burr-XII distribution were estimated. Firstly, we obtain the estimators of parameters, reliability and hazard functions by using the maximum likelihood estimation method. Subsequently, the approximate confidence intervals of the above statistics are obtained by using the asymptotic normality properties of maximum likelihood estimation and the Bootstrap method. Furthermore, based on the square loss function and the LINEX loss function, the point estimators and approximate confidence intervals of parameters are derived by using Bayesian method. A real data is used to illustrate and validate the advantages of the proposed improved adaptive Type-II progressive censoring scheme. Finally, the Monte Carlo simulations are performed to compare the behavior of improved adaptive Type-II progressive censoring scheme with adaptive Type-II progressive censoring scheme for different evaluation methods.

Towards a general theory for nonlinear locally stationary processes

In this paper, some general theory is presented for locally stationary processes based on the stationary approximation and the stationary derivative. Laws of large numbers, central limit theorems as well as deterministic and stochastic bias expansions are proved for processes obeying an expansion in terms of the stationary approximation and derivative. In addition it is shown that this applies to some general nonlinear non-stationary Markov-models. In addition the results are applied to derive the asymptotic properties of maximum likelihood estimates of parameter curves in such models.

Asymptotic properties of maximum likelihood estimators with sample size recalculation.

Consider an experiment in which the primary objective is to determine the significance of a treatment effect at a predetermined type I error and statistical power. Assume that the sample size required to maintain these type I error and power will be re-estimated at an interim analysis. A secondary objective is to estimate the treatment effect. Our main finding is that the asymptotic distributions of standardized statistics are random mixtures of distributions, which are non-normal except under certain model choices for sample size re-estimation (SSR). Monte-Carlo simulation studies and an illustrative example highlight the fact that asymptotic distributions of estimators with SSR may differ from the asymptotic distribution of the same estimators without SSR.

Asymptotic properties of maximum likelihood estimator for the growth rate of a stable CIR process based on continuous time observations

ABSTRACTWe consider a stable Cox–Ingersoll–Ross process driven by a standard Wiener process and a spectrally positive strictly stable Lévy process, and we study asymptotic properties of the maximum likelihood estimator (MLE) for its growth rate based on continuous time observations. We distinguish three cases: subcritical, critical and supercritical. In all cases we prove strong consistency of the MLE in question, in the subcritical case asymptotic normality, and in the supercritical case asymptotic mixed normality are shown as well. In the critical case the description of the asymptotic behaviour of the MLE in question remains open.

On the robustness properties for maximum likelihood estimators of parameters in exponential power and generalized T distributions

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.

Hybrid TDOA/RSS based localization for visible light systems

In a visible light positioning (VLP) system, a receiver can estimate its location based on signals transmitted by light emitting diodes (LEDs). In this manuscript, we investigate a quasi-synchronous VLP system, in which the LED transmitters are synchronous among themselves but are not synchronized with the receiver. In quasi-synchronous VLP systems, position estimation can be performed by utilizing time difference of arrival (TDOA) information together with channel attenuation information, leading to a hybrid localization system. To specify accuracy limits for quasi-synchronous VLP systems, the Cramér–Rao lower bound (CRLB) on position estimation is derived in a generic three-dimensional scenario. Then, a direct positioning approach is adopted to obtain the maximum likelihood (ML) position estimator based directly on received signals from LED transmitters. In addition, a two-step position estimator is proposed, where TDOA and received signal strength (RSS) estimates are obtained in the first step and the position estimation is performed, based on the TDOA and RSS estimates, in the second step. The performance of the two-step positioning technique is shown to converge to that of direct positioning at high signal-to-noise ratios based on asymptotic properties of ML estimation. Finally, CRLBs and performance of the proposed positioning techniques are investigated through simulations.