Conditional Maximum Likelihood Estimator Research Articles

Maximum likelihood estimation of the DDRCINAR(p) model

In this paper, the novel estimating methods and their properties for pth-order dependence-driven random coefficient integer-valued autoregressive time series model (DDRCINAR(p)) are studied as the innovation sequence has a Poisson distribution and the thinning is binomial. Strict stationarity and ergodicity for DDRCINAR(p) model are proved. Conditional maximum likelihood and conditional least squares are used to estimate the model parameters. Asymptotic normality of the proposed estimators are derived. Finite sample properties of the conditional maximum likelihood estimator are examined in relation to the widely used conditional least squares estimator. It is concluded that, if the Poisson assumption can be justified, conditional maximum likelihood method performs better in terms of bias and MSE. Finally, three real data sets are analyzed to demonstrate the practical relevance of the model.

A Dynamic Ordered Logit Model With Fixed Effects

We study a fixed-T panel data logit model for ordered outcomes that accommodates fixed effects and state dependence. We provide identification results for the auto-regressive parameter, regression coefficients, and the threshold parameters in this model. Our results require only four observations on the outcome variable. We provide conditions under which a composite conditional maximum likelihood estimator is consistent and asymptotically normal. We use our estimator to explore the determinants of self-reported health in a panel of European countries over the period 2003-2016. We find that: (i) that the auto-regressive parameter is positive and analogous to a linear AR(1) coefficient of about 0.25, indicating persistence in health status (ii) that the association between income and health becomes insignificant once we control for unobserved heterogeneity and persistence.

Comparison of Item Difficulty Estimates in a Basic Statistics Test using ltm and CTT Software Packages in R

Two free computer software packages “ltm” and “CTT” in the R software environment were tested to demonstrate its usefulness in an item test analysis. The calibration of the item difficulty parameters given the binary responses of two hundred five examinees for the fifteen items multiple choice test were analyzed using the Classical Test Theory (CTT) and Item Response Theory (IRT) methodologies. The software latent trait model “ltm” employed the IRT framework while the software classical test theory functions “CTT” operated under CTT. The IRT Rasch model was used to model the responses of the examinees. The conditional maximum likelihood estimation method was used to estimate the item difficulty parameters for all the items. On the other hand, all the item difficulty indices using the “CTT” software were also calculated. Both the statistical analyses of this study were done in the R software. Results showed that among the fifteen items, the estimates of their item difficulty parameters differed mostly on their values between the two methods. In an IRT framework, items showed extreme difficulty or easy cases as compared to CTT. However, when the estimated values were categorized into intervals and labelled according to its verbal difficulty description, both methodologies showed some similarities in their item difficulties.

Modeling bivariate long‐range dependence with general phase

Bivariate time series models are considered that are suitable for estimation, that have interpretable parameters and that can capture the general semi‐parametric formulation of bivariate long‐range dependence, including a general phase. The models also allow for short‐range dependence and fractional cointegration. A simulation study to test the performance of a conditional maximum likelihood estimation method is carried out, under the proposed models. Finally, an application is presented to the U.S. inflation rates in goods and services where models not allowing for general phase suffer from misspecification.

The family of the bivariate integer-valued autoregressive process (BINAR(1)) with Poisson–Lindley (PL) innovations

ABSTRACTIn the literature of discrete-valued time series modelling, various bivariate integer-valued autoregressive time series models of order 1 (BINAR(1)) have been proposed particularly based on the binomial thinning mechanism and with different innovation distributions. These BINAR(1)s are mostly suitable for modelling bivariate counting series of varied levels of overdispersion. Recently, in the context of overdispersion, the INAR(1) with Poisson–Lindley (PL) innovations have been introduced to provide superior model fitness criteria than other competing INAR(1)s. Thus, this paper proposes to develop classes of BINAR(1)PL, including BINAR(1)PL(I) and BINAR(1)PL(II), under different cross-correlation functions. The parameter estimations are conducted via the conditional maximum likelihood (CML) approach. Monte Carlo simulation experiments are implemented to assess the asymptotic properties of the CML estimators under different combinations of the cross-correlation parameters. The proposed models are also applied to the Pittsburg crimes data and compared with other competing popular overdispersed BINAR(1) models.

A generalized mixture integer-valued GARCH model

We propose a generalized mixture integer-valued generalized autoregressive conditional heteroscedastic model to provide a more flexible modeling framework. This model includes many mixture integer-valued models with different distributions already studied in the literature. The conditional and unconditional moments are discussed and the necessary and sufficient first- and second-order stationary conditions are derived. We also investigate the theoretical properties such as strict stationarity and ergodicity for the mixture process. The conditional maximum likelihood estimators via the EM algorithm are derived and the performances of the estimators are studied via simulation. The model can be selected in terms of both the number of mixture regimes and the number of orders in each regime by several different criteria. A real-life data example is also given to assess the performance of the model.

Noncoherent Symbol Detection of Short CPM Bursts in Frequency-Selective Fading Channels

We consider the detection of short continuous phase modulation (CPM) bursts in a frequency-selective fading channel. The conventional solution comprises training-aided channel estimation followed by coherent detection. However, the performance of the coherent detector is optimal only when the channel is perfectly known, which is practically never the case. In practice, the performance is limited by the quality of the channel estimate. This poses a problem for short bursts, where the number of training symbols must be kept low. When the channel is unknown, the optimal receiver uses available a priori stochastic information to marginalize the channel out of the likelihood function and determine the transmit sequence that maximizes it. Due to the lack of a priori information and the high complexity associated with marginalizing out a multi-tap channel, we derive a suboptimal detector, which replaces the unknown channel by the conditional maximum likelihood (ML) estimate for each hypothesis transmit sequence. From that, we derive a noncoherent soft-input soft-output (SISO) symbol-by-symbol detector. Using Monte-Carlo simulations to estimate the bit error rate (BER), we show the superiority of the proposed approach over the conventional one, especially for extremely short bursts in a time-variant environment.

Soft‐decision schemes for radar estimation of elevation at low grazing angles

In modern radars, the problem of estimating elevation angle at low grazing angles is typically solved using super-resolution techniques. These techniques often require one to provide an estimate of the number of waveforms impinging the array, which one can accomplish using model selection techniques. In this study, the authors investigate the performance of an alternative approach, based on the Bayesian-like model averaging. The Bayesian approach exploits the fact that the parameters of the model related to multipath signals are nuisance ones, which allows one to avoid the estimation of the number of waveforms and improves estimation performance. The method is introduced for the classical conditional maximum-likelihood estimator and extended to its, recently proposed, robustified version. The authors find, however, that the robustified estimator includes its own soft-decision mechanism and benefits from the averaging only for low levels of model uncertainty.

Conditional maximum likelihood estimation for semiparametric transformation models with doubly truncated data

Doubly truncated data arise when a failure time T is observed only if it falls within a subject-specific, possibly random, interval [U,V], where U and V are referred to as left- and right-truncation times, respectively. In this article, we consider the problem of fitting semiparametric transformation regression models to doubly truncated data. Most of the existing approaches in literature, which adjust for double truncation in regression models, require independence between failure times and truncation times, which may not hold in practice. To relax the independence assumption to conditional independence given covariates, we consider a conditional likelihood approach and develop the conditional maximum likelihood estimators (cMLE) for the regression parameters and cumulative hazard function of models. Based on score equations for the regression parameter and the infinite-dimensional function, we propose an iterative algorithm for obtaining the cMLE. The cMLE is shown to be consistent and asymptotically normal. Simulation studies indicate that the cMLE performs well and outperforms the existing estimators when an independence assumption holds. Applications to an AIDS dataset is given to illustrate the proposed method.

On the theory of periodic multivariate INAR processes

In this paper a multivariate integer-valued autoregressive model of order one with periodic time-varying parameters, and driven by a periodic innovations sequence of independent random vectors is introduced and studied in detail. Emphasis is placed on models with periodic multivariate negative binomial innovations. Basic probabilistic and statistical properties of the novel model are discussed. Aiming to reduce computational burden arising from the use of the conditional maximum likelihood method, a composite likelihood-based approach is adopted. The performance of such method is compared with that of some traditional competitors, namely moment estimators and conditional maximum likelihood estimators. Forecasting is also addressed. Furthermore, an application to a real data set concerning the monthly number of fires in three counties in Portugal is presented.

Bootstrap prediction intervals for autoregressive conditional duration models

ABSTRACTIn the recent past, the autoregressive conditional duration (ACD) models have gained popularity in modelling the durations between successive events. The aim of this paper is to propose a simple and distribution free re-sampling procedure for developing the forecast intervals of linear ACD Models. We use the conditional least squares method to estimate the parameters of the ACD Model instead of the conditional Maximum Likelihood Estimation or Quasi-Maximum Likelihood Estimation and show that they are consistent for large samples. The properties of the proposed procedure are illustrated by a simulation study and an application to two real data sets.

Composite likelihood estimation for a Gaussian process under fixed domain asymptotics

We study the problem of estimating the covariance parameters of a one-dimensional Gaussian process with exponential covariance function under fixed-domain asymptotics. We show that the weighted pairwise maximum likelihood estimator of the microergodic parameter can be consistent or inconsistent. This depends on the range of admissible parameter values in the likelihood optimization. On the other hand, the weighted pairwise conditional maximum likelihood estimator is always consistent. Both estimators are also asymptotically Gaussian when they are consistent. Their asymptotic variances are larger or strictly larger than that of the maximum likelihood estimator. A simulation study is presented in order to compare the finite sample behavior of the pairwise likelihood estimators with their asymptotic distributions. For more general covariance functions, an additional inconsistency result is provided, for the weighted pairwise maximum likelihood estimator of a variance parameter.

Binomial AR(1) processes with innovational outliers

Binomial integer-valued AR processes have been well studied in the literature, but there is little progress in modeling bounded integer-valued time series with outliers. In this paper, we first review some basic properties of the binomial integer-valued AR(1) process and then we introduce binomial integer-valued AR(1) processes with two classes of innovational outliers. We focus on the joint conditional least squares (CLS) and the joint conditional maximum likelihood (CML) estimates of models’ parameters and the probability of occurrence of the outlier. Their large-sample properties are illustrated by simulation studies. Artificial and real data examples are used to demonstrate good performances of the proposed models.

The Cox-Aalen model for left-truncated and mixed interval-censored data

ABSTRACTScheike and Zhang [An additive-multiplicative Cox-Aalen regression model. Scand J Stat. 2002;29:75–88] proposed a flexible additive-multiplicative hazard model, called the Cox-Aalen model, by replacing the baseline hazard function in the well-known Cox model with a covariate-dependent Aalen model, which allows for both fixed and dynamic covariate effects. In this paper, based on left-truncated and mixed interval-censored (LT-MIC) data, we consider maximum likelihood estimation for the Cox-Aalen model with fixed covariates. We propose expectation-maximization (EM) algorithms for obtaining the conditional maximum likelihood estimators (cMLE) of the regression coefficients for the Cox-Aalen model. We establish the consistency of the cMLE. Numerical studies show that estimation via the EM algorithms performs well.

Semiparametric regression analysis for left-truncated and interval-censored data without or with a cure fraction

Interval censoring and truncation arise often in cohort studies, longitudinal and sociological research. In this article, we formulate the effects of covariates on left-truncated and mixed case interval-censored (LTIC) data without or with a cure fraction through a general class of semiparametric transformation models. We propose the conditional likelihood approach for statistical inference. For data without a cure fraction, we propose a computationally efficient EM algorithm, facilitated by a novel data augmentation method, to obtain the conditional maximum likelihood estimator (cMLE). For data with a cure fraction, we consider a semiparametric mixture cure model, which combines a logistic regression formula for the uncured probability with the class of transformation models for the failure time of uncured individuals. To overcome the computational complexity due to the presence of a cure fraction, by reparameterization of cure rate in the conditional likelihood function, we propose a computationally stable EM algorithm for obtaining the cMLE. We show that the cMLEs for the regression parameters are consistent and asymptotically normal. Based on the profile likelihood, we apply an EM-aided numerical differentiation method to compute the asymptotic variance estimates. We demonstrate the performance of our procedures through intensive simulation studies and application to the datasets from the Cardiovascular Disease Risk Factors Two-Township Study.

Pedestrian Injury Severity vs. Vehicle Impact Speed: Uncertainty Quantification and Calibration to Local Conditions

This paper describes a method for fitting predictive models that relate vehicle impact speeds to pedestrian injuries, in which results from a national sample are calibrated to reflect local injury statistics. Three methodological issues identified in the literature, outcome-based sampling, uncertainty regarding estimated impact speeds, and uncertainty quantification, are addressed by (i) implementing Bayesian inference using Markov Chain Monte Carlo sampling and (ii) applying multiple imputation to conditional maximum likelihood estimation. The methods are illustrated using crash data from the NHTSA Pedestrian Crash Data Study coupled with an exogenous sample of pedestrian crashes from Minnesota’s Twin Cities. The two approaches produced similar results and, given a reliable characterization of impact speed uncertainty, either approach can be applied in a jurisdiction having an exogenous sample of pedestrian crash severities.

Signed compound poisson integer-valued GARCH processes

We propose signed compound Poisson integer-valued GARCH processes for the modeling of the difference of count time series data. We investigate the theoretical properties of these processes and we state their ergodicity and stationarity under mild conditions. We discuss the conditional maximum likelihood estimator when the series appearing in the difference are INGARCH with geometric distribution and explore its finite sample properties in a simulation study. Two real data examples illustrate this methodology.

Modeling overdispersed or underdispersed count data with generalized Poisson integer-valued autoregressive processes

To accurately and flexibly capture the dispersion features of time series of counts, we introduce the generalized Poisson thinning operation and further define some new integer-valued autoregressive processes. Basic probabilistic and statistical properties of the models are discussed. Conditional least squares and maximum quasi likelihood estimators are investigated via the moment targeting estimation methods for the innovation free case. Also, the asymptotic properties of the estimators are obtained. Conditional maximum likelihood estimation for the parametric cases are also discussed. Finally, some numerical results of the estimates and two real data examples are presented.

Pseudo MLE for semiparametric transformation model with doubly truncated data

In this article, we consider the efficient estimation of the semiparametric transformation model with doubly truncated data. We propose a two-step approach for obtaining the pseudo maximum likelihood estimators (PMLE) of regression parameters. In the first step, the truncation time distribution is estimated by the nonparametric maximum likelihood estimator (Shen, 2010a) when the distribution function K of the truncation time is unspecified or by the conditional maximum likelihood estimator (Bilker and Wang, 1996) when K is parameterized. In the second step, using the pseudo complete-data likelihood function with the estimated distribution of truncation time, we propose expectation–maximization algorithms for obtaining the PMLE. We establish the consistency of the PMLE. The simulation study indicates that the PMLE performs well in finite samples. The proposed method is illustrated using an AIDS data set.

On semiparametric transformation model with LTRC data: pseudo likelihood approach

When the distribution of the truncation time is known up to a finite-dimensional parameter vector, many researches have been conducted with the objective to improve the efficiency of estimation for nonparametric or semiparametric model with left-truncated and right-censored (LTRC) data. When the distribution of truncation times is unspecified, one approach is to use the conditional maximum likelihood estimators (cMLE) (Chen and Shen in Lifetime Data Anal https://doi.org/10.1007/s10985-016-9385-9, 2017). Although the cMLE has nice asymptotic properties, it is not efficient since the conditional likelihood function does not incorporate information on the distribution of truncation time. In this article, we aim to develop a more efficient estimator by considering the full likelihood function. Following Turnbull (J R Stat Soc B 38:290–295, 1976) and Qin et al. (J Am Stat Assoc 106:1434–1449, 2011), we treat the unobserved (left-truncated) subpopulation as missing data and propose a two-stage approach for obtaining the pseudo maximum likelihood estimators (PMLE) of regression parameters. In the first stage, the distribution of left truncation time is estimated by the inverse-probability-weighted (IPW) estimator (Wang in J Am Stat Assoc 86:130–143, 1991). In the second stage, we obtain the pseudo complete-data likelihood function by replacing the distribution of truncation time with the IPW estimator in the full likelihood. We propose an expectation–maximization algorithm for obtaining the PMLE and establish the consistency of the PMLE. Simulation results show that the PMLE outperforms the cMLE in terms of mean squared error. The PMLE can also be used to analyze the length-biased data, where the truncation time is uniformly distributed. We demonstrate that the PMLE works more robust against the support assumption of truncation time for length-biased data compared with the MLE proposed by Qin et al. (2011). We apply our proposed method to the channing house data. While the PMLE is quite appealing under specific cases with independent censoring and time-invariant covariates, its applicability, as shown in simulation study, can be rather restricted for more general settings.