Unobserved Random Variables Research Articles

Multistate models with nested frailty for lifetime analysis: Application to bone marrow transplantation recovery patients

Multistate models allow individuals to move through a series of states over time, making it possible to estimate the transition probabilities and intensities between states and the effect of covariates associated with each transition. Multistate models are useful for studying the evolution of individuals over time, particularly in clustered data, where individuals are grouped together and may share common risk factors. Frailties can be used in multistate models to capture the heterogeneity between groups at risk for different types of transitions, as well as the dependency structure between transitions of individuals in the same group. Frailties are unobserved random variables that are assumed to be independent between different types of transitions but shared by groups of individuals. In this paper, we propose a multistate model with frailties to capture the heterogeneity between groups at risk for different types of transitions, as well as the dependency structure between transitions of individuals in the same group. We assume that the frailties are independent between different types of transitions but shared by groups of individuals. We discuss both parametric and semiparametric estimation of the model parameters, and we evaluate the performance of the model in simulation studies. Finally, we apply the model to a dataset of bone marrow transplantation recovery patients.

Almost sure central limit theorems for the maxima of randomly chosen random variables

In this paper, we give an almost sure central limit theorem (ASCLT) version of a maximum limit theorem (MLT) with an arbitrary sequence {dn, n ≥ 1} of weighted means of max{Xk, k ∈ An}, where {Xn, n ≥ 1} is a sequence of independent random variables, and {An, n ≥ 1} is a sequence of almost surely finite random subsets of positive integers independent of {Xn, n ≥ 1}. Thus we generalize the cases considered in the literature: (i) the nonrandom version of ASCLT for the MLT; (ii) the version of ASCLT for randomly indexed MLT; and (iii) the version of maximum schema of observed and unobserved random variables. We complete the paper with illustrative examples.

Comments on Confidence as Likelihood by Pawitan and Lee in Statistical Science, November 2021

Pawitan and Lee (Statist. Sci. 36 (2021) 509–517) attempt to show a correspondence between confidence and likelihood, specifically, that “confidence is in fact an extended likelihood” (Statist. Sci. 36 (2021) 509–517, abstract). The word “extended” means that the likelihood function can accommodate unobserved random variables such as random effects and future values; see (J. Amer. Statist. Assoc. 91 (1996) 791–806) for details. Here, we argue that the extended likelihood presented by (Statist. Sci. 36 (2021) 509–517) is not the correct extended likelihood and does not justify interpreting confidence as likelihood.

A simulation-based decomposition approach for two-stage staffing optimization in call centers under arrival rate uncertainty

We study a solution approach for a staffing problem in multi-skill call centers. The objective is to find a minimal-cost staffing solution while meeting a target level for the quality of service to customers. We consider a common situation in which the arrival rates are unobserved random variables for which preliminary forecasts are available in a first stage when making the initial staffing decision. In a second stage, more accurate forecasts are obtained and the staffing may have to be modified at a cost, to meet the constraints. This leads to a challenging two-stage stochastic optimization problem in which the quantities involved in the (nonlinear) constraints can only be estimated via simulation, so several independent simulations are required for each first-level scenario. We propose a solution approach that combines sample average approximation with a decomposition method. We provide numerical illustrations to show the practical efficiency of our approach. The proposed method could be adapted to several other staffing problems with uncertain demand, e.g., in retail stores, restaurants, healthcare facilities, and other types of service systems.

Correlation characteristics assessment of construction products reliability parameters

Since the reliability of construction products, by definition, is a probability of failure-free operation, then to assess the correlation characteristics between the parameters of the reliability of construction products that are not observed at the same time, we will use the designations and terms used in the theory of probability and mathematical statistics. Existing statistical methods for assessing the conditional mathematical expectation applied to the construction product reliability are based on the possibility of simultaneous measurement of the value of random variables. Because of this, they get a set of extreme points in the plane of those variables, and then build a regression line estimates based on those parameters. Such methods are obviously unacceptable for the case when the quantities the mentioned random variables are cumulatively unobserved. The solution to this problem will allow us to find estimates (not statistical) of the regression line and a number of other characteristics associated with it for simultaneously unobservable random variables. The algorithm described in the current work illustrates an example of interest in the theory of reliability for finding the extrema of the averaged function over the given initial parameters of the construction product.

A Penalized h-Likelihood Variable Selection Algorithm for Generalized Linear Regression Models with Random Effects

Reinforcement learning is one of the paradigms and methodologies of machine learning developed in the computational intelligence community. Reinforcement learning algorithms present a major challenge in complex dynamics recently. In the perspective of variable selection, we often come across situations where too many variables are included in the full model at the initial stage of modeling. Due to a high-dimensional and intractable integral of longitudinal data, likelihood inference is computationally challenging. It can be computationally difficult such as very slow convergence or even nonconvergence, for the computationally intensive methods. Recently, hierarchical likelihood (h-likelihood) plays an important role in inferences for models having unobservable or unobserved random variables. This paper focuses linear models with random effects in the mean structure and proposes a penalized h-likelihood algorithm which incorporates variable selection procedures in the setting of mean modeling via h-likelihood. The penalized h-likelihood method avoids the messy integration for the random effects and is computationally efficient. Furthermore, it demonstrates good performance in relevant-variable selection. Throughout theoretical analysis and simulations, it is confirmed that the penalized h-likelihood algorithm produces good fixed effect estimation results and can identify zero regression coefficients in modeling the mean structure.

A review of h‐likelihood and hierarchical generalized linear model

AbstractFisher's classical likelihood has become the standard procedure to make inference for fixed unknown parameters. Recently, inferences of unobservable random variables, such as random effects, factors, missing values, etc., have become important in statistical analysis. Because Fisher's likelihood cannot have such unobservable random variables, the full Bayesian method is only available for inference. An alternative likelihood approach is proposed by Lee and Nelder. In the context of Fisher likelihood, the likelihood principle means that the likelihood function carries all relevant information regarding the fixed unknown parameters. Bjørnstad extended the likelihood principle to extended likelihood principle; all information in the observed data for fixed unknown parameters and unobservables are in the extended likelihood, such as the h‐likelihood. However, it turns out that the use of extended likelihood for inferences is not as straightforward as the Fisher likelihood. In this paper, we describe how to extract information of the data from the h‐likelihood. This provides a new way of statistical inferences in entire fields of statistical science.This article is categorized under: Statistical Models > Generalized Linear Models Algorithms and Computational Methods > Maximum Likelihood Methods

Optimal Nonparametric Bayesian Model-Based Multimodal BoVW Creation Using Multilayer pLSA

The main objective of this research paper is to design a system which would generate multimodal, nonparametric Bayesian model, and multilayered probability latent semantic analysis (pLSA)-based visual dictionary (BM-MpLSA). Advancement in technology and the exuberance of sports lovers have necessitated a requirement for automatic action recognition in the live video seed of sports. The fundamental requirement for such model is the creation of visual dictionary for each sports domain. This multimodal nonparametric model has two novel co-occurrence matrix creation—one for image feature vector and the other for textual entities. This matrix provides a basic scaling parameter for the unobserved random variables, and it is an extension of multilayered pLSA-based visual dictionary creation. This paper precisely concentrates on the creation of visual dictionary for Basketball. From the sports event images, the feature vector extracted is modified as SIFT and MPEG 7’s-based dominant color, color layout, scalable color and edge histograms. After quantization and analysis of these vector values, the visual vocabulary would be created by integrating them into the domain specific visual ontology for semantic understanding. The accuracy rate of this work is compared with respect to the action held on image based on performance.

Hierarchical likelihood approach to non-Gaussian factor analysis

ABSTRACTFactor models, structural equation models (SEMs) and random-effect models share the common feature that they assume latent or unobserved random variables. Factor models and SEMs allow well developed procedures for a rich class of covariance models with many parameters, while random-effect models allow well developed procedures for non-normal models including heavy-tailed distributions for responses and random effects. In this paper, we show how these two developments can be combined to result in an extremely rich class of models, which can be beneficial to both areas. A new fitting procedures for binary factor models and a robust estimation approach for continuous factor models are proposed.

Multivariate dynamic Tobit models with lagged observed dependent variables: An effectiveness analysis of highway safety laws

Highway safety laws aim to influence driver behaviors so as to reduce the frequency and severity of crashes, and their outcomes. For one specific highway safety law, it would have different effects on the crashes across severities. Understanding such effects can help policy makers upgrade current laws and hence improve traffic safety. To investigate the effects of highway safety laws on crashes across severities, multivariate models are needed to account for the interdependency issues in crash counts across severities. Based on the characteristics of the dependent variables, multivariate dynamic Tobit (MVDT) models are proposed to analyze crash counts that are aggregated at the state level. Lagged observed dependent variables are incorporated into the MVDT models to account for potential temporal correlation issues in crash data. The state highway safety law related factors are used as the explanatory variables and socio-demographic and traffic factors are used as the control variables. Three models, a MVDT model with lagged observed dependent variables, a MVDT model with unobserved random variables, and a multivariate static Tobit (MVST) model are developed and compared. The results show that among the investigated models, the MVDT models with lagged observed dependent variables have the best goodness-of-fit. The findings indicate that, compared to the MVST, the MVDT models have better explanatory power and prediction accuracy. The MVDT model with lagged observed variables can better handle the stochasticity and dependency in the temporal evolution of the crash counts and the estimated values from the model are closer to the observed values. The results show that more lives could be saved if law enforcement agencies can make a sustained effort to educate the public about the importance of motorcyclists wearing helmets. Motor vehicle crash-related deaths, injuries, and property damages could be reduced if states enact laws for stricter text messaging rules, higher speeding fines, older licensing age, and stronger graduated licensing provisions. Injury and PDO crashes would be significantly reduced with stricter laws prohibiting the use of hand-held communication devices and higher fines for drunk driving.

Do natural catastrophes shake microfinance institutions? Using a new measure of MFI risk

Using financial data for more than 1000 microfinance institutions (MFIs) from about 80 developing and emerging market countries, I explore the impact of natural disasters on the financial fragility of these MFIs. For this purpose, I apply a two-stage approach. First, as MFI risk is a multifaceted concept, it is hard to capture by a single indicator. However, previous studies still depend mainly on single and arbitrary chosen measures. In contrast, my improved measure for MFI risk is constructed using factor analysis including 17 different financial risk measures. Factor analysis is a statistical data reduction technique used to explain variability among observed random variables in terms of fewer unobserved random variables called factors. In the second step, I use the factor scores obtained from the factor analysis as my dependent variable in a dynamic panel model. The main findings suggest that the financial risk faced by MFIs increases significantly when a major disaster strikes. In particular, hydro-meteorological disasters have a larger impact on the financial soundness of MFIs than climatic or geophysical disasters. Finally, it turns out that the effect of most natural disasters depends to some extent on the size and scope of the catastrophe, the level of economic development of a country and the kind of clients a particular MFI serves. Based on these results, I conclude that natural disasters are a serious threat to the solvency and liquidity of a MFI. Regulators should in particular monitor and stress test the impact of disasters on risk measures related to capital adequacy and asset quality.

H‐likelihood Predictive Intervals for Unobservables

SummaryInferences about unobserved random variables, such as future observations, random effects and latent variables, are of interest. In this paper, to make probability statements about unobserved random variables without assuming priors on fixed parameters, we propose the use of the confidence distribution for fixed parameters. We focus on their interval estimators and related probability statements. In random‐effect models, intervals can be formed either for future (yet‐to‐be‐realised) random effects or for realised values of random effects. The consistency of intervals for these two cases requires different regularity conditions. Via numerical studies, their finite sampling properties are investigated.

독립성분 행렬도

Biplot is a useful graphical method to simultaneously explore the rows and columns of a two-way data matrix. In particular, principal component factor biplot is a graphical method to describe the interrelationship among many variables in terms of a few underlying but unobservable random variables called factors. If we consider the unobservable variables (which are mutually independent and also non-Gaussian), we can apply the independent component analysis decomposing a mixture of non-Gaussian in its independent components. In this case, if we apply the principal component factor analysis, we cannot clearly describe the interrelationship among many variables. Therefore, in this study, we apply the independent component analysis of Jutten and Herault (1991) decomposing a mixture of non-Gaussian in its independent components. We suggest an independent component biplot to interpret the independent component analysis graphically.

Variational-Bayes Optical Flow

The Horn-Schunck (HS) optical flow method is widely employed to initialize many motion estimation algorithms. In this work, a variational Bayesian approach of the HS method is presented, where the motion vectors are considered to be spatially varying Student's t-distributed unobserved random variables, i.e., the prior is a multivariate Student's t-distribution, while the only observations available is the temporal and spatial image difference. The proposed model takes into account the residual resulting from the linearization of the brightness constancy constraint by Taylor series approximation, which is also assumed to be a spatially varying Student's t-distributed observation noise. To infer the model variables and parameters we recur to variational inference methodology leading to an expectation-maximization (EM) framework with update equations analogous to the Horn-Schunck approach. This is accomplished in a principled probabilistic framework where all of the model parameters are estimated automatically from the data. Experimental results show the improvement obtained by the proposed model which may substitute the standard algorithm in the initialization of more sophisticated optical flow schemes.

Copula models for insurance claim numbers with excess zeros and time-dependence

This paper develops two copula models for fitting the insurance claim numbers with excess zeros and time-dependence. The joint distribution of the claims in two successive periods is modeled by a copula with discrete or continuous marginal distributions. The first model fits two successive claims by a bivariate copula with discrete marginal distributions. In the second model, a copula is used to model the random effects of the conjoint numbers of successive claims with continuous marginal distributions. Zero-inflated phenomenon is taken into account in the above copula models. The maximum likelihood is applied to estimate the parameters of the discrete copula model. A two-step procedure is proposed to estimate the parameters in the second model, with the first step to estimate the marginals, followed by the second step to estimate the unobserved random effect variables and the copula parameter. Simulations are performed to assess the proposed models and methodologies.

Can the GQARCH Latent Factor Model Improve the Prediction Performance of Multivariate Financial Time Series?

ABSTRACTFactor model is a very useful and popular model in Finance. Such a model attempts to explain the correlation between a larger set of variables in terms of a small number of unobservable or latent random variables called factors. This paper considers a combination of the standard Gaussian factor model and the generalized quadratic autoregressive conditionally heteroskedastic model (GQARCH). We discuss some of its properties and we show that identification problems are alleviated when variation in factor variances is accounted for. The focus is on the iterative estimation procedure of the model parameters. Kalman filter and maximum likelihood methods based on the Expectation-Maximization (EM) algorithm lead to an elegant estimation procedure. A detailed comparison of the forecasts generated by the GQARCH-Factor Model is made with other alternative methods. On the basis of out-of-sample forecast encompassing tests as well as other measures for forecasting accuracy, results indicate that the use of this factorial approach yields overall better forecasts than those generated by competing models.

DECONVOLUTING PREFERENCES AND ERRORS: A MODEL FOR BINOMIAL PANEL DATA

In many stated choice experiments researchers observe the random variablesVt,Xt, andYt= 1{U+δ⊤Xt+ εt<Vt},t≤T, whereδis an unknown parameter andUand εtare unobservable random variables. We show that under weak assumptions the distributions ofUand εtand also the unknown parameterδcan be consistently estimated using a sieved maximum likelihood estimation procedure.

Estimation of prediction error for survival models

When statistical models are used to predict the values of unobserved random variables, loss functions are often used to quantify the accuracy of a prediction. The expected loss over some specified set of occasions is called the prediction error. This paper considers the estimation of prediction error when regression models are used to predict survival times and discusses the use of these estimates. Extending the previous work, we consider both point and confidence interval estimations of prediction error, and allow for variable selection and model misspecification. Different estimators are compared in a simulation study for an absolute relative error loss function, and results indicate that cross-validation procedures typically produce reliable point estimates and confidence intervals, whereas model-based estimates are sensitive to model misspecification. Links between performance measures for point predictors and for predictive distributions of survival times are also discussed. The methodology is illustrated in a medical setting involving survival after treatment for disease.

New M-estimators in semi-parametric regression with errors in variables

In the regression model with errors in variables, we observe $n$ i.i.d. copies of $(Y,Z)$ satisfying $Y=f_{\theta^0}(X)+\xi$ and $Z=X+\epsilon$ involving independent and unobserved random variables $X,\xi,\epsilon$ plus a regression function $f_{\theta^0}$, known up to a finite dimensional $\theta^0$. The common densities of the $X_i$'s and of the $\xi_i$'s are unknown, whereas the distribution of $\epsilon$ is completely known. We aim at estimating the parameter $\theta^0$ by using the observations $(Y_1,Z_1),...,(Y_n,Z_n)$. We propose an estimation procedure based on the least square criterion $\tilde{S}_{\theta^0,g}(\theta)=\m athbb{E}_{\theta^0,g}[((Y-f_{\theta}(X))^2w(X)]$ where $w$ is a weight function to be chosen. We propose an estimator and derive an upper bound for its risk that depends on the smoothness of the errors density $p_{\epsilon}$ and on the smoothness properties of $w(x)f_{\theta}(x)$. Furthermore, we give sufficient conditions that ensure that the parametric rate of convergence is achieved. We provide practical recipes for the choice of $w$ in the case of nonlinear regression functions which are smooth on pieces allowing to gain in the order of the rate of convergence, up to the parametric rate in some cases. We also consider extensions of the estimation procedure, in particular, when a choice of $w_{\theta}$ depending on $\theta$ would be more appropriate.

Robust Bayesian clustering

A new variational Bayesian learning algorithm for Student-t mixture models is introduced. This algorithm leads to (i) robust density estimation, (ii) robust clustering and (iii) robust automatic model selection. Gaussian mixture models are learning machines which are based on a divide-and-conquer approach. They are commonly used for density estimation and clustering tasks, but are sensitive to outliers. The Student-t distribution has heavier tails than the Gaussian distribution and is therefore less sensitive to any departure of the empirical distribution from Gaussianity. As a consequence, the Student-t distribution is suitable for constructing robust mixture models. In this work, we formalize the Bayesian Student-t mixture model as a latent variable model in a different way from Svensén and Bishop [Svensén, M., & Bishop, C. M. (2005). Robust Bayesian mixture modelling. Neurocomputing, 64, 235-252]. The main difference resides in the fact that it is not necessary to assume a factorized approximation of the posterior distribution on the latent indicator variables and the latent scale variables in order to obtain a tractable solution. Not neglecting the correlations between these unobserved random variables leads to a Bayesian model having an increased robustness. Furthermore, it is expected that the lower bound on the log-evidence is tighter. Based on this bound, the model complexity, i.e. the number of components in the mixture, can be inferred with a higher confidence.