Asymptotic normality of conditional density estimation under truncated,censored and dependent data

Abstract

In this paper, we focus on the left-truncated and right-censored model, and construct the local linear and Nadaraya-Watson type estimators of the conditional density. Under suitable conditions, we establish the asymptotic normality of the proposed estimators when the observations are assumed to be a stationary α-mixing sequence. Finite sample behavior of the estimators is investigated via simulations too.     

Parameter Estimation in Conditional Heteroscedastic Models

Abstract

We study asymptotics of parameter estimates in conditional heteroscedastic models. The estimators considered are those obtained by minimizing certain functionals and those obtained by solving estimation equations. We establish consistency and derive asymptotic limit laws of the estimators. Condition under which the limit law is normal is studied. Further, bootstrap for these estimators is discussed. The limiting distribution of the estimators is not necessary always normal, and we present a real data example to illustrate this.     

Bias Correction in the Dynamic Panel Data Model with a Nonscalar Disturbance Covariance Matrix

Abstract

Approximation formulae are developed for the bias of ordinary and generalized Least Squares Dummy Variable (LSDV) estimators in dynamic panel data models. Results from Kiviet [Kiviet, J. F. (1995), on bias, inconsistency, and efficiency of various estimators in dynamic panel data models, J. Econometrics68:53–78; Kiviet, J. F. (1999), Expectations of expansions for estimators in a dynamic panel data model: some results for weakly exogenous regressors, In: Hsiao, C., Lahiri, K., Lee, L‐F., Pesaran, M. H., eds., Analysis of Panels and Limited Dependent Variables, Cambridge: Cambridge University Press, pp. 199–225] are extended to higher‐order dynamic panel data models with general covariance structure. The focus is on estimation of both short‐ and long‐run coefficients. The results show that proper modelling of the disturbance covariance structure is indispensable. The bias approximations are used to construct bias corrected estimators which are then applied to quarterly data from 14 European Union countries. Money demand functions for M1, M2 and M3 are estimated for the EU area as a whole for the period 1991: I–1995: IV. Significant spillovers between countries are found reflecting the dependence of domestic money demand on foreign developments. The empirical results show that in general plausible long‐run effects are obtained by the bias corrected estimators. Moreover, finite sample bias, although of moderate magnitude, is present underlining the importance of more refined estimation techniques. Also the efficiency gains by exploiting the heteroscedasticity and cross‐correlation patterns between countries are sometimes considerable.     

Recursive kernel estimate of the conditional quantile for functional ergodic data

ABSTRACT

In this article, we study the recursive kernel estimator of the conditional quantile of a scalar response variable Y given a random variable (rv) X taking values in a semi-metric space. Two estimators are considered. While the first one is given by inverting the double-kernel estimate of the conditional distribution function, the second estimator is obtained by using the robust approach. We establish the almost complete consistency of these estimates when the observations are sampled from a functional ergodic process. Finally, a simulation study is carried out to illustrate the finite sample performance of these estimators.     

Analysis of simple step-stress model in presence of competing risks

ABSTRACT

In this article, we consider a simple step-stress life test in the presence of exponentially distributed competing risks. It is assumed that the stress is changed when a pre-specified number of failures takes place. The data is assumed to be Type-II censored. We obtain the maximum likelihood estimators of the model parameters and the exact conditional distributions of the maximum likelihood estimators. Based on the conditional distribution, approximate confidence intervals (CIs) of unknown parameters have been constructed. Percentile bootstrap CIs of model parameters are also provided. Optimal test plan is addressed. We perform an extensive simulation study to observe the behaviour of the proposed method. The performances are quite satisfactory. Finally we analyse two data sets for illustrative purposes.     

Estimation in a bivariate integer-valued autoregressive process

ABSTRACT

A bivariate integer-valued autoregressive time series model is presented. The model structure is based on binomial thinning. The unconditional and conditional first and second moments are considered. Correlation structure of marginal processes is shown to be analogous to the ARMA(2, 1) model. Some estimation methods such as the Yule–Walker and conditional least squares are considered and the asymptotic distributions of the obtained estimators are derived. Comparison between bivariate model with binomial thinning and bivariate model with negative binomial thinning is given.     

Estimation of a Conditional Copula and Association Measures

Abstarct. This paper is concerned with studying the dependence structure between two random variables Y ₁ and Y ₂ conditionally upon a covariate X. The dependence structure is modelled via a copula function, which depends on the given value of the covariate in a general way. Gijbels et al. (Comput. Statist. Data Anal., 55, 2011, 1919) suggested two non‐parametric estimators of the ‘conditional’ copula and investigated their numerical performances. In this paper we establish the asymptotic properties of the proposed estimators as well as conditional association measures derived from them. Practical recommendations for their use are then discussed.     

Conditional Quantile Estimation for Truncated and Associated Data

Abstract

In survival or reliability studies, it is common to have data which are not only incomplete but weakly dependent too. Random truncation and censoring are two common forms of such data when they are neither independent nor strongly mixing but rather associated. The focus of this paper is on estimating conditional distribution and conditional quantile functions for randomly left truncated data satisfying association condition. We aim at deriving strong uniform consistency rates and asymptotic normality for the estimators and thereby, extend to association case some results stated under iid and α-mixing hypotheses. The performance of the quantile function estimator is evaluated on simulated data sets.     

Conditional failure occurrence rates for semi-Markov chains*

ABSTRACT

In the present paper, we aim at providing plug-in-type empirical estimators that enable us to quantify the contribution of each operational or/and non-functioning state to the failures of a system described by a semi-Markov model. In the discrete-time and finite state space semi-Markov framework, we study different conditional versions of an important reliability measure for random repairable systems, the failure occurrence rate, which is based on counting processes. The identification of potential failure contributors through the conditional counterparts of the failure occurrence rate is of paramount importance since it could lead to corrective actions that minimize the occurrence of the more important failure modes and therefore improve the reliability of the system. The aforementioned estimators are characterized by appealing asymptotic properties such as strong consistency and asymptotic normality. We further obtain detailed analytical expressions for the covariance matrices of the random vectors describing the conditional failure occurrence rates. As particular cases we present the failure occurrence rates for hidden (semi-) Markov models. We illustrate our results by means of a simulated study. Different applications are presented based on wind, earthquake and vibration data.     

On the efficiency property of the linear estimators in the bivariate pearson type vii distribution

In this paper the efficiency property of the estimators of the parameters of the bivariate Pearson type VII distribution is studied inside the family of linear estimators, assuming that the sample is constituted by dependent random vectors. It is proven that, although there are not efficient linear estimators, the sample mean and the sample covariance matrix (affected by an unbiasedness weighting) are unbiased linear estimators of minimum distance to the Cramér-Rao lower bound. Finally, a numerical simulation example shows that the proposed estimators are computationally feasible.     

A semi-parametric cox’s regression model for zero-inflated left-censored time to event data

Abstract

In some clinical, environmental, or economical studies, researchers are interested in a semi-continuous outcome variable which takes the value zero with a discrete probability and has a continuous distribution for the non-zero values. Due to the measuring mechanism, it is not always possible to fully observe some outcomes, and only an upper bound is recorded. We call this left-censored data and observe only the maximum of the outcome and an independent censoring variable, together with an indicator. In this article, we introduce a mixture semi-parametric regression model. We consider a parametric model to investigate the influence of covariates on the discrete probability of the value zero. For the non-zero part of the outcome, a semi-parametric Cox’s regression model is used to study the conditional hazard function. The different parameters in this mixture model are estimated using a likelihood method. Hereby the infinite dimensional baseline hazard function is estimated by a step function. As results, we show the identifiability and the consistency of the estimators for the different parameters in the model. We study the finite sample behaviour of the estimators through a simulation study and illustrate this model on a practical data example.     

On generalized conditional cumulative residual inaccuracy measure

Abstract

Recently, the notion of cumulative residual Rényi’s entropy has been proposed in the literature as a measure of information that parallels Rényi’s entropy. Motivated by this, here we introduce a generalized measure of it, namely cumulative residual inaccuracy of order α. We study the proposed measure for conditionally specified models of two components having possibly different ages called generalized conditional cumulative residual inaccuracy measure. Several properties of generalized conditional cumulative residual inaccuracy measure including the effect of monotone transformation are investigated. Further, we provide some bounds on using the usual stochastic order and characterize some bivariate distributions using the concept of conditional proportional hazard rate model.     

Extended binomial AR(1) processes with generalized binomial thinning operator

Abstract

In this article, we introduce an extended binomial AR(1) model based on the generalized binomial thinning operator. This operator relaxes the independence assumption of the binomial thinning operator and contains dependent Bernoulli counting series. The new model contains the binomial AR(1) model as a particular case. Some probabilistic and statistical properties are explored. Estimators of the model parameters are derived by conditional maximum likelihood (CML), conditional least squares (CLS) and weighted conditional least squares (WCLS) methods. Some asymptotic properties and numerical results of the estimators are studied. The good performance of the new model is illustrated, among other competitive models in the literature, by an application to the monthly drunken driving counts.     

On mitigating collinearity through mixtures

In linear models having near collinear columns of X, ridge and surrogate estimators often are used to mitigate collinearity. A new class of estimators is based on mixtures, either of X and a design minimal in an ordered class or of the Fisher information and a scalar matrix. Comparisons are drawn among choices for the mixing parameter, and the estimators are found to be admissible relative to ordinary least squares. Case studies demonstrate that selected mixture designs are perturbed from the original design to a lesser extent than are those of the surrogate method, while retaining reasonable efficiency characteristics.     

Confidence Intervals for Conditional Tail Risk Measures in ARMA–GARCH Models

ABSTRACT

ARMA–GARCH models are widely used to model the conditional mean and conditional variance dynamics of returns on risky assets. Empirical results suggest heavy-tailed innovations with positive extreme value index for these models. Hence, one may use extreme value theory to estimate extreme quantiles of residuals. Using weak convergence of the weighted sequential tail empirical process of the residuals, we derive the limiting distribution of extreme conditional Value-at-Risk (CVaR) and conditional expected shortfall (CES) estimates for a wide range of extreme value index estimators. To construct confidence intervals, we propose to use self-normalization. This leads to improved coverage vis-à-vis the normal approximation, while delivering slightly wider confidence intervals. A data-driven choice of the number of upper order statistics in the estimation is suggested and shown to work well in simulations. An application to stock index returns documents the improvements of CVaR and CES forecasts.     

Exponential class of dynamic binary choice panel data models with fixed effects

ABSTRACT

This paper proposes an exponential class of dynamic binary choice panel data models for the analysis of short T (time dimension) large N (cross section dimension) panel data sets that allow for unobserved heterogeneity (fixed effects) to be arbitrarily correlated with the covariates. The paper derives moment conditions that are invariant to the fixed effects which are then used to identify and estimate the parameters of the model. Accordingly, generalized method of moments (GMM) estimators are proposed that are consistent and asymptotically normally distributed at the root-N rate. We also study the conditional likelihood approach and show that under exponential specification, it can identify the effect of state dependence but not the effects of other covariates. Monte Carlo experiments show satisfactory finite sample performance for the proposed estimators and investigate their robustness to misspecification.     

Reliability Characteristics of the Maxwell Distribution: A Bayes Estimation Study

ABSTRACT

This article presents maximum likelihood, Bayes, and empirical Bayes estimators of the truncated first moment and hazard function of the Maxwell distribution. A comparison of the relative efficiency of these three estimators is performed via a Monte Carlo simulation study.     

Estimation of ratio of population means in two-occasion successive sampling

ABSTRACT

This article considers the problem of estimation of the ratio of population means at the current occasion based on the samples selected over two occasions. Using auxiliary information on both the occasions, some classes of estimators have been proposed. Properties of the proposed estimators have been studied and their respective optimum replacement strategies are discussed. The gains in efficiency of the proposed classes of estimators over the direct estimator where no past information is used are computed. Theoretical results have been interpreted through empirical studies which are followed by suitable recommendations.     

A note on nonparametric estimation of circular conditional densities

ABSTRACT

The conditional density offers the most informative summary of the relationship between explanatory and response variables. We need to estimate it in place of the simple conditional mean when its shape is not well-behaved. A motivation for estimating conditional densities, specific to the circular setting, lies in the fact that a natural alternative of it, like quantile regression, could be considered problematic because circular quantiles are not rotationally equivariant. We treat conditional density estimation as a local polynomial fitting problem as proposed by Fan et al. [Estimation of conditional densities and sensitivity measures in nonlinear dynamical systems. Biometrika. 1996;83:189–206] in the Euclidean setting, and discuss a class of estimators in the cases when the conditioning variable is either circular or linear. Asymptotic properties for some members of the proposed class are derived. The effectiveness of the methods for finite sample sizes is illustrated by simulation experiments and an example using real data.     

A new stochastic mixed Liu estimator in linear regression model

Abstract

To overcome multicollinearity, a new stochastic mixed Liu estimator is presented and its efficiency is considered. We also compare the proposed estimators in the sense of matrix mean squared error criteria. Finally a numerical example and a simulation study are given to show the performance of the estimators.