Asymptotic Covariance Matrix Research Articles

Inference in a linear functional relationship with replications

In this paper, we consider a model for data analysis with measurement errors. The main objective of this work is to develop statistical inference tools, such as parameter estimation and hypothesis tests in a linear functional relationship with replicated observations. For this purpose, we use the maximum likelihood method in the presence of incidental parameters, and the unbiased estimating equations approach. Both approaches lead to explicit expressions for the asymptotic covariance matrices of the estimators of the model parameters. A simulation study is performed to assess the empirical behavior of estimators and of a Wald statistic. The methodology is illustrated with a real data set.

General Marginal Regression Models for the Joint Modeling of Event Frequency and Correlated Severities with Applications to Clinical Trials

In many clinical trials, information is collected on both the frequency of event occurrence and the severity of each event. For example, in evaluating a new anti-epileptic medication both the total number of seizures a patient has during the study period as well as the severity (e.g., mild, severe) of each seizure could be measured. In order to arrive at a full picture of drug or treatment performance, one needs to jointly model the number of events and their correlated ordinal severity measures. A separate analysis is not recommended as it is inefficient and can lead to what we define as “zero length bias” in estimates of treatment effect on severity. This paper proposes a general, likelihood based, marginal regression model for jointly modeling the number of events and their correlated ordinal severity measures. We describe parameter estimation issues and derive the Fisher information matrix for the joint model in order to obtain the asymptotic covariance matrix of the parameter estimates. A limited simulation study is conducted to examine the asymptotic properties of the maximum likelihood estimators. Using this joint model, we propose tests that incorporate information from both the number of events and their correlated ordinal severity measures. The methodology is illustrated with two examples from clinical trials: the first concerning a new drug treatment for epilepsy; the second evaluating the effect of a cholesterol lowering medication on coronary artery disease.

PolychoricRM: A computationally efficient R function for estimating polychoric correlations and their asymptotic covariance matrix

ABSTRACT Many applications of structural equation modeling involve ordinal (e.g., Likert) variables. A popular way of dealing with ordinal variables is to estimate the model with polychoric correlations rather than Pearson correlations. Such an estimation also requires the asymptotic covariance matrix of polychoric correlations. It is computationally intensive to estimate polychoric correlations and their asymptotic covariance matrices. We describe a computationally efficient R function PolychoricRM to estimate polychoric correlations and their asymptotic covariance matrix. The function invokes the computing power of modern Fortran and exploits multiple-core (multiple-thread) CPUs on nearly all current computers.

Finite mixture modeling of censored and missing data using the multivariate skew-normal distribution

Finite mixture models have been widely used to model and analyze data from a heterogeneous populations. Moreover, data of this kind can be missing or subject to some upper and/or lower detection limits because of the constraints of experimental apparatuses. Another complication arises when measures of each population depart significantly from normality, such as asymmetric behavior. For such data structures, we propose a robust model for censored and/or missing data based on finite mixtures of multivariate skew-normal distributions. This approach allows us to model data with great flexibility, accommodating multimodality and skewness, simultaneously, depending on the structure of the mixture components. We develop an analytically simple, yet efficient, EM-type algorithm for conducting maximum likelihood estimation of the parameters. The algorithm has closed-form expressions at the E-step that rely on formulas for the mean and variance of the truncated multivariate skew-normal distributions. Furthermore, a general information-based method for approximating the asymptotic covariance matrix of the estimators is also presented. Results obtained from the analysis of both simulated and real datasets are reported to demonstrate the effectiveness of the proposed method. The proposed algorithm and method are implemented in the new R package CensMFM.

Robust estimation for Binomial conditionally nonlinear autoregressive time series based on multivariate conditional frequencies

A topical problem of robust statistical estimation of parameters for binomial conditionally nonlinear autoregressive (BiCNAR) time series under innovation outliers is considered. This problem is solved by means of s-order Markov properties for observed time series and probabilistic properties of multivariate conditional frequencies of the future state under its s-prehistory. The new robust statistical estimator ζˆ called frequencies-based estimator (FBE) is constructed for the BiCNAR parameters under innovation outliers with arbitrary discrete probability distribution having some fixed known expectation. Under mild regularity conditions the constructed FBE is shown to have the robustness properties: consistency and asymptotic normality with obtained asymptotic covariance matrix. FBE also has computational advantages: an explicit form and a fast recursive re-estimation algorithm for extension of the model. Asymptotic risk functional and its minimum are evaluated using Fisher information matrix for the considered model. Sensitivity analysis of the statistical estimator ζ̃ for the BiCNAR parameters, that is constructed for the hypothetical model without outliers, is carried out for the situation with innovation outliers: ζ̃ is shown to be inconsistent in this situation, its bias and the instability coefficient are evaluated and analyzed. The robust estimator ζˆ has a free parameter — weight matrix H. The optimal weight matrix H∗ is found by minimization of the asymptotic risk w.r.t. H. Statistical estimator for H∗ based on the observed time series is constructed. Results of multiple computer experiments on simulated and real data illustrate the theory.

Inference on Nadarajah–Haghighi distribution with constant stress partially accelerated life tests under progressive type-II censoring

This paper presents methods of estimation of the parameters and acceleration factor for Nadarajah–Haghighi distribution based on constant-stress partially accelerated life tests. Based on progressive Type-II censoring, Maximum likelihood and Bayes estimates of the model parameters and acceleration factor are established, respectively. In addition, approximate confidence interval are constructed via asymptotic variance and covariance matrix, and Bayesian credible intervals are obtained based on importance sampling procedure. For comparison purpose, alternative bootstrap confidence intervals for unknown parameters and acceleration factor are also presented. Finally, extensive simulation studies are conducted for investigating the performance of the our results, and two data sets are analyzed to show the applicabilities of the proposed methods.

Estimating FARIMA models with uncorrelated but non-independent error terms

In this paper we derive the asymptotic properties of the least squares estimator (LSE) of fractionally integrated autoregressive moving-average (FARIMA) models under the assumption that the errors are uncorrelated but not necessarily independent nor martingale differences. We relax the independence and even the martingale difference assumptions on the innovation process to extend considerably the range of application of the FARIMA models. We propose a consistent estimator of the asymptotic covariance matrix of the LSE which may be very different from that obtained in the standard framework. A self-normalized approach to confidence interval construction for weak FARIMA model parameters is also presented. All our results are done under a mixing assumption on the noise. Finally, some simulation studies and an application to the daily returns of stock market indices are presented to corroborate our theoretical work.

Backtesting Value-at-Risk and Expected Shortfall in the Presence of Estimation Error

AbstractWe investigate the effect of estimation error on backtests of expected shortfall (ES) forecasts. These backtests are based on first-order conditions of a recently introduced family of jointly consistent loss functions for value-at-risk (VaR) and ES. For both single and multiperiod horizons, we provide explicit expressions for the additional terms in the asymptotic covariance matrix that result from estimation error, and propose robust tests that account for it. Monte Carlo experiments show that the tests that ignore these terms suffer from size distortions, which are more pronounced for higher ratios of out-of-sample to in-sample observations. Robust versions of the backtests perform well with power against common alternatives. We also introduce a novel standardization of the conditional joint test statistic that removes the need to estimate higher-order moments and significantly improves its performance. In an application to VaR and ES forecasts for daily FTSE 100 index returns as generated by (GJR-)GARCH and HEAVY models, we find that estimation error substantially impacts the outcome of the backtests, and is not bound to particular subperiods such as the credit crisis.

Accurate Confidence Regions for Principal Components Factors*

AbstractIn dynamic factor models, factors are often extracted using principal components with their asymptotic confidence regions having empirical coverages below the nominal ones when the temporal dimension is small. We propose a subsampling procedure to compute the factor loadings uncertainty and correct the asymptotic covariance matrix of the extracted factors. We show that the empirical coverages of the modified confidence regions are closer to the nominal ones than those of asymptotic regions and asymptotically valid bootstrap regions. The results are empirically illustrated obtaining confidence intervals of the underlying factor in a system of Spanish macroeconomic variables.

Weighted rank estimation of nonparametric transformation models with case-1 and case-2 interval-censored failure time data

Case-1 and case-2 interval-censored failure time data commonly occur in medical research as well as other fields and many methods have been developed for their analysis under different frameworks. In this paper, we consider regression analysis of such data and present a general class of nonparametric transformation models. One major advantage of these models is their flexibility and generality as they include the linear transformation model as a special case. For estimation of regression parameters, we propose a weighted rank (WR) estimation procedure and establish the consistency and asymptotic normality of the resulting estimator. Furthermore, to estimate the asymptotic covariance matrix of the proposed estimator, a resampling technique, which does not involve nonparametric density estimation or numerical derivatives, is developed. A numerical study is also conducted and suggests that the proposed methodology works well in practice. Finally an application is provided.

Semiparametric Multinomial Logistic Regression for Multivariate Point Pattern Data

We propose a new method for analysis of multivariate point pattern data observed in a heterogeneous environment and with complex intensity functions. We suggest semiparametric models for the intensity functions that depend on an unspecified factor common to all types of points. This is for example well suited for analyzing spatial covariate effects on events such as street crime activities that occur in a complex urban environment. A multinomial conditional composite likelihood function is introduced for estimation of intensity function regression parameters and the asymptotic joint distribution of the resulting estimators is derived under mild conditions. Crucially, the asymptotic covariance matrix depends on ratios of cross pair correlation functions of the multivariate point process. To make valid statistical inference without restrictive assumptions, we construct consistent nonparametric estimators for these ratios. Finally, we construct standardized residual plots, predictive probability plots, and semiparametric intensity plots to validate and to visualize the findings of the model. The effectiveness of the proposed methodology is demonstrated through extensive simulation studies and an application to analyzing the effects of socio-economic and demographical variables on occurrences of street crimes in Washington DC. Supplementary materials for this article are available online.

Statistical inference for mixture GARCH models with financial application

In this paper we consider mixture generalized autoregressive conditional heteroskedastic models, and propose a new iteration algorithm of type EM for the estimation of model parameters. The maximum likelihood estimates are shown to be consistent, and their asymptotic properties are investigated. More precisely, we derive simple expressions in closed form for the asymptotic covariance matrix and the expected Fisher information matrix of the ML estimator. Finally, we study the model selection and propose testing procedures. A simulation study and an application to financial real-series illustrate the results.

On the classical estimation of bivariate copula-based Seemingly unrelated tobit models through the proposed inference function for augmented margins method

This paper extends the analysis of the bivariate Seemingly Unrelated (SUR) Tobit by modeling its nonlinear dependence structure through copula and assuming non-normal marginal error distributions. For model estimation, the use of copula methods enables the use of the (classical) Inference Function for Margins (IFM) method by Joe and Xu (1996), which is more computationally attractive (feasible) than the full maximum likelihood approach. However, our simulation study shows that the IFM method provides a biased estimate of the copula parameter in the presence of censored observations in both margins. In order to obtain an unbiased estimate of the copula association parameter, we propose/develop a modified version of the IFM method, which we refer to as Inference Function for Augmented Margins (IFAM). Since the usual asymptotic approach, that is the computation of the asymptotic covariance matrix of the parameter estimates, is troublesome, we propose the use of resampling procedures (bootstrap methods) to obtain confidence intervals for the copula-based SUR Tobit model parameters. The satisfactory results from the simulation and empirical studies indicate the adequate performance of our proposed model and methods. We illustrate our procedure using bivariate data on consumption of salad dressings and lettuce by U.S. individuals.

On a new generalized lindley distribution: Properties, estimation and applications.

In this study, an extension of the generalized Lindley distribution using the Marshall-Olkin method and its own sub-models is presented. This new model for modelling survival and lifetime data is flexible. Several statistical properties and characterizations of the subject distribution along with its reliability analysis are presented. Statistical inference for the new family such as the Maximum likelihood estimators and the asymptotic variance covariance matrix of the unknown parameters are discussed. A simulation study is considered to compare the efficiency of the different estimators based on mean square error criterion. Finally, a real data set is analyzed to show the flexibility of our proposed model compared with the fit attained by some other competitive distributions.

INSTRUMENTAL VARIABLE ESTIMATION OF STRUCTURAL VAR MODELS ROBUST TO POSSIBLE NONSTATIONARITY

This paper considers the estimation of dynamic causal effects using a proxy structural vector-autoregressive model with possibly nonstationary regressors. We provide general conditions under which the asymptotic normal approximation remains valid. In this case, the asymptotic variance depends on the persistence property of each series. We further provide a consistent asymptotic covariance matrix estimator that requires neither knowledge of the presistence properties of the variables nor pretests for nonstationarity. The proposed consistent covariance matrix estimator is robust and is easy to implement in practice. When all regressors are indeed stationary, the method becomes the same as the standard procedure.

A general expression for second-order covariance matrices—an application to dispersion models

We present a general expression that allows the calculation of both the $n^{-2}$ asymptotic covariance matrices of the maximum likelihood estimator (MLE) and the first-order bias corrected MLE, where $n$ is the sample size. The formula is presented in a matrix notation which has numerical advantages since it requires only simple operations on matrices and vectors. The usefulness of the formula is to construct better Wald statistics. We apply our findings to dispersion models and develop simulation studies which show that modification in the Wald statistic effectively removes size distortions of the type I error probability with no power loss. For illustrative purposes, a real data application is considered to support our theoretical results.

Specification tests for non‐Gaussian maximum likelihood estimators

We propose generalized DWH specification tests which simultaneously compare three or more likelihood‐based estimators in multivariate conditionally heteroskedastic dynamic regression models. Our tests are useful forGarchmodels and in many empirically relevant macro and finance applications involvingVars and multivariate regressions. We determine the rank of the differences between the estimators' asymptotic covariance matrices under correct specification, and take into account that some parameters remain consistently estimated under distributional misspecification. We provide finite sample results through Monte Carlo simulations. Finally, we analyze a structuralVarproposed to capture the relationship between macroeconomic and financial uncertainty and the business cycle.

Optimal Allocation of Excitation and Measurement for Identification of Dynamic Networks

In this paper, the problem of choosing the best allocation of excitations and measurements for the identification of a dynamic network is formally stated and analyzed. The best choice will be one that achieves the most accurate identification with the least costly experiment. Accuracy is assessed by the trace of the asymptotic covariance matrix of the parameters estimates, whereas the cost criterion is the number of excitations and measurements. Analytical and numerical results are presented for two classes of dynamic networks in state space form: branches and cycles. From these results, a number of guidelines for the choice emerge, which are based either on the topology of the network or on the relative magnitude of the modules being identified. An example is given to illustrate that these guidelines can to some extent be applied to networks of more generic topology.

On error analysis of a closed-loop subspace model identification method

This paper studies error analysis and asymptotic variance of a closed-loop subspace model identification method for a system described with the output-error state-space representation. For details, since the procedure of the identification method includes the QR factorization of stacked data Hankel matrices, this study investigates asymptotic properties of block elements of the triangular matrix obtained from the QR factorization. The set of the block elements is separated into two components, namely, the signal-based component and the noise-based component. The contributions are to derive asymptotic properties of both components and to obtain the asymptotic covariance matrix of the vectorization of the noise-based component.

Linear Regression with Stationary Errors: the R Package slm

This paper introduces the R package slm which stands for Stationary Linear Models. The package contains a set of statistical procedures for linear regression in the general context where the error process is strictly stationary with short memory. We work in the setting of Hannan (1973), who proved the asymptotic normality of the (normalized) least squares estimators (LSE) under very mild conditions on the error process. We propose different ways to estimate the asymptotic covariance matrix of the LSE, and then to correct the type I error rates of the usual tests on the parameters (as well as confidence intervals). The procedures are evaluated through different sets of simulations, and two examples of real datasets are studied.