Inference for Box–Cox Transformed Threshold GARCH Models with Nuisance Parameters

Abstract. Generalized autoregressive conditional heteroscedastic (GARCH) models have been widely used for analyzing financial time series with time‐varying volatilities. To overcome the defect of the Gaussian quasi‐maximum likelihood estimator (QMLE) when the innovations follow either heavy‐tailed or skewed distributions, Berkes & Horváth (Ann. Statist., 32, 633, 2004) and Lee & Lee (Scand. J. Statist. 36, 157, 2009) considered likelihood methods that use two‐sided exponential, Cauchy and normal mixture distributions. In this paper, we extend their methods for Box–Cox transformed threshold GARCH model by allowing distributions used in the construction of likelihood functions to include parameters and employing the estimated quasi‐likelihood estimators (QELE) to handle those parameters. We also demonstrate that the proposed QMLE and QELE are consistent and asymptotically normal under regularity conditions. Simulation results are provided for illustration.     

A note on comparison and improvement of estimators based on likelihood

The problem of estimation of a parameter of interest in the presence of a nuisance parameter, which is either location or scale, is considered. Three estimators are taken into account: usual maximum likelihood (ML) estimator, maximum integrated likelihood estimator and the bias-corrected ML estimator. General results on comparison of these estimators w.r.t. the second-order risk based on the mean-squared error are obtained. Possible improvements of basic estimators via the notion of admissibility and methodology given in Ghosh and Sinha [A necessary and sufficient condition for second order admissibility with applications to Berkson's bioassay problem. Ann Stat. 1981;9(6):1334–1338] are considered. In the recent paper by Tanaka et al. [On improved estimation of a gamma shape parameter. Statistics. 2014; doi:10.1080/02331888.2014.915842], this problem was considered for estimating the shape parameter of gamma distribution. Here, we perform more accurate comparison of estimators for this case as well as for some other cases.     

Objective Bayesian Analysis of Skew‐t Distributions

Abstract. We study the Jeffreys prior and its properties for the shape parameter of univariate skew‐t distributions with linear and nonlinear Student's t skewing functions. In both cases, we show that the resulting priors for the shape parameter are symmetric around zero and proper. Moreover, we propose a Student's t approximation of the Jeffreys prior that makes an objective Bayesian analysis easy to perform. We carry out a Monte Carlo simulation study that demonstrates an overall better behaviour of the maximum a posteriori estimator compared with the maximum likelihood estimator. We also compare the frequentist coverage of the credible intervals based on the Jeffreys prior and its approximation and show that they are similar. We further discuss location‐scale models under scale mixtures of skew‐normal distributions and show some conditions for the existence of the posterior distribution and its moments. Finally, we present three numerical examples to illustrate the implications of our results on inference for skew‐t distributions.     

ESTIMATION OF EXPONENTIATED WEIBULL SHAPE PARAMETERS UNDER LINEX LOSS FUNCTION

ABSTRACT

The paper deals with Bayes estimation of the exponentiated Weibull shape parameters under linex loss function when independent non-informative type of priors are available for the parameters. Generalized maximum likelihood estimators have also been obtained. Performances of the proposed Bayes estimator, generalized maximum likelihood estimators, posterior mean (i.e., Bayes estimator under squared error loss function) and maximum likelihood estimators have been studied on the basis of their risks under linex loss function. The comparison is based on a simulation study because the expressions for risk functions of these estimators cannot be obtained in nice closed forms.     

Improved likelihood inference for the shape parameter in Weibull regression

We obtain adjustments to the profile likelihood function in Weibull regression models with and without censoring. Specifically, we consider two different modified profile likelihoods: (i) the one proposed by Cox and Reid [Cox, D.R. and Reid, N., 1987, Parameter orthogonality and approximate conditional inference. Journal of the Royal Statistical Society B, 49, 1–39.], and (ii) an approximation to the one proposed by Barndorff–Nielsen [Barndorff–Nielsen, O.E., 1983, On a formula for the distribution of the maximum likelihood estimator. Biometrika, 70, 343–365.], the approximation having been obtained using the results by Fraser and Reid [Fraser, D.A.S. and Reid, N., 1995, Ancillaries and third-order significance. Utilitas Mathematica, 47, 33–53.] and by Fraser et al. [Fraser, D.A.S., Reid, N. and Wu, J., 1999, A simple formula for tail probabilities for frequentist and Bayesian inference. Biometrika, 86, 655–661.]. We focus on point estimation and likelihood ratio tests on the shape parameter in the class of Weibull regression models. We derive some distributional properties of the different maximum likelihood estimators and likelihood ratio tests. The numerical evidence presented in the paper favors the approximation to Barndorff–Nielsen's adjustment.     

On the calculations of the maximum likelihood estimates for the polynomial spline regression model with unknown knots and AR(1) errors

Asymptotic distributions of the maximum likelihood estimators of the regression coefficients and knot points for the polynomial spline regression models with unknown knots and AR(1) errors have been derived by Chan (1989). Chan showed that under some mild conditions the maximum likelihood estimators, after suitable standardization, asymptotically follow normal distributions as n diverges to infinity. For the calculations of the maximum likelihood estimators, iterative methods must be applied. But this is not easy to implement for the model considered. In this paper, we suggested an alternative method to compute the estimates of the regression parameters and knots. It is shown that the estimates obtained by this method are asymptotically equivalent to the maximum likelihood estimates considered by Chan.     

Maximum likelihood estimator in a multi-phase random regression model

We consider a random regression model with several-fold change-points. The results for one change-point are generalized. The maximum likelihood estimator of the parameters is shown to be consistent, and the asymptotic distribution for the estimators of the coefficients is shown to be Gaussian. The estimators of the change-points converge, with n ^?1 rate, to the vector whose components are the left end points of the maximizing interval with respect to each change-point. The likelihood process is asymptotically equivalent to the sum of independent compound Poisson processes.     

LEVERAGE ADJUSTMENTS FOR DISPERSION MODELLING IN GENERALIZED NONLINEAR MODELS

For normal linear models, it is generally accepted that residual maximum likelihood estimation is appropriate when covariance components require estimation. This paper considers generalized linear models in which both the mean and the dispersion are allowed to depend on unknown parameters and on covariates. For these models there is no closed form equivalent to residual maximum likelihood except in very special cases. Using a modified profile likelihood for the dispersion parameters, an adjusted score vector and adjusted information matrix are found under an asymptotic development that holds as the leverages in the mean model become small. Subsequently, the expectation of the fitted deviances is obtained directly to show that the adjusted score vector is unbiased at least to O(1/n) . Exact results are obtained in the single‐sample case. The results reduce to residual maximum likelihood estimation in the normal linear case.     

Efficient closed-form maximum a posteriori estimators for the gamma distribution

We proposed a new class of maximum a posteriori estimators for the parameters of the Gamma distribution. These estimators have simple closed-form expressions and can be rewritten as a bias-corrected maximum likelihood estimators presented by Ye and Chen [Closed-form estimators for the gamma distribution derived from likelihood equations. Am Statist. 2017;71(2):177–181]. A simulation study was carried out to compare different estimation procedures. Numerical results revels that our new estimation scheme outperforms the existing closed-form estimators and produces extremely efficient estimates for both parameters, even for small sample sizes.     

A new algorithm for maximum likelihood estimation in normal scale-mixture generalized autoregressive conditional heteroskedastic models

In this paper, we propose a new generalized autoregressive conditional heteroskedastic (GARCH) model using infinite normal scale-mixtures which can suitably avoid order selection problems in the application of finite normal scale-mixtures. We discuss its theoretical properties and develop a two-stage algorithm for the maximum likelihood estimator to estimate the mixing distribution non-parametric maximum likelihood estimator (NPMLE) as well as GARCH parameters (two-stage MLE). For the estimation of a mixing distribution, we employ a fast computational algorithm proposed by Wang [On fast computation of the non-parametric maximum likelihood estimate of a mixing distribution. J R Stat Soc Ser B. 2007;69:185–198] under the gradient characterization of the non-parametric mixture likelihood. The GARCH parameters are then estimated either using the expectation-mazimization algorithm or general optimization scheme. In addition, we propose a new forecasting algorithm of value-at-risk (VaR) using the two-stage MLE and the NPMLE. Through a simulation study and real data analysis, we compare the performance of the two-stage MLE with the existing ones including quasi-maximum likelihood estimator based on the standard normal density and the finite normal mixture quasi maximum estimated-likelihood estimator (cf. Lee S, Lee T. Inference for Box–Cox transformed threshold GARCH models with nuisance parameters. Scand J Stat. 2012;39:568–589) in terms of the relative efficiency and accuracy of VaR forecasting.     

Statistical Inference of Type-II Progressively Hybrid Censored Data with Weibull Lifetimes

In this article, we discuss the maximum likelihood estimators and approximate maximum likelihood estimators of the parameters of the Weibull distribution with two different progressively hybrid censoring schemes. We also present the associated expressions of the expected total test time and the expected effective sample size which will be useful for experimental planning purpose. Finally, the efficiency of the point estimation of the parameters based on the two progressive hybrid censoring schemes are compared and the merits of each censoring scheme are discussed.     

Improved two-parameter estimators for the negative binomial and Poisson regression models

Negative binomial regression (NBR) and Poisson regression (PR) applications have become very popular in the analysis of count data in recent years. However, if there is a high degree of relationship between the independent variables, the problem of multicollinearity arises in these models. We introduce new two-parameter estimators (TPEs) for the NBR and the PR models by unifying the two-parameter estimator (TPE) of Özkale and Kaç?ranlar [The restricted and unrestricted two-parameter estimators. Commun Stat Theory Methods. 2007;36:2707–2725]. These new estimators are general estimators which include maximum likelihood (ML) estimator, ridge estimator (RE), Liu estimator (LE) and contraction estimator (CE) as special cases. Furthermore, biasing parameters of these estimators are given and a Monte Carlo simulation is done to evaluate the performance of these estimators using mean square error (MSE) criterion. The benefits of the new TPEs are also illustrated in an empirical application. The results show that the new proposed TPEs for the NBR and the PR models are better than the ML estimator, the RE and the LE.     

A note on the randomized response technique

It is pointed out that the usual estimators for the parameters of a randomized response model are not, contrary to popular belief, maximum likelihood estimators.     

Estimation of the parameters of a truncated gamma distribution

This paper deals with the estimation of the parameters of a truncated gamma distribution over (0,τ), where τ is assumed to be a real number. We obtain a necessary and sufficient condition for the existence of the maximum likelihood estimator(MLE). The probability of nonexistence of MLE is observed to be positive. A simulation study indicates that the modified maximum likelihood estimator and the mixed estimator, which exist with probability one,are to be preferred over MLE. The bias, the mean square error, and the probability of nearness form a basis of our simulation study.     

Estimating the parameters of a doubly truncated normal distribution

This paper deals with the maximum likelihood estimation of parameters for a doubly truncated normal distribution when the truncation points are known. We prove, in this case, that the MLEs are nonexistent (become infinite) with positive probability. For estimators that exist with probability one, the class of Bayes modal estimators or modified maximum likelihood estimators is explored. Another useful estimating procedure, called mixed estimation, is proposed. Simulations compare the behavior of the MLEs, the modified MLEs, and the mixed estimators which reveal that the MLE, in addition to being nonexistent with positive probability, behaves poorly near the upper boundary of the interval of its existence. The modified MLEs and the mixed estimators are seen to be remarkably better than the MLE     

Variable selection in joint mean and variance models of Box–Cox transformation

In many applications, a single Box–Cox transformation cannot necessarily produce the normality, constancy of variance and linearity of systematic effects. In this paper, by establishing a heterogeneous linear regression model for the Box–Cox transformed response, we propose a hybrid strategy, in which variable selection is employed to reduce the dimension of the explanatory variables in joint mean and variance models, and Box–Cox transformation is made to remedy the response. We propose a unified procedure which can simultaneously select significant variables in the joint mean and variance models of Box–Cox transformation which provide a useful extension of the ordinary normal linear regression models. With appropriate choice of the tuning parameters, we establish the consistency of this procedure and the oracle property of the obtained estimators. Moreover, we also consider the maximum profile likelihood estimator of the Box–Cox transformation parameter. Simulation studies and a real example are used to illustrate the application of the proposed methods.     

Estimating a positive normal mean

Let X ₁, X ₂, ..., X _n be a random sample from a normal population with mean μ and variance σ ². In many real life situations, specially in lifetime or reliability estimation, the parameter μ is known a priori to lie in an interval [a, ∞). This makes the usual maximum likelihood estimator (MLE) ̄ an inadmissible estimator of μ with respect to the squared error loss. This is due to the fact that it may take values outside the parameter space. Katz (1961) and Gupta and Rohatgi (1980) proposed estimators which lie completely in the given interval. In this paper we derive some new estimators for μ and present a comparative study of the risk performance of these estimators. Both the known and unknown variance cases have been explored. The new estimators are shown to have superior risk performance over the existing ones over large portions of the parameter space.     

Generalised quasi‐likelihood inference in a semi‐parametric binary dynamic mixed logit model

There exists a recent study where dynamic mixed‐effects regression models for count data have been extended to a semi‐parametric context. However, when one deals with other discrete data such as binary responses, the results based on count data models are not directly applicable. In this paper, we therefore begin with existing binary dynamic mixed models and generalise them to the semi‐parametric context. For inference, we use a new semi‐parametric conditional quasi‐likelihood (SCQL) approach for the estimation of the non‐parametric function involved in the semi‐parametric model, and a semi‐parametric generalised quasi‐likelihood (SGQL) approach for the estimation of the main regression, dynamic dependence and random effects variance parameters. A semi‐parametric maximum likelihood (SML) approach is also used as a comparison to the SGQL approach. The properties of the estimators are examined both asymptotically and empirically. More specifically, the consistency of the estimators is established and finite sample performances of the estimators are examined through an intensive simulation study.     

Maximum likelihood estimation in convex hull models

In this paper we consider models involving the convex hull operation of the parameter and the noise i.e. Y_i = CH(A, X_X). Then we generalize the basic models to ANOVA models; i.e. Y_ij=CH(A∪B_j,X_ij). In some cases the consistent estimators for the J U new parameters are derived. Assuming the existence of density forrandom convex sets, we derive the likelihood for the convex hull model. We then find the maximum Likelihood Estimators for the parameters. Examples for some random convex sets with finite dimensional distributions are derived to show how good these estimators are.     

The role of reversals in order‐restricted inference

A statistical model is said to be an order‐restricted statistical model when its parameter takes its values in a closed convex cone C of the Euclidean space. In recent years, order‐restricted likelihood ratio tests and maximum likelihood estimators have been criticized on the grounds that they may violate a cone order monotonicity (COM) property, and hence reverse the cone order induced by C. The authors argue here that these reversals occur only in the case that C is an obtuse cone, and that in this case COM is an inappropriate requirement for likelihood‐based estimates and tests. They conclude that these procedures thus remain perfectly reasonable procedures for order‐restricted inference.