On double hysteretic heteroskedastic model

ABSTRACT

This paper proposes a hysteretic autoregressive model with GARCH specification and a skew Student's t-error distribution for financial time series. With an integrated hysteresis zone, this model allows both the conditional mean and conditional volatility switching in a regime to be delayed when the hysteresis variable lies in a hysteresis zone. We perform Bayesian estimation via an adaptive Markov Chain Monte Carlo sampling scheme. The proposed Bayesian method allows simultaneous inferences for all unknown parameters, including threshold values and a delay parameter. To implement model selection, we propose a numerical approximation of the marginal likelihoods to posterior odds. The proposed methodology is illustrated using simulation studies and two major Asia stock basis series. We conduct a model comparison for variant hysteresis and threshold GARCH models based on the posterior odds ratios, finding strong evidence of the hysteretic effect and some asymmetric heavy-tailness. Versus multi-regime threshold GARCH models, this new collection of models is more suitable to describe real data sets. Finally, we employ Bayesian forecasting methods in a Value-at-Risk study of the return series.     

Order-preserving Estimators and an Inequality on the Integration of Zonal Polynomial

ABSTRACT

Suppose X , p × p p.d. random matrix, has the distribution which depends on a p × p p.d. parameter matrix Σ and this distribution is orthogonally invariant. The orthogonally invariant estimator of Σ which has the eigenvalues of the same order as the eigenvalues of X is called order-preserving. We conjecture that a non-order-preserving estimator is dominated by modified order-preserving estimators with respect to the entropy (Stein's) loss function. We show that an inequality on the integration of zonal polynomial is sufficient for this conjecture. We also prove this inequality for the case p = 2.     

Testing for parameter constancy in the time series direction in panel data models

We propose tests for parameter constancy in the time series direction in panel data models. We construct a locally best invariant test based on Tanaka [Time series analysis: nonstationary and noninvertible distribution theory. New York: Wiley; 1996] and an asymptotically point optimal test based on Elliott and Müller [Efficient tests for general persistent time variation in regression coefficients. Rev Econ Stud. 2006;73:907–940]. We derive the limiting distributions of the test statistics as T→∞ while N is fixed, and calculate the critical values by applying numerical integration and response surface regression. Simulation results show that the proposed tests perform well if we apply them appropriately.     

Laplace approximations and Bayesian information criteria in possibly misspecified models

We provide general conditions to ensure the valid Laplace approximations to the marginal likelihoods under model misspecification, and derive the Bayesian information criteria including all terms of order O_p(1). Under conditions in theorem 1 of Lv and Liu [J. R. Statist. Soc. B, 76, (2014), 141–167] and a continuity condition for prior densities, asymptotic expansions with error terms of order o_p(1) are derived for the log-marginal likelihoods of possibly misspecified generalized linear models. We present some numerical examples to illustrate the finite sample performance of the proposed information criteria in misspecified models.     

Characterization of discrete scale invariant Markov sequences

ABSTRACT

Some special sampling of discrete scale invariant (DSI) processes are presented to provide a multi-dimensional self-similar process in correspondence. By imposing Markov property we show that the covariance functions of such Markov DSI sequences are characterized by variance, and covariance of adjacent samples in the first scale interval. We also provide a theoretical method for estimating spectral density matrix of corresponding multi-dimensional self-similar Markov process. Some examples such as simple Brownian motion (sBm) with drift and scale invariant autoregressive model are presented and these properties are investigated. We present two new method to estimate Hurst parameter of DSI processes and apply them to some sBm and also to the SP500 indices for some period which has DSI property. We compare our estimates with the maximum-likelihood and rescaled range (R/S) method which are applied to the corresponding multi-dimensional self-similar processes.     

Bayesian estimation for time-series regressions improved with exact likelihoods

We propose an estimation procedure for time-series regression models under the Bayesian inference framework. With the exact method of Wise [Wise, J. (1955). The autocorrelation function and spectral density function. Biometrika, 42, 151–159], an exact likelihood function can be obtained instead of the likelihood conditional on initial observations. The constraints on the parameter space arising from the stationarity conditions are handled by a reparametrization, which was not taken into consideration by Chib [Chib, S. (1993). Bayes regression with autoregressive errors: A Gibbs sampling approach. J. Econometrics, 58, 275–294] or Chib and Greenberg [Chib, S. and Greenberg, E. (1994). Bayes inference in regression model with ARMA(p, q) errors. J. Econometrics, 64, 183–206]. Simulation studies show that our method leads to better inferential results than their results.     

Characterizing Relationships Between Estimations Under a General Linear Model with Explicit and Implicit Restrictions by Rank of Matrix

In the investigation of the restricted linear model ? _r  = {y, X β | A β = b, σ² Σ}, the parameter constraints A β = b are often handled by transforming the model into certain implicitly restricted model. Any estimation derived from the explicitly and implicitly restricted models on the vector β and its functions should be equivalent, although the expressions of the estimation under the two models may be different. However, people more likely want to directly compare different expressions of estimations and yield a conclusion on their equivalence by using some algebraic operations on expressions of estimations. In this article, we give some results on equivalence of the well-known OLSEs and BLUEs under the explicitly and implicitly restricted linear models by using some expansion formulas for ranks of matrices.     

$$D_s$$-optimality in copula models

Optimum experimental design theory has recently been extended for parameter estimation in copula models. The use of these models allows one to gain in flexibility by considering the model parameter set split into marginal and dependence parameters. However, this separation also leads to the natural issue of estimating only a subset of all model parameters. In this work, we treat this problem with the application of the \(D_s\)-optimality to copula models. First, we provide an extension of the corresponding equivalence theory. Then, we analyze a wide range of flexible copula models to highlight the usefulness of \(D_s\)-optimality in many possible scenarios. Finally, we discuss how the usage of the introduced design criterion also relates to the more general issue of copula selection and optimal design for model discrimination.     

Remedying the Neyman–Scott phenomenon in model discrimination

The objective of this paper is to investigate through simulation the possible presence of the incidental parameters problem when performing frequentist model discrimination with stratified data. In this context, model discrimination amounts to considering a structural parameter taking values in a finite space, with k points, k≥2. This setting seems to have not yet been considered in the literature about the Neyman–Scott phenomenon. Here we provide Monte Carlo evidence of the severity of the incidental parameters problem also in the model discrimination setting and propose a remedy for a special class of models. In particular, we focus on models that are scale families in each stratum. We consider traditional model selection procedures, such as the Akaike and Takeuchi information criteria, together with the best frequentist selection procedure based on maximization of the marginal likelihood induced by the maximal invariant, or of its Laplace approximation. Results of two Monte Carlo experiments indicate that when the sample size in each stratum is fixed and the number of strata increases, correct selection probabilities for traditional model selection criteria may approach zero, unlike what happens for model discrimination based on exact or approximate marginal likelihoods. Finally, two examples with real data sets are given.     

A Statistical Method for the Determination of the Appropriate Order in a General Class of Time Series Models

Abstract

We propose a method to determine the order q of a model in a general class of time series models. For the subset of linear moving average models (MA(q)), our method is compared with that of the sample autocorrelations. Since the sample autocorrelation is meant to detect a linear structure of dependence between random variables, it turns out to be more suitable for the linear case. However, our method presents a competitive option in that case, and for nonlinear models (NLMA(q)) it is shown to work better. The main advantages of our approach are that it does not make assumptions on the existence of moments and on the distribution of the noise involved in the moving average models. We also include an example with real data corresponding to the daily returns of the exchange rate process of mexican pesos and american dollars.     

Testing for two states in a hidden Markov model

The authors consider hidden Markov models (HMMs) whose latent process has m ≥ 2 states and whose state‐dependent distributions arise from a general one‐parameter family. They propose a test of the hypothesis m = 2. Their procedure is an extension to HMMs of the modified likelihood ratio statistic proposed by Chen, Chen & Kalbfleisch (2004) for testing two states in a finite mixture. The authors determine the asymptotic distribution of their test under the hypothesis m = 2 and investigate its finite‐sample properties in a simulation study. Their test is based on inference for the marginal mixture distribution of the HMM. In order to illustrate the additional difficulties due to the dependence structure of the HMM, they show how to test general regular hypotheses on the marginal mixture of HMMs via a quasi‐modified likelihood ratio. They also discuss two applications.     

Bayesian estimation and prediction for the inverse weibull distribution under general progressive censoring

Abstract

This paper deals with Bayesian estimation and prediction for the inverse Weibull distribution with shape parameter α and scale parameter λ under general progressive censoring. We prove that the posterior conditional density functions of α and λ are both log-concave based on the assumption that λ has a gamma prior distribution and α follows a prior distribution with log-concave density. Then, we present the Gibbs sampling strategy to estimate under squared-error loss any function of the unknown parameter vector (α, λ) and find credible intervals, as well as to obtain prediction intervals for future order statistics. Monte Carlo simulations are given to compare the performance of Bayesian estimators derived via Gibbs sampling with the corresponding maximum likelihood estimators, and a real data analysis is discussed in order to illustrate the proposed procedure. Finally, we extend the developed methodology to other two-parameter distributions, including the Weibull, Burr type XII, and flexible Weibull distributions, and also to general progressive hybrid censoring.     

Methods for calculating stationary distribution in linear models of time series

The article deals with methods for computing the stationary marginal distribution in linear models of time series. Two approaches are described. First, an algorithm based on approximation of solution of the corresponding integral equation is briefly reviewed. Then, we study the limit behaviour of the partial sums c ₁ η₁+c ₂ η₂+···+c _n η _n where η _i are i.i.d. random variables and c _i real constants. We generalize procedure of Haiman (1998) [Haiman, G., 1998, Upper and lower bounds for the tail of the invariant distribution of some AR(1) processes. Asymptotic Methods in Probability and Statistics, 45, 723–730.] to an arbitrary causal linear process and relax the assumptions of his result significantly. This is achieved by investigating the properties of convolution of densities.     

Best quadratic predictions in finite populations

For a finite population and the resulting linear model Y=Xβ+e, the problem of the optimal invariant quadratic predictors including optimal invariant quadratic unbiased predictor and optimal invariant quadratic (potentially) biased predictor for the population quadratic quantities, f(H)=Y′HY , is of interest and has been previously considered in the literature for the case of HX=0. However, the special case does not contain all of situations at all. So, predicting f(H) in general situations may be of particular interest. In this paper, we make an effort to investigate how to offer a good predictor for f(H), not restricted yet to the mentioned case. Permutation matrix techniques play an important role in handling the process. The expected predictors are finally derived. In addition, we mention that the resulting predictors can be viewed as acceptable in all situations.     

Reliability analysis for stress-strength model from a general family of truncated distributions under censored data

Abstract

Under progressive Type-II censoring, inference of stress-strength reliability (SSR) is studied for a general family of lower truncated distributions. When the lifetime models of the strength and stress variables have arbitrary and common parameters, maximum likelihood and pivotal quantities based generalized estimators of SSR are established, respectively. Confidence intervals are constructed based on generalized pivotal quantities and bootstrap technique under different parameter cases as well. In addition, to compare the equivalence of the strength and stress parameters, likelihood ratio testing of interested parameters is provided as a complementary. Simulation studies and two real-life data examples are provided to investigate the performance of proposed methods.     

Estimating additive models with missing responses

ABSTRACT

For multivariate regressors, the Nadaraya–Watson regression estimator suffers from the well-known curse of dimensionality. Additive models overcome this drawback. To estimate the additive components, it is usually assumed that we observe all the data. However, in many applied statistical analysis missing data occur. In this paper, we study the effect of missing responses on the additive components estimation. The estimators are based on marginal integration adapted to the missing situation. The proposed estimators turn out to be consistent under mild assumptions. A simulation study allows to compare the behavior of our procedures, under different scenarios.     

Sampling hyperparameters in hierarchical models: Improving on Gibbs for high-dimensional latent fields and large datasets

ABSTRACT

A common Bayesian hierarchical model is where high-dimensional observed data depend on high-dimensional latent variables that, in turn, depend on relatively few hyperparameters. When the full conditional distribution over latent variables has a known form, general MCMC sampling need only be performed on the low-dimensional marginal posterior distribution over hyperparameters. This improves on popular Gibbs sampling that computes over the full space. Sampling the marginal posterior over hyperparameters exhibits good scaling of compute cost with data size, particularly when that distribution depends on a low-dimensional sufficient statistic.     

Construction of optimal designs in random coefficient regression models

We consider the problem of optimal experimental design in random coefficient regression models with respect to a quadratic loss function. By application of WHITTLE'S general equivalence theorem we obtain the structure of optimal designs. An alogrithm is given which allows, under certain assumptions, the construction of the information matrix of an optimal design. Moreover, we give conditions on the equivalence of optimal designs with respect to optimality criteria which are analogous to usual A-D- and _E/-optimality.     

Improved estimation of scale parameters of morgenstern type bivariate uniform distribution using ranked set sampling

ABSTRACT

In this article we suggest some improved version of estimators of scale parameter of Morgenstern-type bivariate uniform distribution (MTBUD) based on the observations made on the units of the ranked set sampling regarding the study variable Y which is correlated with the auxiliary variable X, when (X, Y) follows a MTBUD. We also suggest some linear shrinkage estimators of scale parameter of Morgenstern type bivariate uniform distribution (MTBUD). Efficiency comparisons are also made in this work.     

The General Expressions for the Moments of the Stochastic Shrinkage Parameters of the Liu Type Estimator

ABSTRACT

One of the problems with the Liu estimator is the appropriate value for the unknown biasing parameter d. In this article we consider the optimum value for d and give upper bound for the expected value of the estimator of this biasing parameter. We also derive the general expressions for the moments of the stochastic shrinkage parameters of the Liu estimator and the generalized Liu estimator. Numerical calculations are carried out to illustrate the behavior of the mean and variance of the biasing parameter. Also, a numerical example is given to illustrate the effect of the biasing parameter d, on the mean square error of the Liu estimator.