Bayesian estimation based on ranked set sample from Morgenstern type bivariate exponential distribution when ranking is imperfect

In this paper we consider Bayes estimation based on ranked set sample when ranking is imperfect, in which units are ranked based on measurements made on an easily and exactly measurable auxiliary variable X which is correlated with the study variable Y. Bayes estimators under squared error loss function and LINEX loss function for the mean of the study variate Y, when (X, Y) follows a Morgenstern type bivariate exponential distribution, are obtained based on both usual ranked set sample and extreme ranked set sample. Estimation procedures developed in this paper are illustrated using simulation studies and a real data.     

Quasi-maximum likelihood estimation for conditional quantiles

In this paper, we construct a new class of estimators for conditional quantiles in possibly misspecified nonlinear models with time series data. Proposed estimators belong to the family of quasi-maximum likelihood estimators (QMLEs) and are based on a new family of densities which we call ‘tick-exponential’. A well-known member of the tick-exponential family is the asymmetric Laplace density, and the corresponding QMLE reduces to the Koenker and Bassett's (Econometrica 46 (1978) 33) nonlinear quantile regression estimator. We derive primitive conditions under which the tick-exponential QMLEs are consistent and asymptotically normally distributed with an asymptotic covariance matrix that accounts for possible conditional quantile model misspecification and which can be consistently estimated by using the tick-exponential scores and Hessian matrix. Despite its non-differentiability, the tick-exponential quasi-likelihood is easy to maximize by using a ‘minimax’ representation not seen in the earlier work on conditional quantile estimation.     

The sample estimation

Statistical research, as a rule, is based on the sample. Therefore, it is important to evaluate the quality of the sample studied. Our sample evaluation is based on a new interpretation of such concepts as a random factor, disturbance factor, homogeneity, representativeness, and population.     

Berry-Esséen rate in asymptotic normality for perturbed sample quantiles

In estimating quantiles with a sample of sizeN obtained from a distributionF, the perturbed sample quantiles based on a kernel functionk have been investigated by many authors. It is well known that their behaviour depends on the choices of “window-width”, sayw _N. Under suitable and reasonably mild assumptions onF andk, Ralescu and Sun (1993) have recently proven that lim _N→∞ N ^1/4w_N=0 is the necessary and sufficient condition for the asymptotic normality of the perturbed sample quantiles. In this paper, their rate of convergence is investigated. It turns out that the optimal Berry-Esséen rate ofO(N^?1/2) can be achieved by choosing the window-width suitably, sayw _N=O(N^?1/2). The obtained results, in addition to being explicit enough to verify the sufficient condition for the asymptotic normality, improve Ralescu's (1992) result of which the rate is of order (logN)N ^?1/2.     

Bayesian estimation of agent-based models

We consider Bayesian inference techniques for agent-based (AB) models, as an alternative to simulated minimum distance (SMD). Three computationally heavy steps are involved: (i) simulating the model, (ii) estimating the likelihood and (iii) sampling from the posterior distribution of the parameters. Computational complexity of AB models implies that efficient techniques have to be used with respect to points (ii) and (iii), possibly involving approximations. We first discuss non-parametric (kernel density) estimation of the likelihood, coupled with Markov chain Monte Carlo sampling schemes. We then turn to parametric approximations of the likelihood, which can be derived by observing the distribution of the simulation outcomes around the statistical equilibria, or by assuming a specific form for the distribution of external deviations in the data. Finally, we introduce Approximate Bayesian Computation techniques for likelihood-free estimation. These allow embedding SMD methods in a Bayesian framework, and are particularly suited when robust estimation is needed. These techniques are first tested in a simple price discovery model with one parameter, and then employed to estimate the behavioural macroeconomic model of De Grauwe (2012), with nine unknown parameters.     

Bayesian inference in a sample selection model

This paper develops methods of Bayesian inference in a sample selection model. The main feature of this model is that the outcome variable is only partially observed. We first present a Gibbs sampling algorithm for a model in which the selection and outcome errors are normally distributed. The algorithm is then extended to analyze models that are characterized by nonnormality. Specifically, we use a Dirichlet process prior and model the distribution of the unobservables as a mixture of normal distributions with a random number of components. The posterior distribution in this model can simultaneously detect the presence of selection effects and departures from normality. Our methods are illustrated using some simulated data and an abstract from the RAND health insurance experiment.     

Bayesian estimation approaches to first-price auctions

This paper considers Bayesian estimation strategies for first-price auctions within the independent private value paradigm. We develop an ‘optimization’ error approach that allows for estimation of values assuming that observed bids differ from optimal bids. We further augment this approach by allowing systematic over or underbidding by bidders using ideas from the stochastic frontier literature. We perform a simulation study to showcase the appeal of the method and apply the techniques to timber auction data collected in British Columbia. Our results suggest that significant underbidding is present in the timber auctions.     

High quantiles estimation with Quasi-PORT and DPOT: An application to value-at-risk for financial variables

Recurrent ⿿black swans⿿ financial events are a major concern for both investors and regulators because of the extreme price changes they cause, despite their very low probability of occurrence. In this paper, we use unconditional and conditional methods, such as the recently proposed high quantile (HQ) extreme value theory (EVT) models of DPOT (Duration-based Peak Over Threshold) and quasi-PORT (peaks over random threshold), to estimate the Value-at-Risk with very small probability values for an adequately long and major financial time series to obtain a reasonable number of violations for backtesting. We also compare these models and other alternative strategies through an out-of-sample accuracy investigation to determine their relative performance within the HQ context. Policy implications relevant to estimation of risk for extreme events are also provided.     

Bayesian prediction with cointegrated vector autoregressions

A complete procedure for calculating the joint predictive distribution of future observations based on the cointegrated vector autoregression is presented. The large degree of uncertainty in the choice of cointegration vectors is incorporated into the analysis via the prior distribution. This prior has the effect of weighing the predictive distributions based on the models with different cointegration vectors into an overall predictive distribution. The ideas of Litterman [Mimeo, Massachusetts Institute of Technology, 1980] are adopted for the prior on the short run dynamics of the process resulting in a prior which only depends on a few hyperparameters. A straightforward numerical evaluation of the predictive distribution based on Gibbs sampling is proposed. The prediction procedure is applied to a seven-variable system with a focus on forecasting Swedish inflation.     

Bayesian averaging,prediction and nonnested model selection

This paper studies the asymptotic relationship between Bayesian model averaging and post-selection frequentist predictors in both nested and nonnested models. We derive conditions under which their difference is of a smaller order of magnitude than the inverse of the square root of the sample size in large samples. This result depends crucially on the relation between posterior odds and frequentist model selection criteria. Weak conditions are given under which consistent model selection is feasible, regardless of whether models are nested or nonnested and regardless of whether models are correctly specified or not, in the sense that they select the best model with the least number of parameters with probability converging to 1. Under these conditions, Bayesian posterior odds and BICs are consistent for selecting among nested models, but are not consistent for selecting among nonnested models and possibly overlapping models. These findings have important bearing for applied researchers who are frequent users of model selection tools for empirical investigation of model predictions.     

Improved estimation of extreme quantiles in the multivariate Lomax (Pareto II) distribution

Estimation of a quantile of the common marginal distribution in a multivariate Lomax (Pareto II) distribution with unknown location and scale parameters is considered. For quadratic loss and specified extreme quantiles, it is established that the best affine equivariant procedure is inadmissible by constructing a better estimator.     

Bayesian estimation of a random coefficient model

This paper carries out a Bayesian analysis of the Hildreth-Houck (1968) random coefficient model and applies it to some cross-section production function data. Posterior distributions for mean coefficients, actual coefficients, variances and variance ratios are derived. The variance ratio posteriors are largely uninformative but they do lead to relatively informative densities on the variances, and the problem of negative variance estimates, obtained with previous techniques, is overcome. Posterior densities for the mean coefficients are not extremely sensitive to the variance ratios.     

Bayesian point estimation of the cointegration space

A neglected aspect of the otherwise fairly well developed Bayesian analysis of cointegration is point estimation of the cointegration space. It is pointed out here that, due to the well known non-identification of the cointegration vectors, the parameter space is not Euclidean and the loss functions underlying the conventional Bayes estimators are therefore questionable. We present a Bayes estimator of the cointegration space which takes the curved geometry of the parameter space into account. This estimate has the interpretation of being the posterior mean cointegration space and is invariant to the order of the time series, a property not shared with many of the Bayes estimators in the cointegration literature. An overall measure of cointegration space uncertainty is also proposed. Australian interest rate data are used for illustration. A small simulation study shows that the new Bayes estimator compares favorably to the maximum likelihood estimator.     

Bayesian prediction in doubly stochastic Poisson process

A stochastic marked point process model based on doubly stochastic Poisson process is considered in the problem of prediction for the total size of future marks in a given period, given the history of the process. The underlying marked point process \((T_{i},Y_{i})_{i\ge 1}\) , where \(T_{i}\) is the time of occurrence of the \(i\) th event and the mark \(Y_{i}\) is its characteristic (size), is supposed to be a non-homogeneous Poisson process on \(\mathbb {R}_{+}^{2}\) with intensity measure \(P\times \varTheta \) , where \(P\) is known, whereas \(\varTheta \) is treated as an unknown measure of the total size of future marks in a given period. In the problem of prediction considered, a Bayesian approach is used assuming that \(\varTheta \) is random with prior distribution presented by a gamma process. The best predictor with respect to this prior distribution is constructed under a precautionary loss function. A simulation study for comparing the behavior of the predictors under various criteria is provided.     

An ergodic theorem for proportions of observations that fall into random sets determined by sample quantiles

Assume that a sequence of observations \((X_n; n\ge 1)\) forms a strictly stationary process with an arbitrary univariate cumulative distribution function. We investigate almost sure asymptotic behavior of proportions of observations in the sample that fall into a random region determined by a given Borel set and a sample quantile. We provide sufficient conditions under which these proportions converge almost surly and describe the law of the limiting random variable.     

Distribution-free and link-free estimation for the sample selection model

The prevalent estimation methods for the sample selection model rely heavily on parametric assumptions and are sensitive to departures from the underlying parametric assumptions [see, e.g., Goldberger (1983)]. We propose an alternative estimation method, the corrected maximum likelihood estimate, which is consistent for the slope vector in the outcome equation up to a multiplicative scalar, even through the parametric model on which the estimate is based might be misspecified. As an important corollary, it follows from our result that Olsen's (1980) corrected ordinary least squares estimate is consistent if the outcome equation is linear, without requiring Olsen's assumptions on the joint error distribution.     

Bayesian estimation of explained variance in ANOVA designs

We propose to use the squared multiple correlation coefficient as an effect size measure for experimental analysis‐of‐variance designs and to use Bayesian methods to estimate its posterior distribution. We provide the expressions for the squared multiple, semipartial, and partial correlation coefficients corresponding to four commonly used analysis‐of‐variance designs and illustrate our contribution with two worked examples.