53,851 research outputs found
Closed-Form Bayesian Inferences for the Logit Model via Polynomial Expansions
Articles in Marketing and choice literatures have demonstrated the need for
incorporating person-level heterogeneity into behavioral models (e.g., logit
models for multiple binary outcomes as studied here). However, the logit
likelihood extended with a population distribution of heterogeneity doesn't
yield closed-form inferences, and therefore numerical integration techniques
are relied upon (e.g., MCMC methods).
We present here an alternative, closed-form Bayesian inferences for the logit
model, which we obtain by approximating the logit likelihood via a polynomial
expansion, and then positing a distribution of heterogeneity from a flexible
family that is now conjugate and integrable. For problems where the response
coefficients are independent, choosing the Gamma distribution leads to rapidly
convergent closed-form expansions; if there are correlations among the
coefficients one can still obtain rapidly convergent closed-form expansions by
positing a distribution of heterogeneity from a Multivariate Gamma
distribution. The solution then comes from the moment generating function of
the Multivariate Gamma distribution or in general from the multivariate
heterogeneity distribution assumed.
Closed-form Bayesian inferences, derivatives (useful for elasticity
calculations), population distribution parameter estimates (useful for
summarization) and starting values (useful for complicated algorithms) are
hence directly available. Two simulation studies demonstrate the efficacy of
our approach.Comment: 30 pages, 2 figures, corrected some typos. Appears in Quantitative
Marketing and Economics vol 4 (2006), no. 2, 173--20
A multivariate gamma distribution applied to compositional data analysis
Parametric compositional data analysis in a high dimensional simplex can be performed by employing the Dirichlet distribution, or alternatively, through the logistic normal distribution if the Dirichlet is not appropriate. In this paper, a multivariate gamma (MGAM) distribution is proposed as an alternative distribution for compositional data. In addition, the MGAM distribution is extended to a multivariate extreme value (MEV) distribution and goodness of fit statistics are calculated for comparison against the logistic normal distribution. An application is considered where the amount of gas produced from a coal gasication facility depends crucially on the size distribution of the coal, which is measured as compositional data and characterised by six variables. The observed sample space is divided into three regions of high (H), standard (S) and low (L) gas production by choosing appropriate thresholds, and new observations are classified among the regions
Miscellaneous results related to the Gaussian product inequality conjecture for the joint distribution of traces of Wishart matrices
This note reports partial results related to the Gaussian product inequality
(GPI) conjecture for the joint distribution of traces of Wishart matrices. In
particular, several GPI-related results from Wei (2014) and Liu et al. (2015)
are extended in two ways: by replacing the power functions with more general
classes of functions, and by replacing the usual Gaussian and multivariate
gamma distributional assumptions by the more general trace-Wishart distribution
assumption. These findings suggest that a Kronecker product form of the GPI
holds for diagonal blocks of any Wishart distribution.Comment: 10 pages, 0 figure
Bayesian Boundary Trend Filtering
Estimating boundary curves has many applications such as economics, climate
science, and medicine. Bayesian trend filtering has been developed as one of
locally adaptive smoothing methods to estimate the non-stationary trend of
data. This paper develops a Bayesian trend filtering for estimating the
boundary trend. To this end, the truncated multivariate normal working
likelihood and global-local shrinkage priors based on the scale mixtures of
normal distribution are introduced. In particular, well-known horseshoe prior
for difference leads to locally adaptive shrinkage estimation for boundary
trend. However, the full conditional distributions of the Gibbs sampler involve
high-dimensional truncated multivariate normal distribution. To overcome the
difficulty of sampling, an approximation of truncated multivariate normal
distribution is employed. Using the approximation, the proposed models lead to
an efficient Gibbs sampling algorithm via the P\'olya-Gamma data augmentation.
The proposed method is also extended by considering a nearly isotonic
constraint. The performance of the proposed method is illustrated through some
numerical experiments and real data examples.Comment: 25 pages, 6 figure
Aggregation of Dependent Risks with Specific Marginals by the Family of Koehler-Symanowski Distributions
Many problems in Finance, such as risk management, optimal asset allocation, and derivative pricing, require an understanding of the volatility and correlations of assets returns. In these cases, it may be necessary to represent empirical data with a parametric distribution. In the literature, many distributions can be found to represent univariate data, but few can be extended to multivariate populations. The most widely used multivariate distribution in the aggregation of dependent risks in a portfolio is the Normal distribution, which has the drawbacks of inflexibility and frequent inappropriateness. Here, we consider modelling assets and measuring portfolio risks using the family of Koehler-Symanowski multivariate distributions with specific marginals, as, for example, the generalized lambda distribution. This family of distributions can be defined using the cdf along with interaction terms in the independence case. This family can be derived using a particular transformation of exponential random variables and an independent gamma. This distribution has the advantage of allowing models with complex dependence structures, as we demonstrate with Monte Carlo simulations and the analysis of the risk of a portfolioRisk Management, Monte Carlo Method, Generalized Lambda Distribution, Koehler-Symanowski Distribution
Simplified Pair Copula Constructions --- Limits and Extensions
So called pair copula constructions (PCCs), specifying multivariate
distributions only in terms of bivariate building blocks (pair copulas),
constitute a flexible class of dependence models. To keep them tractable for
inference and model selection, the simplifying assumption that copulas of
conditional distributions do not depend on the values of the variables which
they are conditioned on is popular. In this paper, we show for which classes of
distributions such a simplification is applicable, significantly extending the
discussion of Hob{\ae}k Haff et al. (2010). In particular, we show that the
only Archimedean copula in dimension d \geq 4 which is of the simplified type
is that based on the gamma Laplace transform or its extension, while the
Student-t copula is the only one arising from a scale mixture of Normals.
Further, we illustrate how PCCs can be adapted for situations where conditional
copulas depend on values which are conditioned on
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