83,546 research outputs found
Information and Covariance Matrices for Multivariate Burr III and Logistic distributions
Main result of this paper is to derive the exact analytical expressions of
information and covariance matrices for multivariate Burr III and logistic
distributions. These distributions arise as tractable parametric models in
price and income distributions, reliability, economics, populations growth and
survival data. We showed that all the calculations can be obtained from one
main moment multi dimensional integral whose expression is obtained through
some particular change of variables. Indeed, we consider that this calculus
technique for improper integral has its own importance in applied probability
calculus.Comment: submitted to Communications in Statistic
On the Independence Jeffreys prior for skew--symmetric models with applications
We study the Jeffreys prior of the skewness parameter of a general class of
scalar skew--symmetric models. It is shown that this prior is symmetric about
0, proper, and with tails under mild regularity conditions.
We also calculate the independence Jeffreys prior for the case with unknown
location and scale parameters. Sufficient conditions for the existence of the
corresponding posterior distribution are investigated for the case when the
sampling model belongs to the family of skew--symmetric scale mixtures of
normal distributions. The usefulness of these results is illustrated using the
skew--logistic model and two applications with real data
A new specification of generalized linear models for categorical data
Regression models for categorical data are specified in heterogeneous ways.
We propose to unify the specification of such models. This allows us to define
the family of reference models for nominal data. We introduce the notion of
reversible models for ordinal data that distinguishes adjacent and cumulative
models from sequential ones. The combination of the proposed specification with
the definition of reference and reversible models and various invariance
properties leads to a new view of regression models for categorical data.Comment: 31 pages, 13 figure
Asymptotic inference for semiparametric association models
Association models for a pair of random elements and (e.g., vectors)
are considered which specify the odds ratio function up to an unknown parameter
\bolds\theta. These models are shown to be semiparametric in the sense that
they do not restrict the marginal distributions of and . Inference for
the odds ratio parameter \bolds\theta may be obtained from sampling either
conditionally on or vice versa. Generalizing results from Prentice and
Pyke, Weinberg and Wacholder and Scott and Wild, we show that asymptotic
inference for \bolds\theta under sampling conditional on is the same as
if sampling had been conditional on . Common regression models, for example,
generalized linear models with canonical link or multivariate linear,
respectively, logistic models, are association models where the regression
parameter \bolds\beta is closely related to the odds ratio parameter
\bolds\theta. Hence inference for \bolds\beta may be drawn from samples
conditional on using an association model.Comment: Published in at http://dx.doi.org/10.1214/07-AOS572 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Reflecting about Selecting Noninformative Priors
Following the critical review of Seaman et al. (2012), we reflect on what is
presumably the most essential aspect of Bayesian statistics, namely the
selection of a prior density. In some cases, Bayesian inference remains fairly
stable under a large range of noninformative prior distributions. However, as
discussed by \citet{Hd}, there may also be unintended consequences of a choice
of a noninformative prior and, these authors consider this problem ignored in
Bayesian studies. As they based their argumentation on four examples, we
reassess these examples and their Bayesian processing via different prior
choices. Our conclusion is to lower the degree of worry about the impact of the
prior, exhibiting an overall stability of the posterior distributions. We thus
consider that the warnings of Seaman et al. (2012), while commendable, do not
jeopardize the use of most noninformative priors.Comment: 15 pages, 8 figures, 5 table
On the correspondence from Bayesian log-linear modelling to logistic regression modelling with -priors
Consider a set of categorical variables where at least one of them is binary.
The log-linear model that describes the counts in the resulting contingency
table implies a specific logistic regression model, with the binary variable as
the outcome. Within the Bayesian framework, the -prior and mixtures of
-priors are commonly assigned to the parameters of a generalized linear
model. We prove that assigning a -prior (or a mixture of -priors) to the
parameters of a certain log-linear model designates a -prior (or a mixture
of -priors) on the parameters of the corresponding logistic regression. By
deriving an asymptotic result, and with numerical illustrations, we demonstrate
that when a -prior is adopted, this correspondence extends to the posterior
distribution of the model parameters. Thus, it is valid to translate inferences
from fitting a log-linear model to inferences within the logistic regression
framework, with regard to the presence of main effects and interaction terms.Comment: 27 page
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