Bayesian Binomial Regression: Predicting Survival at a Trauma Center

Standard methods for analyzing binomial regression data rely on asymptotic inferences. Bayesian methods can be performed using simple computations, and they apply for any sample size. We provide a relatively complete discussion of Bayesian inferences for binomial regression with emphasis on inferences for the probability of “success.” Furthermore, we illustrate diagnostic tools, perform model selection among nonnested models, and examine the sensitivity of the Bayesian methods.     

Simulating from irregular data: Kernel Carlo Simulation

In this paper, we address the problem of simulating from a data-generating process for which the observed data do not follow a regular probability distribution. One existing method for doing this is bootstrapping, but it is incapable of interpolating between observed data. For univariate or bivariate data, in which a mixture structure can easily be identified, we could instead simulate from a Gaussian mixture model. In general, though, we would have the problem of identifying and estimating the mixture model. Instead of these, we introduce a non-parametric method for simulating datasets like this: Kernel Carlo Simulation. Our algorithm begins by using kernel density estimation to build a target probability distribution. Then, an envelope function that is guaranteed to be higher than the target distribution is created. We then use simple accept–reject sampling. Our approach is more flexible than others, can simulate intelligently across gaps in the data, and requires no subjective modelling decisions. With several univariate and multivariate examples, we show that our method returns simulated datasets that, compared with the observed data, retain the covariance structures and have distributional characteristics that are remarkably similar.     

Conditional Generalized Liouville Distributions on the Simplex

Liouville and generalized Liouville distributions on the simplex have been proposed for modeling compositional data and have been shown to be free from the extreme independence structure that characterizes the Dirichlet class. In this article, generalized Liouville distributions are shown to be rich enough to distinguish some lesser modes of independence as well. Unfortunately, it is noted that the applicability of the Liouville family will be limited, owing to the lack of invariance with respect to the chosen fill-up value. As an alternative, a new family of simplex distributions is proposed, one that admits invariance with respect to choice of fill-up value, as well as the ability to differentiate among many forms of independence.     

Power-divergence-type measure of departure from diagonals-parameter symmetry for square contingency tables with ordered categories

For the analysis of square contingency tables with ordered categories, Goodman considered the diagonals-parameter symmetry (DPS) model. This paper proposes a measure to represent the degree of departure from the DPS model. The proposed measure is expressed by applying Read and Cressie’s power-divergence or Patil and Taillie’s diversity index. The measure would be useful for comparing the degree of departure from the DPS model in several tables. Examples are given.     

Parameter selection in modified histogram estimates

A multivariate modified histogram density estimate depending on a reference density g and a partition P has been proved to have good consistency properties according to several information theoretic criteria. Given an i.i.d. sample, we show how to select automatically both g and P so that the expected L ₁ error of the corresponding selected estimate is within a given constant multiple of the best possible error plus an additive term which tends to zero under mild assumptions. Our method is inspired by the combinatorial tools developed by Devroye and Lugosi [Devroye, L. and Lugosi, G., 2001, Combinatorial Methods in Density Estimation (New York, NY: Springer–Verlag)] and it includes a wide range of reference density and partition models. Results of simulations are also presented.     

Asymptotically minimax non–parametric function estimation with positivity constraints I

The spatially inhomogeneous smoothness of the non-parametric density or regression-function to be estimated by non-parametric methods is often modelled by Besov- and Triebel-type smoothness constraints. For such problems, Donoho and Johnstone [D.L. Donoho and I.M. Johnstone, Minimax estimation via wavelet shrinkage. Ann. Stat. 26 (1998), pp. 879–921.], Delyon and Juditsky [B. Delyon and A. Juditsky, On minimax wavelet estimators, Appl. Comput. Harmon. Anal. 3 (1996), pp. 215–228.] studied minimax rates of convergence for wavelet estimators with thresholding, while Lepski et al. [O.V. Lepski, E. Mammen, and V.G. Spokoiny, Optimal spatial adaptation to inhomogeneous smoothness: an approach based on kernel estimators with variable bandwidth selectors, Ann. Stat. 25 (1997), pp. 929–947.] proposed a variable bandwidth selection for kernel estimators that achieved optimal rates over Besov classes. However, a second challenge in many real applications of non-parametric curve estimation is that the function must be positive. Here, we show how to construct estimators under positivity constraints that satisfy these constraints and also achieve minimax rates over the appropriate smoothness class.     

Random coefficients regression models. a review 1

A review is given of random regression coefficients models. The emphasis is put on the problem of estimating the mean regression coefficients and the covariance matrix of the coefficients. Prediction of the individual random coefficients is not discussed. The main purpose of the review is to point to the practical aspects of the models and the problem of statistical inference in finite samples. Some problems for future research are indicated.     

A note on ∞manque in variance covariance components models

Stimualted by C.R. Rao's MINQUE J. Focke and G. Dewess introduced the so called r-and ∞ MINQUE. Although they developed a unique charecterization of ∞-MINQUE, they did not give explicite formulas for its computation. The goal this paper is to close this lack and to etend the concept to more general models.     

Bivariate exponential-type distributions with linear regression

Let Xbe a bivariate exponential-type random vector (BIDLIKAR, PATIL (1968)), than it is proved:

1. If P(X ≥0) = 1 is valid, then Xhas linear regression to both directions if and only if Xpossesses a symmetric Γ-distribution.

2. Xpossesses linear regression to both directions with constant regression coefficients (independent of the parameter vector ? of the exponential-type distribution (BIDLIKAR, PATIL (1968)) if and only if Xis normal distributed.