Inverse stable prior for exponential models

We consider a class of non-conjugate priors as a mixing family of distributions for a parameter (e.g., Poisson or gamma rate, inverse scale or precision of an inverse-gamma, inverse variance of a normal distribution) of an exponential subclass of discrete and continuous data distributions. The prior class is proper, nonzero at the origin (unlike the gamma and inverted beta priors with shape parameter less than one and Jeffreys prior for a Poisson rate), and is easy to generate random numbers from. The prior class also provides flexibility in capturing a wide array of prior beliefs (right-skewed and left-skewed) as modulated by a bounded parameter $\alpha \in (0, 1).$ The resulting posterior family in the single-parameter case can be expressed in closed-form and is proper, making calibration unnecessary. The mixing induced by the inverse stable family results to a marginal prior distribution in the form of a generalized Mittag-Leffler function, which covers a broad array of distributional shapes. We derive closed-form expressions of some properties like the moment generating function and moments. We propose algorithms to generate samples from the posterior distribution and calculate the Bayes estimators for real data analysis. We formulate the predictive prior and posterior distributions. We test the proposed Bayes estimators using Monte Carlo simulations. The extension to hierarchical modeling and inverse variance components models is straightforward. We can find $\alpha$ (which acts like a smoothing parameter) values for which the inverse stable can provide better shrinkage than the inverted beta prior in many cases. We illustrate the methodology using a real data set, introduce a hyperprior density for the hyperparameters, and extend the model to a heavy-tailed distribution

The McDonald Gompertz Distribution: Properties and Applications

This paper introduces a five-parameter lifetime model with increasing, decreasing, upside -down bathtub and bathtub shaped failure rate called as the McDonald Gompertz (McG) distribution. This new distribution extend the Gompertz, generalized Gompertz, generalized exponential, beta Gompertz and Kumaraswamy Gompertz distributions, among several other models. We obtain several properties of the McG distribution including moments, entropies, quantile and generating functions. We provide the density function of the order statistics and their moments. The parameter estimation is based on the usual maximum likelihood approach. We also provide the observed information matrix and discuss inferences issues. In the end, the flexibility and usefulness of the new distribution is illustrated by means of application to two real data sets

Some Reliability Properties of Transformed-Transformer Family of Distributions

The Transformed-Transformer family of distributions are the resulting family of distributions as transformed from a random variable $T$ through another transformer random variable $X$ using a weight function $\omega$ of the cumulative distribution function of $X$. In this paper, we study different stochastic ageing properties, as well as different stochastic orderings of this family of distributions. We discuss the results with several well known distributions

Beta generated Kumaraswamy-G and other new families of distributions

A new generalization of the family of Kumaraswamy-G (Cordeiro and de Castro, 2011) distribution that includes three recently proposed families namely the Garhy generated family (Elgarhy et al., 2016), Beta-Dagum and Beta-Singh-Maddala distribution (Domma and Condino, 2016) is proposed by constructing beta generated Kumaraswamy-G distribution. Useful expansions of the pdf and the cdf of the proposed family is derived and seen as infinite mixtures of the Kumaraswamy-G distribution. Order statistics, Probability weighted moments, moment generating function, R\'enyi entropies, quantile power series, random sample generation, asymptotes and shapes are also investigated. Two methods of parameter estimation are presented. Suitability of the proposed model in comparisons to its sub models is carried out considering two real life data sets. Finally, some new classes of beta generated families are proposed for future investigations.Comment: 35 pages, 4 figures, 2 Tables. Version-II. Preprin

The I-Function Distribution and its Extensions

In this paper we introduce a new probability distribution on (0,1), associated with the I-function, namely, the I-function distribution. This distribution generalizes several known distributions with positive support. It is also shown that the distribution of products, quotients and powers of independent I-function variates are I-function variates. Another distribution called the I-function inverse Gaussian distribution is also introduced and studied.Comment: 17 page

Bayesian Ensembles of Binary-Event Forecasts: When Is It Appropriate to Extremize or Anti-Extremize?

Many organizations face critical decisions that rely on forecasts of binary events. In these situations, organizations often gather forecasts from multiple experts or models and average those forecasts to produce a single aggregate forecast. Because the average forecast is known to be underconfident, methods have been proposed that create an aggregate forecast more extreme than the average forecast. But is it always appropriate to extremize the average forecast? And if not, when is it appropriate to anti-extremize (i.e., to make the aggregate forecast less extreme)? To answer these questions, we introduce a class of optimal aggregators. These aggregators are Bayesian ensembles because they follow from a Bayesian model of the underlying information experts have. Each ensemble is a generalized additive model of experts' probabilities that first transforms the experts' probabilities into their corresponding information states, then linearly combines these information states, and finally transforms the combined information states back into the probability space. Analytically, we find that these optimal aggregators do not always extremize the average forecast, and when they do, they can run counter to existing methods. On two publicly available datasets, we demonstrate that these new ensembles are easily fit to real forecast data and are more accurate than existing methods

Estimation with Binned Data

Variables such as household income are sometimes binned, so that we only know how many households fall in each of several bins such as $0-10,000,$10,000-15,000, or $200,000+. We provide a SAS macro that estimates the mean and variance of binned data by fitting the extended generalized gamma (EGG) distribution, the power normal (PN) distribution, and a new distribution that we call the power logistic (PL). The macro also implements a "best-of-breed" estimator that chooses from among the EGG, PN, and PL estimates on the basis of likelihood and finite variance. We test the macro by estimating the mean family and household incomes of approximately 13,000 US school districts between 1970 and 2009. The estimates have negligible bias (0-2%) and a root mean squared error of just 3-6%. The estimates compare favorably with estimates obtained by fitting the Dagum, generalized beta (GB2), or logspline distributions.Comment: 16 pages + 2 tables + 4 figure

Set-Based Tests for Genetic Association Using the Generalized Berk-Jones Statistic

Studying the effects of groups of Single Nucleotide Polymorphisms (SNPs), as in a gene, genetic pathway, or network, can provide novel insight into complex diseases, above that which can be gleaned from studying SNPs individually. Common challenges in set-based genetic association testing include weak effect sizes, correlation between SNPs in a SNP-set, and scarcity of signals, with single-SNP effects often ranging from extremely sparse to moderately sparse in number. Motivated by these challenges, we propose the Generalized Berk-Jones (GBJ) test for the association between a SNP-set and outcome. The GBJ extends the Berk-Jones (BJ) statistic by accounting for correlation among SNPs, and it provides advantages over the Generalized Higher Criticism (GHC) test when signals in a SNP-set are moderately sparse. We also provide an analytic p-value calculation procedure for SNP-sets of any finite size. Using this p-value calculation, we illustrate that the rejection region for GBJ can be described as a compromise of those for BJ and GHC. We develop an omnibus statistic as well, and we show that this omnibus test is robust to the degree of signal sparsity. An additional advantage of our method is the ability to conduct inference using individual SNP summary statistics from a genome-wide association study. We evaluate the finite sample performance of the GBJ though simulation studies and application to gene-level association analysis of breast cancer risk.Comment: Corrected typos in abstrac

Bootstrap Bartlett correction in inflated beta regression

The inflated beta regression model aims to enable the modeling of responses in the intervals $(0,1]$, $[0,1)$ or $[0,1]$. In this model, hypothesis testing is often performed based on the likelihood ratio statistic. The critical values are obtained from asymptotic approximations, which may lead to distortions of size in small samples. In this sense, this paper proposes the bootstrap Bartlett correction to the statistic of likelihood ratio in the inflated beta regression model. The proposed adjustment only requires a simple Monte Carlo simulation. Through extensive Monte Carlo simulations the finite sample performance (size and power) of the proposed corrected test is compared to the usual likelihood ratio test and the Skovgaard adjustment already proposed in the literature. The numerical results evidence that inference based on the proposed correction is much more reliable than that based on the usual likelihood ratio statistics and the Skovgaard adjustment. At the end of the work, an application to real data is also presented.Comment: 17 pages, 2 figures, 3 table

Asymptotic regime for impropriety tests of complex random vectors

Impropriety testing for complex-valued vector has been considered lately due to potential applications ranging from digital communications to complex media imaging. This paper provides new results for such tests in the asymptotic regime, i.e. when the vector dimension and sample size grow commensurately to infinity. The studied tests are based on invariant statistics named impropriety coefficients. Limiting distributions for these statistics are derived, together with those of the Generalized Likelihood Ratio Test (GLRT) and Roy's test, in the Gaussian case. This characterization in the asymptotic regime allows also to identify a phase transition in Roy's test with potential application in detection of complex-valued low-rank subspace corrupted by proper noise in large datasets. Simulations illustrate the accuracy of the proposed asymptotic approximations.Comment: 11 pages, 8 figures, submitted to IEEE TS