Adaptive Percolation Using Subjective Likelihoods

A phenomenon that I call “adaptive percolation” commonly arises in biology, business, economics, defense, finance, manufacturing, and the social sciences. Here one wishes to select a handful of entities from a large pool of entities via a process of screening through a hierarchy of sieves. The process is not unlike the percolation of a liquid through a porous medium. The probability model developed here is based on a nested and adaptive Bayesian approach that results in the product of beta-binomial distributions with common parameters. The common parameters happen to be the observed data. I call this the percolated beta-binomial distribution . The model turns out to be a slight generalization of the probabilistic model used in percolation theory. The generalization is a consequence of using a subjectively specified likelihood function to construct a probability model. The notion of using likelihoods for constructing probability models is not a part of the conventional toolkit of applied probabilists. To the best of my knowledge, a use of the product of beta-binomial distributions as a probability model for Bernoulli trials appears to be new. The development of the material of this article is illustrated via data from the 2009 astronaut selection program, which motivated this work.     

Linear regression model with new symmetric distributed errors

Regression models play a dominant role in analyzing several data sets arising from areas like agricultural experiment, space experiment, biological experiment, financial modeling, etc. One of the major strings in developing the regression models is the assumption of the distribution of the error terms. It is customary to consider that the error terms follow the Gaussian distribution. However, there are some drawbacks of Gaussian errors such as the distribution being mesokurtic having kurtosis three. In many practical situations the variables under study may not be having mesokurtic but they are platykurtic. Hence, to analyze these sorts of platykurtic variables, a two-variable regression model with new symmetric distributed errors is developed and analyzed. The maximum likelihood (ML) estimators of the model parameters are derived. The properties of the ML estimators with respect to the new symmetrically distributed errors are also discussed. A simulation study is carried out to compare the proposed model with that of Gaussian errors and found that the proposed model performs better when the variables are platykurtic. Some applications of the developed model are also pointed out.     

On the generalized process capability under simple and mixture models

Process capability (PC) indices measure the ability of a process of interest to meet the desired specifications under certain restrictions. There are a variety of capability indices available in literature for different interest variables such as weights, lengths, thickness, and the life time of items among many others. The goal of this article is to study the generalized capability indices from the Bayesian view point under different symmetric and asymmetric loss functions for the simple and mixture of generalized lifetime models. For our study purposes, we have covered a simple and two component mixture of Maxwell distribution as a special case of the generalized class of models. A comparative discussion of the PC with the mixture models under Laplace and inverse Rayleigh are also included. Bayesian point estimation of maintenance performance of the system is also part of the study (considering the Maxwell failure lifetime model and the repair time model). A real-life example is also included to illustrate the procedural details of the proposed method.     

Coverage‐adjusted Confidence Intervals for a Binomial Proportion

We consider the classic problem of interval estimation of a proportion p based on binomial sampling. The ‘exact’ Clopper–Pearson confidence interval for p is known to be unnecessarily conservative. We propose coverage adjustments of the Clopper–Pearson interval that incorporate prior or posterior beliefs into the interval. Using heatmap‐type plots for comparing confidence intervals, we show that the coverage‐adjusted intervals have satisfying coverage and shorter expected lengths than competing intervals found in the literature.     

Analysis of survival data by a Weibull-generalized Poisson distribution

In life-testing and survival analysis, sometimes the components are arranged in series or parallel system and the number of components is initially unknown. Thus, the number of components, say Z, is considered as random with an appropriate probability mass function. In this paper, we model the survival data with baseline distribution as Weibull and the distribution of Z as generalized Poisson, giving rise to four parameters in the model: increasing, decreasing, bathtub and upside bathtub failure rates. Two examples are provided and the maximum-likelihood estimation of the parameters is studied. Rao's score test is developed to compare the results with the exponential Poisson model studied by Kus [17] and the exponential-generalized Poisson distribution with baseline distribution as exponential and the distribution of Z as generalized Poisson. Simulation studies are carried out to examine the performance of the estimates.     

Eliciting expert judgements about a set of proportions

Eliciting expert knowledge about several uncertain quantities is a complex task when those quantities exhibit associations. A well-known example of such a problem is eliciting knowledge about a set of uncertain proportions which must sum to 1. The usual approach is to assume that the expert's knowledge can be adequately represented by a Dirichlet distribution, since this is by far the simplest multivariate distribution that is appropriate for such a set of proportions. It is also the most convenient, particularly when the expert's prior knowledge is to be combined with a multinomial sample since then the Dirichlet is the conjugate prior family. Several methods have been described in the literature for eliciting beliefs in the form of a Dirichlet distribution, which typically involve eliciting from the expert enough judgements to identify uniquely the Dirichlet hyperparameters. We describe here a new method which employs the device of over-fitting, i.e. eliciting more than the minimal number of judgements, in order to (a) produce a more carefully considered Dirichlet distribution and (b) ensure that the Dirichlet distribution is indeed a reasonable fit to the expert's knowledge. The method has been implemented in a software extension of the Sheffield elicitation framework (SHELF) to facilitate the multivariate elicitation process.     

Some notes on robust sure independence screening

Sure independence screening (SIS) proposed by Fan and Lv [4 J. Fan and R. Li, Variable selection via nonconcave penalized likelihood and its oracle properties, J. Amer. Statist. Assoc. 96 (2001), pp. 1348–1360. doi: 10.1198/016214501753382273[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] uses marginal correlations to select important variables, and has proven to be an efficient method for ultrahigh-dimensional linear models. This paper provides two robust versions of SIS against outliers. The two methods, respectively, replace the sample correlation in SIS with two robust measures, and screen variables by ranking them. Like SIS, the proposed methods are simple and fast. In addition, they are highly robust against a substantial fraction of outliers in the data. These features make them applicable to large datasets which may contain outliers. Simulation results are presented to show their effectiveness.     

A new model selection procedure based on dynamic quantile regression

In this article, we propose a novel robust data-analytic procedure, dynamic quantile regression (DQR), for model selection. It is robust in the sense that it can simultaneously estimate the coefficients and the distribution of errors over a large collection of error distributions even those that are heavy-tailed and may not even possess variances or means; and DQR is easy to implement in the sense that it does not need to decide in advance which quantile(s) should be gathered. Asymptotic properties of related estimators are derived. Simulations and illustrative real examples are also given.     

On the Exact and Near-Exact Distributions of the Product of Generalized Gamma Random Variables and the Generalized Variance

In this article, the authors first obtain the exact distribution of the logarithm of the product of independent generalized Gamma r.v.’s (random variables) in the form of a Generalized Integer Gamma distribution of infinite depth, where all the rate and shape parameters are well identified. Then, by a routine transformation, simple and manageable expressions for the exact distribution of the product of independent generalized Gamma r.v.’s are derived. The method used also enables us to obtain quite easily very accurate, manageable and simple near-exact distributions in the form of Generalized Near-Integer Gamma distributions. Numerical studies are carried out to assess the precision of different approximations to the exact distribution and they show the high accuracy of the approximations provided by the near-exact distributions. As particular cases of the exact distributions obtained we have the distribution of the product of independent Gamma, Weibull, Frechet, Maxwell-Boltzman, Half-Normal, Rayleigh, and Exponential distributions, as well as the exact distribution of the generalized variance, the exact distribution of discriminants or Vandermonde determinants and the exact distribution of any linear combination of generalized Gumbel distributions, as well as yet the distribution of the product of any power of the absolute value of independent Normal r.v.’s.     

Baker- Lin-Huang Type Bivariate Distributions Based on Order Statistics

Baker (2008 Baker, R. (2008). An order-statistics-based method for constructing multivariate distributions with fixed marginals. J. Multivariate Anal. 99: 23122327.[Crossref], [Web of Science ®] , [Google Scholar]) introduced a new class of bivariate distributions based on distributions of order statistics from two independent samples of size n. Lin and Huang (2010 Lin, G.D., Huang, J.S. (2010). A note on the maximum correlation for Baker’s bivariate distributions with fixed marginals. J. Multivariate Anal. 101: 22272233.[Crossref], [Web of Science ®] , [Google Scholar]) discovered an important property of Baker’s distribution and showed that the Pearson’s correlation coefficient for this distribution converges to maximum attainable value, i.e., the correlation coefficient of the Fréchet upper bound, as n increases to infinity. Bairamov and Bayramoglu (2013 Bairamov, I., Bayramoglu, K. (2013). From Huang-Kotz distribution to Baker’s distribution. J. Multivariate Anal. 113: 106115.[Crossref], [Web of Science ®] , [Google Scholar]) investigated a new class of bivariate distributions constructed by using Baker’s model and distributions of order statistics from dependent random variables, allowing higher correlation than that of Baker’s distribution. In this article, a new class of Baker’s type bivariate distributions with high correlation are constructed based on distributions of order statistics by using an arbitrary continuous copula instead of the product copula.