Bayesian estimation and prediction based on generalized Type-II hybrid censored sample

ABSTRACT

In this paper, we consider a general form for the underlying distribution and a general conjugate prior, and develop a general procedure for deriving the maximum likelihood and Bayesian estimators based on an observed generalized Type-II hybrid censored sample. The problems of predicting the future order statistics from the same sample and that from a future sample are also discussed from a Bayesian viewpoint. For the illustration of the developed results, the exponential and Pareto distributions are used as examples. Finally, two numerical examples are presented for illustrating all the inferential procedures developed here.     

Bayesian Analysis of Contingency Tables

ABSTRACT

The display of the data by means of contingency tables is used in different approaches to statistical inference, for example, to broach the test of homogeneity of independent multinomial distributions. We develop a Bayesian procedure to test simple null hypotheses versus bilateral alternatives in contingency tables. Given independent samples of two binomial distributions and taking a mixed prior distribution, we calculate the posterior probability that the proportion of successes in the first population is the same as in the second. This posterior probability is compared with the p-value of the classical method, obtaining a reconciliation between both results, classical and Bayesian. The obtained results are generalized for r × s tables.     

Graphical prior elicitation in univariate models

ABSTRACT

Standard prior elicitation procedures require experts to explicitly quantify their beliefs about parameters in the form of multiple summaries. In this article, we draw on recent advances in the statistical graphics and information visualization communities to propose a novel elicitation scheme that implicitly learns an expert’s opinions through their sequential selection of graphics of carefully constructed hypothetical future samples. While the scheme can be applied to a broad array of models, we use it to construct procedures for elicitation in data models commonly used in practice: Bernoulli, Poisson, and Normal. We also provide open-source, web-based Shiny implementations of the procedures.     

Bayes and robust Bayes predictions in a subfamily of scale parameters under a precautionary loss function

ABSTRACT

This paper deals with Bayes, robust Bayes, and minimax predictions in a subfamily of scale parameters under an asymmetric precautionary loss function. In Bayesian statistical inference, the goal is to obtain optimal rules under a specified loss function and an explicit prior distribution over the parameter space. However, in practice, we are not able to specify the prior totally or when a problem must be solved by two statisticians, they may agree on the choice of the prior but not the values of the hyperparameters. A common approach to the prior uncertainty in Bayesian analysis is to choose a class of prior distributions and compute some functional quantity. This is known as Robust Bayesian analysis which provides a way to consider the prior knowledge in terms of a class of priors Γ for global prevention against bad choices of hyperparameters. Under a scale invariant precautionary loss function, we deal with robust Bayes predictions of Y based on X. We carried out a simulation study and a real data analysis to illustrate the practical utility of the prediction procedure.     

Expert Knowledge Elicitation: Subjective but Scientific

ABSTRACT

Expert opinion and judgment enter into the practice of statistical inference and decision-making in numerous ways. Indeed, there is essentially no aspect of scientific investigation in which judgment is not required. Judgment is necessarily subjective, but should be made as carefully, as objectively, and as scientifically as possible.

Elicitation of expert knowledge concerning an uncertain quantity expresses that knowledge in the form of a (subjective) probability distribution for the quantity. Such distributions play an important role in statistical inference (for example as prior distributions in a Bayesian analysis) and in evidence-based decision-making (for example as expressions of uncertainty regarding inputs to a decision model). This article sets out a number of practices through which elicitation can be made as rigorous and scientific as possible.

One such practice is to follow a recognized protocol that is designed to address and minimize the cognitive biases that experts are prone to when making probabilistic judgments. We review the leading protocols in the field, and contrast their different approaches to dealing with these biases through the medium of a detailed case study employing the SHELF protocol.

The article ends with discussion of how to elicit a joint probability distribution for multiple uncertain quantities, which is a challenge for all the leading protocols. Supplementary materials for this article are available online.     

Impact of selection bias on the evaluation of clusters of chemical compounds in the drug discovery process

Expert opinion plays an important role when selecting promising clusters of chemical compounds in the drug discovery process. Indeed, experts can qualitatively assess the potential of each cluster, and with appropriate statistical methods, these qualitative assessments can be quantified into a success probability for each of them. However, one crucial element often overlooked is the procedure by which the clusters are assigned to/selected by the experts for evaluation. In the present work, the impact such a procedure may have on the statistical analysis and the entire evaluation process is studied. It has been shown that some implementations of the selection procedure may seriously compromise the validity of the evaluation even when the rating and selection processes are independent. Consequently, the fully random allocation of the clusters to the experts is strongly advocated. Copyright © 2014 John Wiley & Sons, Ltd.     

Optimal designs of multivariate synthetic |S| control chart based on median run length

The multivariate synthetic generalized sample variance |S| (synthetic |S|) chart is a combination of the |S| sub-chart and the conforming run length sub-chart. A procedure for optimal designs of the synthetic |S| chart, based on the median run length (MRL), for both zero and steady-state modes are provided by minimizing the out-of-control MRL. The comparative results show that the synthetic |S| chart performs better than the standard |S| chart for detecting shifts in the covariance matrix of a multivariate normally distributed process, in terms of the MRL. An example is given to illustrate the operation of the synthetic |S| chart.     

Estimating and Testing Nonlinear Local Dependence Between Two Time Series

ABSTRACT

The most common measure of dependence between two time series is the cross-correlation function. This measure gives a complete characterization of dependence for two linear and jointly Gaussian time series, but it often fails for nonlinear and non-Gaussian time series models, such as the ARCH-type models used in finance. The cross-correlation function is a global measure of dependence. In this article, we apply to bivariate time series the nonlinear local measure of dependence called local Gaussian correlation. It generally works well also for nonlinear models, and it can distinguish between positive and negative local dependence. We construct confidence intervals for the local Gaussian correlation and develop a test based on this measure of dependence. Asymptotic properties are derived for the parameter estimates, for the test functional and for a block bootstrap procedure. For both simulated and financial index data, we construct confidence intervals and we compare the proposed test with one based on the ordinary correlation and with one based on the Brownian distance correlation. Financial indexes are examined over a long time period and their local joint behavior, including tail behavior, is analyzed prior to, during and after the financial crisis. Supplementary material for this article is available online.     

LAD-Lasso variable selection for doubly censored median regression models

ABSTRACT

A variable selection procedure based on least absolute deviation (LAD) estimation and adaptive lasso (LAD-Lasso for short) is proposed for median regression models with doubly censored data. The proposed procedure can select significant variables and estimate the parameters simultaneously, and the resulting estimators enjoy the oracle property. Simulation results show that the proposed method works well.     

A Note on the Lower Bounds on Maximum Number of Clear Two-Factor Interactions for (2 m−p III) and (2 m−p IV) Designs

ABSTRACT

In the reliability analysis of mechanical repairable equipment subjected to reliability deterioration with operating time, two forms of the non-homogeneous Poisson processes, namely the Power-Law (PL) and the Log-Linear (LL) model, have found general acceptance in the literature. Inferential procedures, conditioned on the assumption of the PL or LL model, underestimate the overall uncertainty about a quantity of interest because the PL and LL models can provide different estimates of the quantity of interest, even when both of them adequately fit the observed data. In this paper, a composite estimation procedure, which uses the PL and LL models as competing models, is proposed in the framework of Bayesian statistics, thus allowing the uncertainty involved in model selection to be considered. A model-free approach is then proposed for incorporating technical information on the failure mechanism into the inferential procedure. Such an approach, which is based on two model-free quantities defined irrespectively of the functional form of the failure model, prevents that the prior information on the failure mechanism can improperly introduce prior probabilities on the adequacy of each model to fit the observed data. Finally, numerical applications are provided to illustrate the proposed procedures.     

Bayesian Inference on Multivariate Normal Covariance and Precision Matrices in a Star-Shaped Model with Missing Data

In this article, we study Bayesian estimation for the covariance matrix Σ and the precision matrix Ω (the inverse of the covariance matrix) in the star-shaped model with missing data. Based on a Cholesky-type decomposition of the precision matrix Ω = Ψ′Ψ, where Ψ is a lower triangular matrix with positive diagonal elements, we develop the Jeffreys prior and a reference prior for Ψ. We then introduce a class of priors for Ψ, which includes the invariant Haar measures, Jeffreys prior, and reference prior. The posterior properties are discussed and the closed-form expressions for Bayesian estimators for the covariance matrix Σ and the precision matrix Ω are derived under the Stein loss, entropy loss, and symmetric loss. Some simulation results are given for illustration.     

Identification of technical analysis patterns with smoothing splines for bitcoin prices

ABSTRACT

This research studies automatic price pattern search procedure for bitcoin cryptocurrency based on 1-min price data. To achieve this, search algorithm is proposed based on nonparametric regression method of smoothing splines. We investigate some well-known technical analysis patterns and construct algorithmic trading strategy to evaluate the effectiveness of the patterns. We found that method of smoothing splines for identifying the technical analysis patterns and that strategies based on certain technical analysis patterns yield returns that significantly exceed results of unconditional trading strategies.     

A Bayesian Approach to the Estimation of Expected Cell Counts by Using Log-Linear Models

ABSTRACT

In this article, Bayesian estimation of the expected cell counts for log-linear models is considered. The prior specified for log-linear parameters is used to determine a prior for expected cell counts, by means of the family and parameters of prior distributions. This approach is more cost-effective than working directly with cell counts because converting prior information into a prior distribution on the log-linear parameters is easier than that of on the expected cell counts. While proceeding from the prior on log-linear parameters to the prior of the expected cell counts, we faced with a singularity problem of variance matrix of the prior distribution, and added a new precision parameter to solve the problem. A numerical example is also given to illustrate the usage of the new parameter.     

A hypothesis testing procedure for the evaluation on the lifetime performance index of products with Burr XII distribution under progressive type I interval censoring

ABSTRACT

It is a very important topic these days to assessing the lifetime performance of products in manufacturing or service industries. Lifetime performance indices C_L is used to measure the larger-the-better type quality characteristics to evaluate the process performance for the improvement of quality and productivity. The lifetimes of products are assumed to have Burr XII distribution. The maximum likelihood estimator is used to estimate the lifetime performance index based on the progressive type I interval censored sample. The asymptotic distribution of this estimator is also developed. We use this estimator to build the new hypothesis testing algorithmic procedure with respect to a lower specification limit. Finally, two practical examples are given to illustrate the use of this testing algorithmic procedure to determine whether the process is capable.     

Analysis of over-dispersed count data with extra zeros using the Poisson log-skew-normal distribution

ABSTRACT

Mixed Poisson distributions are widely used in various applications of count data mainly when extra variation is present. This paper introduces an extension in terms of a mixed strategy to jointly deal with extra-Poisson variation and zero-inflated counts. In particular, we propose the Poisson log-skew-normal distribution which utilizes the log-skew-normal as a mixing prior and present its main properties. This is directly done through additional hierarchy level to the lognormal prior and includes the Poisson lognormal distribution as its special case. Two numerical methods are developed for the evaluation of associated likelihoods based on the Gauss–Hermite quadrature and the Lambert's W function. By conducting simulation studies, we show that the proposed distribution performs better than several commonly used distributions that allow for over-dispersion or zero inflation. The usefulness of the proposed distribution in empirical work is highlighted by the analysis of a real data set taken from health economics contexts.     

MOSUM monitoring for variance change in nonparametric regression models

ABSTRACT

This article considers the monitoring for variance change in nonparametric regression models. First, the local linear estimator of the regression function is given. A moving square cumulative sum procedure is proposed based on residuals of the estimator. And the asymptotic results of the statistic under the null hypothesis and the alternative hypothesis are obtained. Simulations and Application support our procedure.     

A Simple Method to Study Sensitivity of BMP's

In this article we measure the local or infinitesimal sensitivity of a kind of Bayes estimates which appear in bonus–malus systems. Bonus–malus premiums can be viewed as a functional depending on the prior distribution. To measure when small changes in the prior cause large changes in the premium we compute the norm of the Fréchet derivative and propose a simple procedure to decide if a bonus–malus premium is robust. As an application, an example where the risk has a Poisson distribution and its parameter follows a Gamma prior distribution is presented under the net and variance premium principles.     

Regression with an infinite number of observations applied to estimating the parameters of the stable distribution using the empirical characteristic function

ABSTRACT

The parameters of stable law parameters can be estimated using a regression based approach involving the empirical characteristic function. One approach is to use a fixed number of points for all parameters of the distribution to estimate the characteristic function. In this work the results are derived where all points in an interval is used to estimate the empirical characteristic function, thus least squares estimators of a linear function of the parameters, using an infinite number of observations. It was found that the procedure performs very good in small samples.     

Maximum likelihood estimation for germination-growth processes with application to neurotransmitters data

A maximum likelihood procedure is given for estimating parameters in a germination-growth process, based on germination times only or on both times and locations. The process is assumed to be driven by a Poisson process whose intensity is of known analytical form. The procedure is shown to perform well on simulated data with unnormalised gamma intensity and is also applied to data on release of neurotransmitter at a synapse.