Bayesian Interval Research Articles

Bayesian prediction of order statistics with fixed and random sample sizes based on k-record values from Pareto distribution

ABSTRACTIn this paper, the two-parameter Pareto distribution is considered and the problem of prediction of order statistics from a future sample and that of its geometric mean are discussed. The Bayesian approach is applied to construct predictors based on observed k-record values for the cases when the future sample size is fixed and when it is random. Several Bayesian prediction intervals are derived. Finally, the results of a simulation study and a numerical example are presented for illustrating all the inferential procedures developed here.

A nonparametric Bayesian prediction interval for a finite population mean

ABSTRACTGiven a sample from a finite population, we provide a nonparametric Bayesian prediction interval for a finite population mean when a standard normal assumption may be tenuous. We will do so using a Dirichlet process (DP), a nonparametric Bayesian procedure which is currently receiving much attention. An asymptotic Bayesian prediction interval is well known but it does not incorporate all the features of the DP. We show how to compute the exact prediction interval under the full Bayesian DP model. However, under the DP, when the population size is much larger than the sample size, the computational task becomes expensive. Therefore, for simplicity one might still want to consider useful and accurate approximations to the prediction interval. For this purpose, we provide a Bayesian procedure which approximates the distribution using the exchangeability property (correlation) of the DP together with normality. We compare the exact interval and our approximate interval with three standard intervals, namely the design-based interval under simple random sampling, an empirical Bayes interval and a moment-based interval which uses the mean and variance under the DP. However, these latter three intervals do not fully utilize the posterior distribution of the finite population mean under the DP. Using several numerical examples and a simulation study we show that our approximate Bayesian interval is a good competitor to the exact Bayesian interval for different combinations of sample sizes and population sizes.

QTL mapping in multiple populations and development stages reveals dynamic quantitative trait loci for fruit size in cucumbers of different market classes

QTL analysis in multi-development stages with different QTL models identified 12 consensus QTLs underlying fruit elongation and radial growth presenting a dynamic view of genetic control of cucumber fruit development. Fruit size is an important quality trait in cucumber (Cucumis sativus L.) of different market classes. However, the genetic and molecular basis of fruit size variations in cucumber is not well understood. In this study, we conducted QTL mapping of fruit size in cucumber using F2, F2-derived F3 families and recombinant inbred lines (RILs) from a cross between two inbred lines Gy14 (North American picking cucumber) and 9930 (North China fresh market cucumber). Phenotypic data of fruit length and diameter were collected at three development stages (anthesis, immature and mature fruits) in six environments over 4years. QTL analysis was performed with three QTL models including composite interval mapping (CIM), Bayesian interval mapping (BIM), and multiple QTL mapping (MQM). Twenty-nine consistent and distinct QTLs were detected for nine traits from multiple mapping populations and QTL models. Synthesis of information from available fruit size QTLs allowed establishment of 12 consensus QTLs underlying fruit elongation and radial growth, which presented a dynamic view of genetic control of cucumber fruit development. Results from this study highlighted the benefits of QTL analysis with multiple QTL models and different mapping populations in improving the power of QTL detection. Discussion was presented in the context of domestication and diversifying selection of fruit length and diameter, marker-assisted selection of fruit size, as well as identification of candidate genes for fruit size QTLs in cucumber.

Comparison of Bayesian Credible Intervals to Frequentist Confidence Intervals

Frequentist confidence intervals were compared with Bayesian credible intervals under a variety of scenarios to determine when Bayesian credible intervals outperform frequentist confidence intervals. Results indicated that Bayesian interval estimation frequently produces results with precision greater than or equal to the frequentist method.

Measurements of neutrino oscillation in appearance and disappearance channels by the T2K experiment with6.6×1020protons on target

We report on measurements of neutrino oscillation using data from the T2K long-baseline neutrino experiment collected between 2010 and 2013. In an analysis of muon neutrino disappearance alone, we find the following estimates and 68% confidence intervals for the two possible mass hierarchies: Normal Hierarchy: $\sin^2\theta_{23}=0.514^{+0.055}_{-0.056}$ and $\Delta m^2_{32}=(2.51\pm0.10)\times 10^{-3}$ eV$^2$/c$^4$ Inverted Hierarchy: $\sin^2\theta_{23}=0.511\pm0.055$ and $\Delta m^2_{13}=(2.48\pm0.10)\times 10^{-3}$ eV$^2$/c$^4$ The analysis accounts for multi-nucleon mechanisms in neutrino interactions which were found to introduce negligible bias. We describe our first analyses that combine measurements of muon neutrino disappearance and electron neutrino appearance to estimate four oscillation parameters and the mass hierarchy. Frequentist and Bayesian intervals are presented for combinations of these parameters, with and without including recent reactor measurements. At 90% confidence level and including reactor measurements, we exclude the region: $\delta_{CP}=[0.15,0.83]\pi$ for normal hierarchy and $\delta_{CP}=[-0.08,1.09]\pi$ for inverted hierarchy. The T2K and reactor data weakly favor the normal hierarchy with a Bayes Factor of 2.2. The most probable values and 68% 1D credible intervals for the other oscillation parameters, when reactor data are included, are: $\sin^2\theta_{23}=0.528^{+0.055}_{-0.038}$ and $|\Delta m^2_{32}|=(2.51\pm0.11)\times 10^{-3}$ eV$^2$/c$^4$.

Genome-wide QTL mapping of saltwater tolerance in sibling species of Anopheles (malaria vector) mosquitoes

Although freshwater (FW) is the ancestral habitat for larval mosquitoes, multiple species independently evolved the ability to survive in saltwater (SW). Here, we use quantitative trait locus (QTL) mapping to investigate the genetic architecture of osmoregulation in Anopheles mosquitoes, vectors of human malaria. We analyzed 1134 backcross progeny from a cross between the obligate FW species An. coluzzii, and its closely related euryhaline sibling species An. merus. Tests of 2387 markers with Bayesian interval mapping and machine learning (random forests) yielded six genomic regions associated with SW tolerance. Overlap in QTL regions from both approaches enhances confidence in QTL identification. Evidence exists for synergistic as well as disruptive epistasis among loci. Intriguingly, one QTL region containing ion transporters spans the 2Rop chromosomal inversion that distinguishes these species. Rather than a simple trait controlled by one or a few loci, our data are most consistent with a complex, polygenic mode of inheritance.

A Comparison of Bayesian and Frequentist Interval Estimators in Regression that Utilize Uncertain Prior Information

SummaryConsider a linear regression model with independent normally distributed errors. Suppose that the scalar parameter of interest is a specified linear combination of the components of the regression parameter vector. Also suppose that we have uncertain prior information that a distinct specified linear combination of these components takes the value zero. We provide succinct and informative descriptions of interval estimators for the parameter of interest using the new concepts of scaled offset and scaled half‐length. We describe the Bayesian equi‐tailed and shortest credible intervals for the parameter of interest that result from a prior density for the parameter about which we have uncertain prior information that is a mixture of a rectangular ‘slab’ and a Dirac delta function ‘spike’, combined with noninformative prior densities for the other parameters of the model. This prior belongs to the class of ‘slab and spike’ priors, which have been used for Bayesian variable selection. We compare these credible intervals with Kabaila and Giri's frequentist confidence interval for the parameter of interest that utilizes this uncertain prior information. We show that these frequentist and Bayesian interval estimators depend on the data in very different ways. We also consider some close variants of this prior distribution that lead to Bayesian and frequentist interval estimators with greater similarity. Nonetheless, as we show, substantial differences between these interval estimators remain.

One-sample Bayesian prediction intervals based on progressively type-II censored data from the half-logistic distribution under progressive stress model

Based on progressively type-II censored sample, we discuss Bayesian interval prediction under progressive stress accelerated life tests. The lifetime of a unit under use condition stress is assumed to follow the half-logistic distribution with a scale parameter satisfying the inverse power law. Prediction bounds of future order statistics are obtained. A simulation study is performed and numerical computations are carried out, based on two different progressive censoring schemes. The coverage probabilities and average interval lengths of the confidence intervals are computed via a Monte Carlo simulation.

Comparison of the Frequentist MATA Confidence Interval with Bayesian Model-Averaged Confidence Intervals

Model averaging is a technique used to account for model uncertainty, in both Bayesian and frequentist multimodel inferences. In this paper, we compare the performance of model-averaged Bayesian credible intervals and frequentist confidence intervals. Frequentist intervals are constructed according to the model-averaged tail area (MATA) methodology. Differences between the Bayesian and frequentist methods are illustrated through an example involving cloud seeding. The coverage performance and interval width of each technique are then studied using simulation. A frequentist MATA interval performs best in the normal linear setting, while Bayesian credible intervals yield the best coverage performance in a lognormal setting. The use of a data-dependent prior probability for models improved the coverage of the model-averaged Bayesian interval, relative to that using uniform model prior probabilities. Data-dependent model prior probabilities are philosophically controversial in Bayesian statistics, and our results suggest that their use is beneficial when model averaging.

An expert opinion elicitation method based on binary search and Bayesian intervals

This paper proposes an indirect method to elicit experts' opinions. The approach is an attempt to balance accuracy and time consumption (efficiency) in elicitation exercises. Each step of the proposed method involves two types of questions: first, the expert is asked about which of two subintervals most likely holds the unknown of interest (a procedure similar to the binary search algorithm); then, the expert assigns a credibility level on which he bases his decision. A set of Bayesian intervals of the unknown is thereby obtained, and the probability distribution underlying expert's beliefs is approached. The usefulness of the method is analysed by means of: 1) a comparison with established methods from the literature (e.g., the direct and bisection methods); 2) its application to a real world case within the context of probabilistic risk analysis.

BAYESIAN TWO-SAMPLE PREDICTION OF THE GENERALIZED PARETO DISTRIBUTION WITH FIXED AND RANDOM SAMPLE SIZES BASED ON GENERALIZED ORDER STATISTICS

Bayesian predictive intervals for future observations from a future sample from the generalized Pareto distribution (GPD) based on generalized order statistics (GOS) are obtained when the shape parameter θ is unknown. We consider two cases: (i) fixed sample size (FSS), and (ii) random sample size (RSS).Some closed forms for the Bayesian predictive functions are obtained. Finally examples are calculated for the lower and the upper bounds of the future observations in cases when the future sample is ordinary order statistics (OOS), record values and progressive type II censoring with different values for the scale parameter σ.

On Bayes predictor of times to failure of Type-II progressively censored sample

The purpose of this paper is Bayesian prediction, robust Bayesian prediction and two-sided predictive interval for both one-sample and two-sample prediction problems. In one-sample prediction problem, we discuss the prediction of the life-lengths (j=1, 2, … , ri; i=1, 2, … , m) of all censored units in all m stages of censoring, where denotes the jth-order statistics out of ri removed units at stage i. While in two-sample prediction problem, we discuss the prediction of the life-lengths (T1, … , Ts) which is a second independent ordered sample of size s of future observation from the same population of Y, where Y is the progressively Type-II right-censored order statistics. Finally, we conduct a simulation study, and consider an illustrative example for our proposed predictors.

Comparison of Interval Estimators of Pr(X < Y) in the Two-parameter Exponential Distribution

We consider confidence intervals for the stress–strength reliability Pr(X< Y) in the two-parameter exponential distribution. We have derived the Bayesian highest posterior density interval using non-informative prior distributions. We have compared its performance with the intervals based on the generalized pivot variable intervals in terms of their coverage probabilities and expected lengths. Our simulation study shows that the Bayesian interval performs better according to the criteria used, especially when the sample sizes are very small. An example is given.

Bayesian inference and prediction for modified Weibull distribution under generalized order statistics

Based on generalized order statistics (gos), Bayes estimators have been obtained for parameters, reliability and hazard function from modified Weibull distribution. The Bayes estimators are studied under squared error loss and LINEX error loss functions. The Bayesian prediction intervals for gos from the modified Weibull distribution are obtained in one sample case. Finally, a real life data set and simulation data are used to illustrate the theoretical results.

Exact Likelihood Inference for k Exponential Populations Under Joint Type-II Censoring

In this paper, when a jointly Type-II censored sample arising from k independent exponential populations is available, the conditional MLEs of the k exponential mean parameters are derived. The moment generating functions and the exact densities of these MLEs are obtained using which exact confidence intervals are developed for the parameters. Moreover, approximate confidence intervals based on the asymptotic normality of the MLEs and credible confidence regions from a Bayesian viewpoint are also discussed. An empirical comparison of the exact, approximate, bootstrap, and Bayesian intervals is also made in terms of coverage probabilities. Finally, an example is presented in order to illustrate all the methods of inference developed here.

Bayesian Interval Estimation for the Difference in TPRs and FPRs of Two Diagnostic Tests with Unverified Negatives

We derive Bayesian interval estimators for the differences in the true positive rates and false positive rates of two dichotomous diagnostic tests applied to the members of two distinct populations. The populations have varying disease prevalences with unverified negatives. We compare the performance of the Bayesian credible interval to the Wald interval using Monte Carlo simulation for a spectrum of different TPRs, FPRs, and sample sizes. For the case of a low TPR and low FPR, we found that a Bayesian credible interval with relatively noninformative priors performed well. We obtain similar interval comparison results for the cases of a high TPR and high FPR, a high TPR and low FPR, and of a high TPR and mixed FPR after incorporating mildly informative priors.

Exact Likelihood Inference for k Exponential Populations Under Joint Progressive Type-II Censoring

Comparative lifetime experiments are of great importance when the interest is in ascertaining the relative merits of k competing products with regard to their reliability. In this paper, when a joint progressively Type-II censored sample arising from k independent exponential populations is available, the conditional MLEs of the k exponential mean parameters are derived. Their conditional moment generating functions and exact densities are obtained, using which exact confidence intervals are developed for the parameters. Moreover, approximate confidence intervals based on the asymptotic normality of the MLEs and credible confidence regions from a Bayesian viewpoint are discussed. An empirical evaluation of the exact, approximate, bootstrap, and Bayesian intervals is also made in terms of coverage probabilities and average widths. Finally, an example is presented in order to illustrate all the methods of inference developed here.

Interval Prediction of Future Order Statistics in Two-Component Mixture Inverse Lomax Model: A Bayesian Approach

SYNOPTIC ABSTRACTThe inverse Lomax distribution is studied as a probability distribution following a time-to-failure system. This article is based on the study of Bayesian prediction intervals for certain future ordered observations in samples from the two-component inverse Lomax mixture distribution. Specifically, the lower and upper prediction bounds are calculated for five-number summary, i.e., minimum, first quartile, median, third quartile, and maximum observation in the future sample of specified size. The goal is achieved by the application of the posterior predictive distribution and Type-I right censoring scheme. Real-data application is provided for the illustration of algebraic results.

Bayesian Accelerated Life Testing under Competing Weibull Causes of Failure

Consider a J-component series system which is put on Accelerated Life Test (ALT) involving K stress variables. First, a general formulation of ALT is provided for log-location-scale family of distributions. A general stress translation function of location parameter of the component log-lifetime distribution is proposed which can accommodate standard ones like Arrhenius, power-rule, log-linear model, etc., as special cases. Later, the component lives are assumed to be independent Weibull random variables with a common shape parameter. A full Bayesian methodology is then developed by letting only the scale parameters of the Weibull component lives depend on the stress variables through the general stress translation function. Priors on all the parameters, namely the stress coefficients and the Weibull shape parameter, are assumed to be log-concave and independent of each other. This assumption is to facilitate Gibbs sampling from the joint posterior. The samples thus generated from the joint posterior is then used to obtain the Bayesian point and interval estimates of the system reliability at usage condition.

Interval estimations in metrology

This paper investigates interval estimation for a measurand that is known to be positive. Both the Neyman and Bayesian procedures are considered and the difference between the two, not always perceived, is discussed in detail. A solution is proposed to a paradox originated by the frequentist assessment of the long-run success rate of Bayesian intervals.