Bayesian Prediction Intervals Research Articles

Bayesian Prediction for Progressive Censored Data From the Weibull-Geometric Model

SYNOPTIC ABSRACTThis article is concerned with the problem of predicting future observables from the Weibull-geometric model based on progressively Type-II censored data. The Bayes point predictors and the Bayesian prediction intervals are obtained. The one and two-sample prediction techniques are considered. Numerical computations are given to illustrate the performance of the procedures.

Effects of temperature and PAR on the photosynthesis of Kappaphycus sp. (Solieriaceae, Rhodophyta) from Okinawa, Japan, at the northern limit of native Kappaphycus distribution in the western Pacific

:The successful cultivation of the Japanese Kappaphycus sp. (K. striatus auctorum japonicorum) in Okinawa, Japan, requires the detailed understanding of its physiological response to environmental factors. The photosynthetic performance of this native red alga was investigated under a variety of temperature and photosynthetic active radiation (PAR) conditions by employing methods based on pulse amplitude modulation–chlorophyll fluorometry and dissolved oxygen sensors. The net photosynthesis–irradiance (P–E) curve at 24°C revealed that the compensation (Ec) and saturation (Ek) irradiances were 26 [95% Bayesian prediction interval (BPI), 11–29 μmol photons m−2 s−1] and 140 (95% BPI, 98–192) μmol photons m−2 s−1, respectively. No inhibition in oxygenic evolution and quantum yield was observed at the highest PAR of 1000 μmol photons m−2 s−1. However, the ability of the seaweed to recover from photoinhibition was complicated following long-term PAR exposures at 18°C but not at 28°C. The Japanese Kappaphycus sp. showed photosynthetic optima at 17.4–29.1°C, derived from the Fv/Fm and gross photosynthesis–temperature models, respectively. These characteristic results were closely related to the depth of its habitat and its northern limit of distribution in Okinawa as influenced primarily by seawater temperature. Findings of this study should be useful to the development of culture systems adapted for subtropical waters, whether for commercial purpose or for conservation of natural communities.

Bayesian Analysis of a 3-Component Mixture of Rayleigh Distributions under Type-I Right Censoring Scheme

Since the last few decades, constructing flexible parametric classes of probability distributions has been the most popular approach in the Bayesian analysis. As compared to simple probability models, a mixture model of some suitable lifetime distributions may be more capable of capturing the heterogeneity of the nature. In this study, a 3-component mixture of Rayleigh distributions is investigated by considering type-I right censoring scheme to obtain data from a heterogeneous population. The closed form expressions for the Bayes estimators and posterior risks assuming the non-informative (uniform and Jeffreys’) priors under squared error loss function, precautionary loss function and DeGroot loss function are derived. The performance of the Bayes estimators for different sample sizes, test termination times and parametric values under different loss functions is investigated. The posterior predictive distribution for a future observation and the Bayesian predictive interval are constructed. In addition, the limiting expressions for the Bayes estimators and posterior risks are derived. Simulated data sets are used for the different comparisons and the model is finally illustrated using the real data.

Light and temperature effects on photosynthetic activity of Eucheuma denticulatum and Kappaphycus alvarezii (brown and green color morphotypes) from Sulawesi Utara, Indonesia

SUMMARYKnowledge concerning the effects of several abiotic factors on the physiology of carrageenophytes is essential both in ecological and economic standpoints, to ensure their sufficient supply for the sustainability of seaweed‐based industries. This paper presents the photosynthetic characteristics of farmed carrageenophytes, E ucheuma denticulatum and K appaphycus alvarezii [brown (BRN) and green (GRN) color morphotypes] from Sulawesi Utara (Sulawesi Island), Indonesia, as determined by examining their photosynthetic response across different temperatures and irradiances using dissolved oxygen measurements and pulse‐amplitude modulated fluorometer. Net photosynthesis–irradiance ( P – E ) curves at 26°C revealed that net photosynthetic rates of the three seaweeds gradually increased until the estimated saturation irradiances ( E k ) of 58 μmol photons m− 2 s−1 (49–68 μmol photons m− 2 s−1, 95% Bayesian prediction intervals; BPI) for E . denticulatum, and 158 and 143 μmol photons m− 2 s−1 (134–185 and 99–203 μmol photons m− 2 s−1, 95% BPI) for BRN and GRN K . alvarezii, respectively; and that no photoinhibition was observed at the highest irradiance of 1000 μmol photons m− 2 s−1. All seaweed samples exhibited photosynthetic tolerance to high PAR as shown by their recovery in maximum quantum yields (Fv / Fm ) following chronic exposures; as well as tolerance over a broad range of temperature, which is from 19 to 33°C for E . denticulatum, 20–29°C for BRN K . alvarezii, and 17–32°C for GRN K . alvarezii. Temperature responses of these carrageenophytes indicated that they were well‐adapted to the annual seawater temperatures in the cultivation site; however, they are also likely close to threshold levels for thermal inhibition, given the decline in Fv / Fm above 30°C.

Bayesian analysis of type-I right censored data using the 3-component mixture of Burr distributions

This study is concerned with the problem of estimating the parameters of a 3-component mixture of Burr distributions using type-I right censored data.The closed form expressions for the Bayes estimators and their posterior risks assuming the non-informative (uniform and Jeffreys’) priors under squared error loss function, precautionary loss function and DeGroot loss function are derived.Performance of the Bayes estimators for different sample sizes, test termination times(a point of time after which all other tests are terminated)and parametric values under different loss functions is investigated. The posterior predictive distribution for a future observation and the Bayesian predictive interval are constructed. In addition, the limiting expressions for the Bayes estimatorsand posterior risks are derived.Simulated data sets are designed for thecomparisons and the model is finally illustrated using the real data.

Thermal and PAR effects on the photosynthesis of Eucheuma denticulatum and Kappaphycus striatus (so-called Sacol strain) cultivated in shallow bottom of Bali, Indonesia

Photosynthetic responses to various temperature and PAR conditions of two carrageenophytes, Eucheuma denticulatum and Kappaphycus striatus (so-called Sacol strain), cultured in shallow areas of Bali, Indonesia, were examined using dissolved oxygen measurements and pulse–amplitude-modulated (PAM) fluorometry. Net photosynthesis–irradiance (P–E) curves at 26 °C revealed that the values of Pmax for E. denticulatum and K. striatus were 13.76 and 5.02 μg O2 gfw−1 min−1 (12.38–15.12 and 4.55–5.50 μg O2 gfw−1 min−1, 95 % Bayesian prediction interval (BPI)), respectively. Photoinhibition was not observed even at 1000 μmol photons m−2 s−1. Gross photosynthesis and respiration characteristics over a range of temperatures (8–36 °C) revealed optimum temperature for photosynthesis to be 31.1 °C for E. denticulatum, and 31.6 °C for K. striatus. These characteristics of photosynthesis indicate that both farmed seaweeds are well adapted to the light and temperature conditions of the cultivation site. However, studies on longer timescales for photochemical efficiency, photosynthesis, and respiration are still needed to determine their limits of tolerance. We also hypothesize that higher values of photosynthetic parameters for E. denticulatum suggest that this species is relatively superior in productivity under optimal conditions, conditional on the farming method.

Bayesian estimation of finite3-component mixture of Burr Type-XII distributions assuming Type-I right censoring scheme

As compared to simple models, the mixture models of underlying lifetime distributions are intuitively more appropriate and appealing to model the heterogeneous nature of process. This study focuses on the problem of estimating the parameters of a newly developed 3-component mixture of Burr Type-XII distributions using Type-I right censored data. Firstly, considering a Bayesian structure, some mathematical properties of a 3-component mixture of Burr Type-XII distributions are discussed. These mathematical properties include Bayes estimators and posterior risks for the unknown component and proportion parameters using the non-informative and the informative priors under squared error loss function, precautionary loss function and DeGroot loss function. Secondly, in case when no or little prior information is available, elicitation of hyperparameters is given. Also, the posterior predictive distribution for a future observation and the Bayesian predictive interval are constructed. Moreover, the limiting expressions for the Bayes estimators and posterior risks are derived. In addition, the performance of the Bayes estimators for different sample sizes, test termination times and parametric values under different loss functions is investigated. Finally, simulated datasets are designed for the different comparisons and the model is illustrated using the real data.

Statistical inference for the Gompertz distribution based on Type-II progressively hybrid censored data

ABSTRACTIn this article, the statistical inference for the Gompertz distribution based on Type-II progressively hybrid censored data is discussed. The estimation of the parameters for Gompertz distribution is obtained using maximum likelihood method (MLE) and Bayesian method under three different loss functions. We also proved the existence and uniqueness of the MLE. The one-sample Bayesian prediction intervals are obtained. The work is done for different values of the parameters. We apply the Monto Carlo simulation to compare the proposed methods, also an example is discussed to construct the Prediction intervals.

Bayesian Prediction Bounds for the Exponential-Type Distribution Based on Ordered Ranked Set Sampling

Abstract We suggest a ranked set sample method to improve Bayesian prediction intervals. The paper deals with the Bayesian prediction intervals in the context of an ordered ranked set sample from a certain class of exponential-type distributions. A proper general prior density function is used and the predictive cumulative function is obtained in the two-sample case. The special case of linear exponential distributed observations is considered and completed with numerical results.

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.

Bayesian Prediction Based on Type-I Hybrid Censored Data from a General Class of Distributions

One and two-sample Bayesian prediction intervals based on Type-I hybrid censored for a general class of distribution 1-<i>F</i>(x)=[<i>ah</i> (x)+<i>b</i>]<i>^c</i> are obtained. For the illustration of the developed results, the inverse Weibull distribution with two unknown parameters and the inverted exponential distribution are used as examples. Using the importance sampling technique and Markov Chain Monte Carlo (MCMC) to compute the approximation predictive survival functions. Finally, a real life data set and a generated data set are used to illustrate the results derived here.

Statistical inference for the generalized inverted exponential distribution based on upper record values

In this paper, non-Bayesian and Bayesian estimators for the unknown parameters are obtained based on records from the generalized inverted exponential distribution. Bayes’ estimators of the unknown parameters are obtained under symmetric and asymmetric loss functions using gamma priors on both the shape and the scale parameters. The Bayes estimators cannot be obtained in explicit forms. So we propose Markov Chain Monte Carlo (MCMC) techniques to generate samples from the posterior distributions and in turn computing the Bayes estimators. We have also derived the Bayes interval of this distribution and discussed both frequentist and the Bayesian prediction intervals of the future record values based on the observed record values. Monte Carlo simulations are performed to compare the performances of the proposed methods, and a data set has been analyzed for illustrative purposes.

Non-linear Bayesian prediction of generalized order statistics for liftime models

In this paper, we obtain Bayesian prediction intervals as well as Bayes predictive estimators under square error loss for generalized order statistics when the distribution of the underlying population belongs to a family which includes several important distributions.

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.

Bayesian Prediction of Future Generalized Order Statistics from a Class of Finite Mixture Distributions

This article is concerned with the problem of prediction for the future generalized order statistics from a mixture of two general components based on doubly type II censored sample. We consider the one sample prediction and two sample prediction techniques. Bayesian prediction intervals for the median of future sample of generalized order statistics having odd and even sizes are obtained. Our results are specialized to ordinary order statistics and ordinary upper record values. A mixture of two Gompertz components model is given as an application. Numerical computations are given to illustrate the procedures.

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.

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.

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.