Keywords: Rayleigh model, Bayes estimator, Progressive Type-II right censoring
scheme, ISELF, LLF, OSBPBL.
The Rayleigh distribution is considered as a useful life distribution. It plays an important role in statistics and operations research. Rayleigh model is applied in several areas such as health, agriculture, biology and physics. It often used in physics, related fields to model processes such as sound and light radiation, wave heights, as well as in communication theory to describe hourly median and instantaneous peak power of received radio signals. The model for frequency of different wind speeds over a year at wind turbine sites and daily average wind speed are considered under the Rayleigh model.
of a random variable given in the data, y, already observed. Dependent on the observed data, it is dened as the distribution of a new independent and identical future observation drawn from the same population. It is normally used in a Bayesian framework. Using the entire posterior distribution of the parameter(s), a probability distribution over an interval is derived as a posterior predictive distribution of the future observa- tion conditional on the observed data. To be more specic, by marginalizing the posterior distribution over the parameter(s), posterior predictive distribution of future observation can be derived. Arnold and Press , Al-Hussaini et al. , Al-Hussaini and Ahmad , Bolstad , and Bansal  have given a detailed discussion on prediction and predictive distri- bution under the Bayesian paradigm. We, now, present the derivation of posterior predictive distribution and Bayesian predictive interval.
Simulations can be helpful and an illuminating way to approach problems in Bayesiananalysis. Bayesian problems of updating estimates can be handled easily and straight forwardly with simulation. Because the distribution function of the Gumbel type II distribution can be expressed, as well as its inverse in closed form, the inversion method of simulation is straightforward to implement. The study was carried out for different values of (n, r) using τ ∊ 2.5 and υ = 0.5. Censoring rates are assumed to be 5% and 10%.
Seguro and Lambert (2000) proposed modified maximum likelihood method to calculate the parameters of the Weibull wind speed distribution for wind energy analysis. They recommended their method for use with wind data in frequency distribution format. Ng et al. (2012) used maximum likelihood estimators (MLEs), corrected MLEs, weighted MLEs, maximum product spacing estimators and least squares estimators to estimate three parameter Weibull distribution based on progressively Type-II right censored sample. In addition they proposed the use of a censored estimation method with one-step bias-correction to obtain estimates for iterative procedures.
In statistical analysis a lifetime or failure time data is wildly used in many areas. Then the lifetime can be defined by having time scale, time origin and an event, which noted as failure or death. In this study we are interested in censored lifetime data especially a progressively type-II censoreddata. The lifetime data is called censored when an information about an individual survival time is available, but the survival time is not known exactly. The progressively type-II censoreddata will be approached if the deletions are carried out at an observed failure time. The analysis of this type of data is important in many sciences like the biomedical, engineering, and social sciences. For more explanation, lifetime distribution methodology applications are mainly used to investigate the manufactured items' durability or to study human diseases and their treatment. The interest in analyzing such data is not new. In about 1970, dealing with this type of data had been expanded rapidly depending on methodology, theory, and fields of application. Since about 1980, software packages for lifetime data had been developed widely with a lot of new features and packages.
The aim of this study, as compared with above studies, is to present new group Lasso for selecting groups regarding the significant variables of left-censored regression. After that, novel Gibbs algorithm for sampling parameters with regard to the variable selection has been conducted. Simulation results as well as real dataanalysis indicated that the proposed new approach executed excellently in superior results as compared to the present approaches in the literatures.
In Chapter V, simulation results for biases of the point estimators, coverage probability and mean confidence lengths of the interval estimators, and true type-I error rate control, unadjusted and adjusted power of the test are extensively and intensively discussed. In addition, extensive and intensive dataanalysis was performed for a real-world data set, which represents the monthly water capacity of Shasta reservoir with code name USBR SHA that is operated and maintained by the U.S. Federal Bureau of Reclamation under The United States Department of the Interior, California, USA. The data set is available in the link
(2) where is the scale parameter of the distribution. A great deal of research has been done on estimating the parameter of Rayleigh distribution using both classical and Bayesian techniques, and a very good summary of this work can be found in Sinha and Howlader , Ariyawansa and Templeton , Howlader , Howlader and Hossian , Lalitha and Mishra , Abd Elfattah et al. , Hendi et. al , Dey and Das  and Dey . Statistical prediction was the earliest and most prevalent form of statistical inference. Prediction has its uses in variety of disciplines such as medicine, engineering and business. For more details on the history of statistical prediction analysis and examples, see Aitchison , Dunsmore , Aitchison and Dunsmore , Bain , Chhikara and Guttman , Geisser . In this paper we consider the estimation of the posterior predictive density of a future observation based on the current data and construct predictive intervals of a future observation.
The rest of the article is organized as follows: In Section 2 , competing risks model under progressive type-II censoring is described. Section 3 presents Bayesian one-sample prediction of future observables. The half-logistic distribution (HLD) is considered in Section 4 . In Section 5 , numerical computations and simulations are given. Concluding remarks are ﬁnally pre- sented in Section 6 .
Several authors have studied the Rayleigh distribution, Howlader and Hossain (1995) studied the problem of the estimation of the parameter and the reliability function with censoreddata and squared loss function. Dyer and Whisenand, (1973), provided the best linear unbiased estimator of σ based on complete sample, censored sample and selected order statistics. Bayesian estimation and prediction problems for the Rayleigh distribution based on doubly censored sample have been considered by Balakrishnan, (1989), Fernandez, (2000), Raqab and Madi, (2011). Bayesian estimation problems for the Rayleigh distribution based on progressivelycensored sample have been considered by Kim and Han, (2009), Raqab and Madi,(2011), and Dey and Dey, (2014).
4. BayesianAnalysis: Simulation with Laplace’s Demon
Based on some reviews in the area of approximating a Laplace distribution in the literature which has a very effective response for decades and also, in recent years based on Log-gamma estimation of parameters using different approach like Bayes estimate, MLE, Lindley, Newton Raphson’s method of optimization etc. (Akhtar, 2014; Bilal, Khan, Hasan & Khan, 2003; Khan & Bhat, 2002; Khan & Khan, 2013). Actually to find the posterior results summaries of such functions with their mean and variances, it is a very intricate case to handle, more especially when more covariate were involved as incorporate variables. In such cases, we use the Bayesian frame-work approach using the Metropolis-Hastings sampling algorithm in MCMC methods to solve and find the posterior result.
Environmental studies often include some observations falling below a level of de- tection (LOD). These values are reported as < LOD, where the LOD is a speciﬁed value for each observation. The values reported as < LOD are either left censored or interval censored 0 < x < LOD. Censored spatial data are often analyzed by ignoring spatial correlation and using one of many methods available for independent observa- tions (Helsel, 2005; Gibbons, 1995; Porter, Ward and Bell, 1988). Or, the censoring is ignored by substituting some function of the level of detection (e.g. LOD/2, LOD) for the censored values and then using a commonly available spatial method, e.g. variogram estimation and kriging. This substitution simpliﬁes the spatial analysis but results in biased estimates of the mean and variance (Helsel 2005), and, as we show later, a biased estimate of the overall spatial variability.
This paper develops Bayesiananalysis in the context of progressively Type II censoreddata from the two- parameter compound Rayleigh distribution. The maximum likelihood and Bayes estimates along with the associated posterior risks are derived for unknown reliability parameters under the balanced logarithmic loss and balanced general entropy loss functions. A practical example and simulation study have been considered to illustrate the proposed estimation methods and compare the performance of derived estimates based on maximum likelihood and Bayesian frameworks. The study indicates that Bayesian approach is more preferable over the maximum likelihood approach for estimation of the reliability parameters, while in Bayesian approach, a balance general entropy loss function can effectively be employed.
The main purpose of this work is to draw comparisons between the classi- cal maximum likelihood and the Bayesian estimators on the parameters, the survival function and hazard rate of the Weibull distribution when the dataunder consideration are right and interval censored. We have considered the survival data to follow Weibull distribution due to its adaptability in fitting time-to-failure of a very widespread multiplicity to multifaceted mechanisms in the field of life-testing and survival analysis.
The problem of censoring is more commonly encountered in life-time data because no experiment may remain sustained for an infinite time due to restrictions on the available time or cost for testing. There are different kinds of censoring schemes which include right, left and interval censoring, single or multiple censoring and type-I or type-II censoring. Type-I and type-II censoring schemes are most popular among them. Saleem et al. (2010) considered the Bayesiananalysis of the power function mixture distribution using type-I censoreddata. Shi and Yan (2010) discussed the empirical Bayes estimates of two-parameter exponential distribution under type-I censoring.
Abstract. Considering a number of Paired Comparison (PC) models existing in the literature, the posterior distribution for the parameters of the Rayleigh PC model is derived in this paper using the informative priors: Conjugate and Dirichlet. The values of the hyperparameters are elicited using prior predictive distribution. The preferences for the data of cigarette brands, such as Goldleaf (GL), Marlboro (ML), Dunhill (DH), and Benson & Hedges (BH), are collected based on university students' opinions. The posterior estimates of the parameters are obtained under the loss functions: Quadratic Loss Function (QLS), Weighted Loss Function (WLS), and Squared Error Loss Function (SELF) with their risks. The preference and predictive probabilities are investigated. The posterior probabilities are evaluated with respect to the hypotheses of two parameters comparison. In this respect, the graphs of marginal posterior distributions are presented, and appropriateness of the model is tested by Chi-Square.
In this section, the performance of the MLE and the proposed Bayes estimates of and is investigated through a simulation study based on both complete and various Type-I censored schemes. The simulation study is carried out for various values of the combination , , n, T ; for ( , ) ( 0 . 5 , 2 ) , ( 3 , 0 . 3 ) , ( 1 , 1 ) and ( 2 , 0 ) , the termination time T is assumed arbitrary to equal 8 and 10 . The hyper parameters A and B are assumed to equal one over the available sample mean and the minimum observed value, x ( 1 ) respectively. 1000
removed from the experiment. The work on progressive censoring has become popular in life-testing and reliability studies. Kim and Han ( 2009 ) studied the problem of estimating the scale parameter of the Rayleigh distribution under general progressive censoring. Krishna and Kumar ( 2011 ) discussed reliability estimation for the Lindley distribution with progressive type-II censoreddata. Lee et al. ( 2011 ) obtained a Bayes estimator under the Rayleigh distribution with a progressive type-II right censored sample. Raqab and Madi ( 2011 ) addressed inference for the generalized Rayleigh distribution based on progressivelycensoreddata. Azimi et al. ( 2012 ) considered the Bayesian estimation of the parameter and reliability function of Rayleigh distribution based on a progressively type-II censored sample. Rastogi and Tripathi ( 2012 ) studied parameter estimation of the Burr type XII distribution on the basis of a progressively type-II censored sample. Pradhan and Kundu ( 2009 ) considered the statistical inference of the unknown parameters of the generalized exponential distribution in presence of progressive censoring. A recent account on progressive censoring schemes can be obtained in the monograph by Balakrishnan and Aggarwala ( 2000 ) or in the excellent review article by Balakrishnan ( 2007 ).
In our simulation study, we chose different samples of size of 25, 50 and 100 to represent small, medium and large data set. The shape parameter is estimated for generalized Rayleigh distribution by using Bayesian method of estimation under various types of priors. The simulation study was conducted using R-software to examine and compare the performance of the estimates for different sample sizes by using various types of priors. The results are presented in tables from table 1 to 3 given below: