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Electronic Journal of Applied Statistical Analysis
The two-parameter Weibull distribution is a particularly difficult case, because it requires a two-dimensional joint prior distribution of the Weibull parameters.
The novelty of the procedure suggested here is that the prior information can be presented in the form of the interval assessment of the reliability function as opposed to that on the Weibull parameterswhich is generally easier to obtain. Based on this prior information, the procedure allows constructing the continuous joint prior distribution of Weibull parameters as well as the posterior estimates of the mean and standard deviation of the estimated reliability function or the CDF at any given value of the exposure variable.
A numeric example is discussed as an illustration. We additionally elaborate on a new parametric form of the prior distribution for the scale parameter of the exponential distribution. This distribution is not a Gamma as might intuitively be expected ; its mode is available in a closed form, and the mean is obtainable through a series approximation.
Article :. Date of Publication: 05 December DOI: Need Help?Issue 55, September Introduction to the Bayesian-Weibull Distribution. In this article, another school of thought in statistical analysis will be covered, namely Bayesian statistics. The premise of Bayesian statistics is to incorporate prior knowledge along with a given set of current observations in order to make statistical inferences. The prior information could come from operational or observational data, from previous comparable experiments or from engineering knowledge.
This type of analysis is particularly useful if there is a lack of current test data and when there is a strong prior understanding about the parameter of the assumed life model and a distribution can be used to model the parameter. By incorporating prior information about parameter sa posterior distribution for the parameter s can be obtained and inferences on the model parameters and their functions can be made.
This is expressed with the following posterior pdf :. The integral in Eqn. Generally, the integral in Eqn. It can be seen from Eqn. First, the idea of prior information does not exist in classical statistics. All inferences in classical statistics are based on the sample data. On the other hand, in the Bayesian framework, prior information constitutes the basis of the theory. Another difference is in the overall approach of making inferences and in their interpretation.
For example, in Bayesian analysis the parameters of the distribution to be ''fitted'' are the random variables. In reality, there is no distribution fitted to the data in the Bayesian case.
For instance, consider the case where a data set is obtained from a reliability test. This can be achieved by using the Bayes theorem. In this example, the range of values for the shape parameter is the prior distribution, which in this case is Uniform. By applying Eqn. So the analyst will end up with a distribution for the parameter rather than an estimate of the parameter as in classical statistics. This model considers prior knowledge on the beta parameter of the Weibull distribution when it is chosen to be fitted on a given set of data.
There are many practical applications for this model, particularly when dealing with small sample sizes and some prior knowledge for the shape parameter is available.Lecture 47A: MLE and Bayesian Estimation -3
For example, when a test is performed, there is often a good understanding about the behavior of the failure mode under investigation, primarily through historical data. At the same time, most reliability tests are performed on a limited number of samples. Under these conditions, it would be very useful to use this prior knowledge with the goal of making more accurate predictions.To browse Academia.
Skip to main content. Log In Sign Up. Ishfaq Ahmad Assistant Professor. Syed Shah. The properties of Bayes estimators of the parameters are studied under different loss functions.
A comprehensive simulation scheme is conducted using non-informative and informative priors. The loss functions and priors are compared for both shape and scale parameters through the posterior risks. It is named after the It is well known that in general if the proper information is Waloddi Weibull. As a result of flexibility in time-to-failure available, it is better to use the informative prior s than the of a very widespread diversity to versatile mechanisms, the non-informative prior ssee for example Berger  in this two-parameter Weibull distribution has been used quite respect.
The main objective of this paper is to use extensively in reliability and survival analysis particularly informative along with non-informative prior s to compute when the data are not censored. Much of the attractiveness the Bayes estimators and Bayes posterior risks under of the Weibull distribution is due to the wide variety of different loss functions.
It is also focus on the Bayes shapes which can assume by altering its parameters. The estimation for the Weibull distribution when scale and shape two-parameter Weibull distribution has one shape and one parameter s are unknown by using joint informative priors scale parameter.
The random variable X follows Weibull i. Lognormal-Inverted Gamma, Exponential-Gamma, distribution with the shape and scale parameters asGamma-Gamma and joint non-informative priors i.
Posterior distribution is the workbench of Bayesian statisticians. It is obtained when Even though, there is an enormous literature accessible on prior information is combined with the likelihood. Therefore estimation of the Weibull distribution using the frequentist prior information is necessary for Bayesian approach. The approach, but not much work has been done on the Bayesian prior information is a purely subjective assessment of an inference of the Weibull parameter s.
Wu  use the expert before any data has been observed. Therefore it is maximum likelihood method to obtain the point estimates of very difficult to consider the informative priors due to the parameters of the two parameters Weibull distribution. Also Berger  argues Zhang and Meeker  explains Bayesian methods for life that when information is not in compact form, the Bayesian testing planning with type-II censored data from the Weibull analysis using non-informative priors is single most suitable distribution when the Weibull shape parameter is given.
So here we utilize two non-informative priors Kundu  has explained the Bayesian inference of unknown along with three informative priors for our purposed model. Berger and Sun  have discussed the Bayesian is not possible to obtain the Bayes estimates in explicit form. The likelihood is defined in Section 2. Sections 3 misspecification of the shape parameter of Weibull deals with the posterior distributions using non-informative distribution by transforming the data to the case of priors while Section 4 elaborates the posterior distributions exponential distribution.
Ahmed et al. Section 6 estimate and Bayesian estimates for the Weibull distribution. The Zaidin et al.Scientific Research An Academic Publisher. In certain practical problems, actual measurements of a variable interest are costly or time-consuming, but the ranking items according to the variable is relatively easy with- out actual measurement. Under such circumstances McIntyre  proposed a sampling scheme called ranked-set sampling RSS which can be employed to gain more information than simple random sampling SRSwhile keeping the cost of, or the time constraint on, the sampling about the same.
In RSS; one first draws units at random from the population and partition them into m sets of m units. The m units in each set are ranked without making, an actual measurement.
The first set of m units are ranked and the smallest is selected for actual quantification. From the second set of m units, the unit ranked and the second smallest is measured, and so on. This method of selection continues until the unit ranked largest is measured from the m-th set.
If a large sample is required, then the procedure can be repeated r times to obtain a sample of size. These chosen elements are called ranked set sampling. The mathe- matical support and statistical theory was provided by Takahasi and Wakimoto . Dell and Clutter  studied theoretical aspects of this technique on the assumption of perfect and imperfect judgment ranking. Shaibu and Muttlak  used median and extreme ranked set sampling method for estimating the parameters of normal, expo- nential and gamma distributions.
Al-Omari et al. Islam et al. Ibrahim and Syam  used stratified median ranked set sampling method for estimating the population mean. Some research works have investigated ranked set sampling from a Bayesian point of view. Varian  and Zellner  introduced Bayesian estimation by using asymmetric loss functions. Al-Saleh and Muttlak  obtained the Bayesian estimates of the exponential distribution.
Sadek et al. Hassan  obtained the maximum likelihood estimator and Bayesian estimates of shape and scale parameters of the exponentiated exponential distribution based on SRS and RSS. For more research work on Bayesian one may refer to Mohammadi and Pazira Ghafoori et al. In Section 2, the preliminaries are discussed.
Simulation results and Conclusions are presented in Section 4 and 5 respectively. Let be a sequence of independent and identically distributed iid random variables from a Weibull distribution with probability density function pdf. In order to derive, and to measure the performance of an estimator we use squared error, loss function SEL see, Berger  and Linex loss function.
The Linex loss function for the parameter can be expressed as. The sign and magnitude of the shape parameter c indicate that the direction and degree of symmetry, respectively. When the value of c is zero, the Linex loss function is approximately squared error loss, when c is less than zero, the Linex loss function gives more weight to under-estimation against over-estimation, and it is reversed when c value is greater than zero.
The conjugate prior foris considered, whose pdf is given by. Ifthen becomes the Jeffreys prior. In this section, we derive the Bayes estimates of the Weibull parameter based on simple random sampling and maximum ranked set sampling with unequal samples by assuming that the shape parameter is known.
In each case, we use both conjugate and non-informative prior for the scale parameter.
Also, we use the symmetric loss function squared error loss and asymmetric loss function Linear-exponential, Linex to derive the corresponding Bayesian estimates.This article appears in the Life Data Analysis Reference book. The Bayesian methods presented next are for the 2-parameter Weibull distribution. Bayesian concepts were introduced in Parameter Estimation. This model considers prior knowledge on the shape parameter of the Weibull distribution when it is chosen to be fitted to a given set of data.
There are many practical applications for this model, particularly when dealing with small sample sizes and some prior knowledge for the shape parameter is available.
For example, when a test is performed, there is often a good understanding about the behavior of the failure mode under investigation, primarily through historical data. At the same time, most reliability tests are performed on a limited number of samples.
Under these conditions, it would be very useful to use this prior knowledge with the goal of making more accurate predictions.
A common approach for such scenarios is to use the 1-parameter Weibull distribution, but this approach is too deterministic, too absolute you may say and you would be right. Applying Bayes's rule on the 2-parameter Weibull distribution and assuming the prior distributions of and are independent, we obtain the following posterior pdf :.
In this model, is assumed to follow a noninformative prior distribution with the density function. This is called Jeffrey's prior, and is obtained by performing a logarithmic transformation on.
Specifically, since is always positive, we can assume that ln follows a uniform distribution, Applying Jeffrey's rule as given in Gelman et al. The prior distribution ofdenoted ascan be selected from the following distributions: normal, lognormal, exponential and uniform. The procedure of performing a Bayesian-Weibull analysis is as follows:. In other words, a distribution the posterior pdf is obtained, rather than a point estimate as in classical statistics i. Therefore, if a point estimate needs to be reported, a point of the posterior pdf needs to be calculated.
Typical points of the posterior distribution used are the mean expected value or median. The expected value of is obtained by:. Similarly, the expected value of is obtained by:. The median points are obtained by solving the following equations for and respectively:.
Of course, other points of the posterior distribution can be calculated as well. For example, one may want to calculate the 10th percentile of the joint posterior distribution w. The procedure for obtaining other points of the posterior distribution is similar to the one for obtaining the median values, where instead of 0. As explained in Parameter Estimationin Bayesian analysis, all the functions of the parameters are distributed.
In other words, a posterior distribution is obtained for functions such as reliability and failure rate, instead of point estimate as in classical statistics.
Therefore, in order to obtain a point estimate for these functions, a point on the posterior distributions needs to be calculated.The analysis of progressively censored data has received considerable attention in the last few years. In this paper, we consider the joint progressive censoring scheme for two populations. It is assumed that the lifetime distribution of the items from the two populations follows Weibull distribution with the same shape but different scale parameters.
Based on the joint progressive censoring scheme, first, we consider the maximum likelihood estimators of the unknown parameters whenever they exist. We provide the Bayesian inferences of the unknown parameters under a fairly general priors on the shape and scale parameters. The Bayes estimators and the associated credible intervals cannot be obtained in closed form, and we propose to use the importance sampling technique to compute the same. Further, we consider the problem when it is known a priori that the expected lifetime of one population is smaller than the other.
We provide the order-restricted classical and Bayesian inferences of the unknown parameters. Monte Carlo simulations are performed to observe the performances of the different estimators and the associated confidence and credible intervals.
One real data set has been analyzed for illustrative purpose. Download to read the full article text. Balakrishnan N. Springer, New York. Google Scholar. Berger J. Bayesian analysis of poly-Weibull model.
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