a SpringerOpen Journal
RE SE A RCH
Bayesian analysis of generalized log-Burr family with R Md Tanwir Akhtar* and Athar Ali Khan Abstract Log-Burr distribution is a generalization of logistic and extreme value distributions, which are important reliability models. In this paper, Bayesian approach is used to model reliability data for log-Burr model using analytic and simulation tools. Laplace approximation is implemented for approximating posterior densities of the parameters. Moreover, parallel simulation tools are also implemented using ‘LaplacesDemon’ package of R. Keywords: Bayesian; Log-Burr; Laplace approximation; Simulation; Laplace’s Demon; Posterior density
Introduction The log-Burr distribution is a generalization of two important reliability models, that is, logistic distribution and extreme value distribution. The non-Bayesian analysis of generalized log-Burr distribution is a very difficult task, whereas it can be a routine analysis when dealing in a Bayesian paradigm. In this paper, an attempt has been made with the following objectives: • To define a Bayesian model, that is, specification of likelihood and prior distribution. • To write down the R code for approximating posterior densities with Laplace approximation and simulation tools (R Core Team, 2013). • To illustrate numeric as well as graphic summaries of the posterior densities.
The probability density function of a parametric locationscale model for a random variable y on (−∞, ∞) with location parameter μ (−∞ < μ < ∞) and scale parameter σ (> 0) is given by 0002
F0 (y; μ, σ ) =
f0 (t)dt 0004 y R0 (y; μ, σ ) = 1 − f0 (t)dt = 1 − F0 (y; μ, σ ) −∞
The standardized random variable z = (y − μ)/σ clearly has density and reliability functions f0 (z) and R0 (z) respectively, and Equation (1) with μ = 0 and σ = 1 is called the standard form of the distribution. The lifetime distribution, that is, exponential, Weibull, all have the property that y = logt has a location scale distribution: the Weibull, log-normal, and log-logistic distribution for t correspond to extreme value, normal, and logistic distributions for y. The reliability functions for z = (y − μ)/σ on (−∞, ∞) are respectively, R0 (z) = exp(ez )
The log location-scale model
1 f (y; μ, σ ) = f0 σ
The corresponding distribution and reliability function for y are
R0 (z) = 1 − 0003(z) R0 (z) = (1 + e )
−∞< y 0) is a shape parameter. The special case, k = 1 gives the logistic distribution and k → ∞ gives the extreme value distribution. Since the generalized log-Burr family includes log-logistic and Weibull distributions, it allows discrimination between them. It is also a flexible model for fitting the lifetime data (e.g., Lawless 2003). Figure 1 shows probability density functions for log-Burr distributions with different values of k.
R0 (z; k) = (1 + ez /k)−k
and the corresponding reliability function
−∞ 0, α > 0. f (x) = π(x2 + α 2 ) The mean and variance of the half-Cauchy distribution do not exist, but its mode is equal to 0. The half-Cauchy distribution with scale α = 25 is a recommended, default, noninformative prior distribution for a scale parameter. At this scale α = 25, the density of half-Cauchy is nearly flat but not completely (see Figure 2), prior distributions
k→∞ k = 10 k=1 k = 0.5
Figure 1 Probability density function of log-Burr distribution for k = 0.5, 1, 10, ∞.
that are not completely flat provide enough information for the numerical approximation algorithm to continue to explore the target density, the posterior distribution. The inverse-gamma is often used as a noninformative prior distribution for scale parameter, however, this model creates problem for scale parameters near zero, Gelman and Hill (2007) recommend that, the uniform, or if more information is necessary the half-Cauchy is a better choice. Thus, in this paper, the half-Cauchy distribution with scale parameter α = 25 is used as a noninformative prior distribution.
The Laplace approximation Many simple Bayesian analyses based on noninformative prior distribution give similar results to standard nonBayesian approaches, for example, the posterior t-interval for the normal mean with unknown variance. The extent to which a noninformative prior distribution can be justified as an objective assumption depends on the amount of information available in the data; in the simple cases as the sample size n increases, the influence of the prior distribution on posterior inference decreases. These ideas, sometime referred to as asymptotic approximation theory because they refer to properties that hold in the limit as n becomes large. Thus, a remarkable method of asymptotic approximation is the Laplace approximation which accurately approximates the unimodal posterior moments and marginal posterior densities in many cases. In this section we introduce a brief, informal description of Laplace approximation method.
Akhtar and Khan SpringerPlus 2014, 3:185 http://www.springerplus.com/content/3/1/185
Page 3 of 10
Suppose −h(θ ) is a smooth, bounded unimodal function , with a maximum at θˆ , and θ is a scalar. By Laplace’s method (e.g., Tierney and Kadane 1986), the integral 0004
f (θ)exp[−nh(θ)] dθ
can be approximated by
0007 2π ˆI = f (θˆ ) σ exp[−nh(θˆ )] , n
∂2h ˆ ∂θ 2 θ
As presented in Mosteller and Wallace (1964), Laplace’s method is to expand about θˆ to obtain:
0005 0006 0004 (θ − θˆ )2 00030003 ˆ 0003 ˆ ˆ ˆ ˆ h (θ ) dθ. I ≈ f (θ ) exp −n h(θ)+(θ − θ )h (θ )+ 2
Recalling that h0003 (θˆ) = 0, we have 0005
0006 (θ − θˆ )2 00030003 ˆ exp −n h(θ) ˆ + ˆ h (θ) I ≈ f (θ) dθ 2
0004 ˆ 2 −n(θ − θ) ˆ ˆ = f (θ ) exp[ −nh(θ )] exp dθ 2σ 2 0007 2π σ exp[ −nh(θˆ )] . = f (θˆ ) n 0004
function(Model, parm, Data, Interval = 1e-06, Iterations = 100, Method= 'LBFGS', Samples= 1000, sir= TRUE, Stop.Tolerance= 1e-05) NULL
Intuitively, if exp[−nh(θ )] is very peaked about θˆ , then the integral can be well approximated by the behavior of the integrand near θˆ . More formally, it can be shown that 0002 0003 1 . I = Iˆ 1 + O n
To calculate moments of posterior distributions, we need to evaluate expressions such as:
E[ g(θ)] =
g(θ) exp[−nh(θ)] dθ , exp[−nh(θ)] dθ
For getting posterior modes of the log-posteriors, a number of optimization algorithms are implemented. This includes Levenberg-Marquardt (LM) algorithm which is default. However, we find that the Limited-Memory BFGS (L-BFGS) is a better alternative in Bayesian scenario. The limited-memory BFGS (Broyden-Fletcher-GoldfarbShanno) algorithm is a quasi-Newton optimization algorithm that compactly approximates the Hessian matrix. Rather than storing the dense Hessian matrix, L-BFGS stores only a few vectors that represent the approximation. It may be noted that Newton-Raphson is the last choice as it is very sensitive to the starting values, it creates problems when starting values are far from the targets, and calculating and inverting the Hessian matrix can be computationally expensive, although it is also implemented in LaplaceApproximation for the sake of completion. The main arguments of LaplaceApproximation can be seen by using the function args as
where exp[−nh(θ )] = L(θ y)p(θ ) (e.g., Tanner 1996).
Fitting of intercept model Fitting with LaplaceApproxomation
The Laplace approximation is a family of asymptotic techniques used to approximate integrals (Statisticat LLC 2013). It seems to accurately approximate uni-modal posterior moments and marginal posterior densities in many cases. Here, for fitting of linear regression model we use the function LaplaceApproximation which is an implementation of Laplace’s approximations of the integrals involved in the Bayesian analysis of the parameters in the modeling process. This function deterministically maximizes the logarithm of the unnormalized joint posterior density using one of the several optimization techniques. The aim of Laplace approximation is to estimate posterior mode and variance of each parameter.
First argument Model defines the model to be implemented, which contains specification of likelihood and prior. LaplaceApproximation passes two argument to the model function, parm and Data, and receives five arguments from the model function: LP (the logarithm of the unnormalized joined posterior density), Dev (the deviance), Monitor (the monitored variables), yhat (the variables for the posterior predictive checks), and parm, the vector of parameters, which may be constrained in the model function. The argument parm requires a vector of initial values equal in length to the number of parameters, and LaplaceApproximation will attempt to optimize these initial values for the parameters, where the optimized values are the posterior modes. The Data argument requires a listed data which must be include variable names and parameter names. The argument sir=TRUE stands for implementation of sampling importance resampling algorithm, which is a bootstrap procedure to draw independent sample with replacement from the posterior sample with unequal sampling probabilities. Contrary to sir of LearnBayes package, here proposal density is multivariate normal and not t. Locomotive controls data
Let us introduce a failure times dataset taken from Lawless (2003), so that all the concepts and computations will be discussed around that data. The same data were discussed by Schmee and Nelson (1977). This data set contains the number of thousand miles at which different locomotive
Akhtar and Khan SpringerPlus 2014, 3:185 http://www.springerplus.com/content/3/1/185
controls failed, in a life test involving 96 controls. The test was terminated after 135,000 miles, by which time 37 failures had occurred. The failure times for the 37 failed units are 22.5, 37.5, 46.0, 48.5, 51.5, 53.0, 54.5, 57.5, 66.5, 68.0, 69.5, 76.5, 77.0, 78.5, 80.0, 81.5, 82.0, 83.0, 84.0, 91.5, 93.5, 102.5, 107.0, 108.5, 112.5, 113.5, 116.0, 117.0, 118.5, 119.0, 120.0, 122.5, 123.0, 127.5, 131.0, 132.5, 134.0. In addition, there are 59 censoring times, all equal to 135.0. Creation of data
The function LaplaceApproximation requires data that is specified in a list. For locomotive controls data the logarithm of failTime will be the response variable. Since intercept is the only term in the model, a vector of 1’s is inserted into designed matrix X. Thus, J=1 indicates only column of 1’s in the matrix. failTime
- Burr Pdf Free Download Windows 10
- Burr Pdf free download. software
- Burr Pdf Free Download 64 Bit
- Pdf Download
- Burr PDF Free Download
Free PDF Download Books by Vivien Burr. This accessible, yet scholarly, textbook aims to introduce students to the area of social science theory and research that has come to be known as social constr.
Burr Pdf Free Download Windows 10
Are you facing problem to convert PDF documents to Word? You don’t know how to convert PDF to Word? Don’t Worry! there are many free software available to convert PDF to Word. We explain in detail about PDF to Word converter software full version. You can download these PDF to Word converter software full version. These software are easy to install and use. Our suggested software are free and power PDF to word converter. We had break the article in two parts Free PDF to Word converter for computer & Free online PDF to Word Converter.
Free PDF to Word Converter Software Full Version for Installing
Download Burr by Gore Vidal in PDF EPUB format complete free. Brief Summary of Book: Burr by Gore Vidal Here is a quick description and cover image of book Burr written. Burr is a portrait of perhaps the most complex and misunderstood of the Founding Fathers. In 1804, while serving as vice president, Aaron Burr fought a duel with his political nemesis, Alexander Hamilton, and killed him. In 1807, he was arrested, tried, and acquitted of treason. In 1833, Burr is newly married, an aging statesman considered a.
Below are some free full version PDF to Word converter for your computer:-
1.) Nemo PDF to Word Converter Software Full Version Free Download
Nemo PDF – Free Word to PDF converter full version to download for your computer. You can easily convert PDF to Word & RTF format. It supports password protected files. It can support PDF file crated with Adobe PDF 9.4 & its previous version. It can also support the PDF files created with other software.
It will preserve the 100% original file format. It will preserve the text, images, tables at their original location while converting PDF to Word. You can also select the pages in a PDF file for conversion. This tool can handle multiple PDF files for PDF to Word conversion.
You can install Nemo PDF to Word converter on Windows 7, Windows XP, Vista, 2003& 2000. I am using windows 8 and install this software on windows 8. It is working fine for me & you can also give this software a try if you have windows 8.
If you are facing any problem while converting documents or how to use this tool, read this guide for help.
Free PDF to Word Converter Software Full Version Download
2.) Hello PDF – Free PDF to Word Converter Full Version for Download
Hello PDF, another free full version PDF to Word converter for your computer. You can convert complete or partial PDF file to Word format. Only drawback of this software is this that it can support only up to PDF format 1.6. This means that PDf files created up to Adobe Acrobat 7 is supported by this software. Still it is good enough and it is in developing stage for support up to Adobe Acrobat 9.
For any help or problem using this software see the FAQ here.
Burr Pdf free download. software
3.) BullZip – Paid & Free PDF to Word Converter Software
Burr Pdf Free Download 64 Bit
BullZip, is another PDf to Word Converter software. BullZip comes in Free & Paid version. Free Version is with limited features while Paid version have many advance features.
Free Online PDF to Word Converter Software
Burr PDF Free Download
When we talk about free online PDF to Word converter, then we have option for BCL Research Converter, Very PDF & Nitro PDF. All these online PDF to Word converter are very powerful & convert you PDF to word in seconds. Only sometimes we face delay with Nitro PDF, about delivery of converted files through email. However, Nitro PDF is most popular service among all online PDF to Word converter software.