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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

y−μ σ

0004

F0 (y; μ, σ ) =

y

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

extreme value

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.

0.1

R0 (z; k) = (1 + ez /k)−k

0.2

and the corresponding reliability function

0.3

−∞ 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

0.2 0.0

0.1

f(z)

0.3

k→∞ k = 10 k=1 k = 0.5

−4

−2

0 z

2

4

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

I=

f (θ)exp[−nh(θ)] dθ

can be approximated by

0007 2π ˆI = f (θˆ ) σ exp[−nh(θˆ )] , n

where

σ=

∂2h ˆ ∂θ 2 θ

−1/2 .

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

(2)

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

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