Computing the Bayesian estimators of the Birnbaum-Saunders (BS) distribution.

Description

Computing the Bayesian estimators of the BS distribution based on approximated Jeffreys prior proposed by Achcar (1993). The approximated Jeffreys piors is \pi_{j}(\alpha,\beta)\propto\frac{1}{\alpha\beta}\sqrt{\frac{1}{\alpha^2}+\frac{1}{4}}.

Usage

Jeffreysbs(x, CI = 0.95, M0 = 800, M = 1000)

Arguments

Value

A list including summary statistics of a Gibbs sampler for the Bayesian inference including point estimation for the parameter, its standard error, and the corresponding 100(1-\alpha)\% credible interval, goodness-of-fit measures, asymptotic 100(1-\alpha)\% confidence interval (CI) and corresponding standard errors, and Fisher information matix.

Author(s)

Mahdi Teimouri

References

J. A. Achcar 1993. Inferences for the Birnbaum-Saunders fatigue life model using Bayesian methods, Computational Statistics \& Data Analysis, 15 (4), 367-380.

Examples

data(fatigue)
x <- fatigue
Jeffreysbs(x, CI = 0.95, M0 = 800, M = 1000)

Bone mineral content data

Description

The mineral density of three dominant and nondominant of bones measured in g/cm^2 johnson1999.

Usage

data(bone)

Format

A text file with 6 columns.

References

R. A. ArnoldJohnson and D. W. Wichern 1999. Applied Multivariate Analysis, Prentice-Hall, New Jersey.

Examples

data(bone)

Computing the Bayesian estimators of the Birnbaum-Saunders (BS) distribution.

Description

Computing the Bayesian estimators of the BS distribution using conjugate prior, that is, conjugate and reference priors. The probability density function of generalized inverse Gaussian (GIG) distribution is given by good1953population

f_{{GIG}}(x|\lambda,\chi,\psi)=\frac{1}{2{K}_{\lambda}(\sqrt{\psi \chi})}\Bigl(\frac{\psi}{\lambda}\Bigr)^{\lambda/2}x^{\lambda-1}\exp\biggl\{-\frac{\chi}{2x}-\frac{\psi x}{2}\biggr\},

where x>0, -\infty<\lambda <+\infty, \psi>0, and \chi>0 are parameters of this family. The pdf of a inverse gamma (IG) distribution denoted as {IG}(\gamma,\theta) is given by

f_{{IG}}(x|\gamma,\theta)=\frac{\theta^{\gamma} x^{-\gamma-1}}{\Gamma(\gamma)}\exp\left\{-\frac{\theta}{x}\right\},

where x>0, \gamma>0, and \theta>0 are the shape and scale parameters, respectively.

Usage

conjugatebs(x,gamma0=1,theta0=1,lambda0=0.001,chi0=0.001,psi0=0.001,CI=0.95,M0=800,M=1000)

Arguments

Value

A list including summary statistics of a Gibbs sampler for Bayesian inference including point estimation for the parameter, its standard error, and the corresponding 100(1-\alpha)\% credible interval, goodness-of-fit measures, asymptotic 100(1-\alpha)\% confidence interval (CI) and corresponding standard errors, and Fisher information matix.

Author(s)

Mahdi Teimouri

References

I. J. Good 1953. The population frequencies of species and the estimation of population parameters. Biometrika, 40(3-4):237-264.

Examples

data(fatigue)
x <- fatigue
conjugatebs(x,gamma0=1,theta0=1,lambda0=0.001,chi0=0.001,psi0=0.001,CI=0.95,M0=800,M=1000)

Fatigue data

Description

A set of 101 observations obtained by Birnbaum and Saunders(1969) from fatigue life of 6061-T6 aluminum coupons cut parallel to the direction of rolling and oscillated at 18 cycles per second (cps).

Usage

data(fatigue)

Format

A text file with 1 column.

References

Z. W. Birnbaum and S. C. Saunders 1969. Estimation for a family of life distributions with applications to fatigue. Journal of Applied Probability, 328-347.

Examples

data(fatigue)

Computing the maximum likelihood (ML) estimator for the generalized Birnbaum-Saunders (GBS) distribution.

Description

Computing the ML estimator for the GBS distribution proposed by Owen (2006) whose density function is given by

f_{{GBS}}(t|\alpha,\beta,\nu)=\frac{(1-\nu)t +\nu \beta}{\sqrt{2\pi}\alpha \sqrt{\beta}t^{\nu+1}} \exp\left\{-\frac{(t-\beta)^2}{2\alpha^2\beta t^{2\nu}}\right\},

where t>0. The parameters of GBS distribution are \alpha>0, \beta>0, and 0<\nu<1. For \nu=0.5, the GBS distribution turns into the ordinary Birnbaum-Saunders distribution.

Usage

mlebs(x, start, method = "Nelder-Mead", CI = 0.95)

Arguments

Value

A list including the ML estimator, goodness-of-fit measures, asymptotic 100(1-\alpha)\% confidence interval (CI) and corresponding standard errors, and Fisher information matix.

Author(s)

Mahdi Teimouri

Examples

data(fatigue)
x <- fatigue
mlebs(x, start = c(1, 29), method = "Nelder-Mead", CI = 0.95)

Plasma survival data

Description

The plasma survival data contains the Survival times of plasma cell myeloma for 112 patients, see Carbone et al. (1967).

Usage

data(plasma)

Format

A text file with 4 columns.

References

P. P. Carbone, L. E. Kellerhouse, and E. A. Gehan 1967. Plasmacytic myeloma: A study of the relationship of survival to various clinical manifestations and anomalous protein type in 112 patients. The American Journal of Medicine, 42 (6), 937-48.

Examples

data(plasma)

Simulating from Birnbaum-Saunders (BS) distribution.

Description

Simulating from BS distribution whose density function is given by

f_{{BS}}(t|\alpha,\beta)=\frac{0.5t +0.5 \beta}{\sqrt{2\pi}\alpha \sqrt{\beta}t^{\frac{3}{2}}} \exp\left\{-\frac{(t-\beta)^2}{2\alpha^2\beta t}\right\},

where t>0. The parameters of GBS distribution are \alpha>0 and \beta>0.

Usage

rbs(n, alpha, beta)

Arguments

Value

A vector of n realizations from distribution.

Author(s)

Mahdi Teimouri

Examples

rbs(n = 100, alpha = 1, beta = 2)

Computing the Bayesian estimators of the Birnbaum-Saunders (BS) distribution.

Description

Computing the Bayesian estimators of the BS distribution using reference prior proposed by Berger and Bernardo(1989). The joint distribution of the priors is \pi(\alpha,\beta)=1/(\alpha,\beta).

Usage

referencebs(x, CI = 0.95, M0 = 800, M = 1000)

Arguments

Value

A list including summary statistics of a Gibbs sampler for Bayesian inference including point estimation for the parameter, its standard error, and the corresponding 100(1-\alpha)\% credible interval, goodness-of-fit measures, asymptotic 100(1-\alpha)\% confidence interval (CI) and corresponding standard errors, and Fisher information matix.

Author(s)

Mahdi Teimouri

References

J. O. Berger and J. M. Bernardo 1989. Estimating a product of means: Bayesian analysis with reference priors. Journal of the American Statistical Association, 84(405), 200-207.

Examples

data(fatigue)
x <- fatigue
referencebs(x, CI = 0.95, M0 = 800, M = 1000)

Bayesian estimator for the Birnbaum-Saunders family under progressive type-II censoring scheme.

Description

Estimates parameters of the Birnbaum-Saunders family in a Bayesian framework through the Metropolis-Hasting algorithm when subjects are placed on progressive type-II censoring scheme with likelihood function

l(\alpha,\beta|x_{1:m:n},\dots,x_{m:m:n})=\log L(\Theta) \propto C \sum_{i=1}^{m} \log f(x_{i:m:n}{{;}}|\alpha,\beta) + \sum_{i=1}^{m} R_i \log \bigl[1-F(x_{i:m:n}{{;}}|\alpha,\beta)\bigr],

in which F(.|\alpha,\beta) is cumulative distribution function of the Birnbaum-Saunders family with C=n(n-R_1-1)(n-R_1-R_2-2)\dots (n-R_1-R_2-\dots-R_{m-1}-m+1). The acceptance for each new sample of \alpha and \beta, respectively, becomes

A_{\alpha}=\min \left\{1,\prod_{i=1}^{m}\frac{\bigl[1-F_{BS}(t_{i:m:n}|1/(\alpha^{new})^2,\beta)\bigr]^{R_{i}}}{\bigl[1-F_{BS}(t_{i:m:n}|1/(\alpha_{old})^2,\beta)\bigr]^{R_{i}}}\right\}

,

A_{\beta}=\min \left\{1,\prod_{i=1}^{m}\frac{\bigl[1-F_{BS}(t_{i:m:n}|\alpha,\beta^{new})\bigr]^{R_{i}}}{\bigl[1-F_{BS}(t_{i:m:n}|\alpha,\beta_{old})\bigr]^{R_{i}}}\right\}.

Usage

typeIIbs(plan, M0 = 4000, M = 6000, CI = 0.95)

Arguments

Value

A list including summary statistics after burn-in point including: mean, median, standard deviation, 100(1 - CI)/2 percentile, 100(1/2 + CI/2) percentile.

Author(s)

Mahdi Teimouri

References

M. Teimouri and S. Nadarajah 2016. Bias corrected MLEs under progressive type-II censoring scheme, Journal of Statistical Computation and Simulation, 86 (14), 2714-2726.

N. Balakrishnan and R. Aggarwala 2000. Progressive Censoring: Theory, Methods, and Applications. Springer Science \& Business Media, New York.

Examples

data(plasma)
typeIIbs(plan = plasma, M0 = 100, M = 200, CI = 0.95)

Starting message when loading package bibs

Description

It contains a welcome message for user of package bibs.

Value

Welcome message for user of bibs package.