| Title: | Birnbaum-Saunders Model |
|---|---|
| Description: | It provides the density, distribution function, quantile function, random number generator, reliability function, failure rate, likelihood function, moments and EM algorithm for Maximum Likelihood estimators, also empirical quantile and generated envelope for a given sample, all this for the three parameter Birnbaum-Saunders model based on Skew-Normal Distribution. Also, it provides the random number generator for the mixture of Birnbaum-Saunders model based on Skew-Normal distribution. Additionally, we incorporate the EM algorithm based on the assumption that the error term follows a finite mixture of Sinh-normal distributions. |
| Authors: | Luis Benites Sanchez[cre, aut] <[email protected]>, Rocio Paola Maehara[cre, aut] <[email protected]> and Paulo Jos<c3><a9> Alejandro Aybar Flores[ctb] <[email protected]> |
| Maintainer: | Rocio Paola Maehara <[email protected]> |
| License: | GPL (>= 2) |
| Version: | 1.0 |
| Built: | 2024-12-17 06:52:20 UTC |
| Source: | CRAN |
It provides the density, distribution function, quantile function, random number generator, reliability function, failure rate, likelihood function, moments and EM algorithm for Maximum Likelihood estimators, also empirical quantile and generated envelope for a given sample, all this for the three parameter Birnbaum-Saunders model based on Skew-Normal Distribution. Also, it provides the random number generator for the mixture of Birbaum-Saunders model based on Skew-Normal distribution. Additionally, we incorporate the EM algorithm based on the assumption that the error term follows a finite mixture of Sinh-normal distributions.
| Package: | bssn |
| Type: | Package |
| Version: | 1.5 |
| Date: | 2020-02-12 |
| License: | GPL (>=2) |
Rocio Maehara [email protected] and Luis Benites [email protected]
Vilca, Filidor; Santana, L. R.; Leiva, Victor; Balakrishnan, N. (2011). Estimation of extreme percentiles in Birnbaum Saunders distributions. Computational Statistics & Data Analysis (Print), 55, 1665-1678.
Santana, Lucia; Vilca, Filidor; Leiva, Victor (2011). Influence analysis in skew-Birnbaum Saunders regression models and applications. Journal of Applied Statistics, 38, 1633-1649.
bssn, EMbssn, momentsbssn, ozone, reliabilitybssn, FMshnReg
#See examples for the bssnEM function linked above.#See examples for the bssnEM function linked above.
It provides the density, distribution function, quantile function, random number generator, likelihood function, moments and EM algorithm for Maximum Likelihood estimators for a given sample, all this for the three parameter Birnbaum-Saunders model based on Skew-Normal Distribution. Also, we have the random number generator for the mixture of Birbaum-Saunders model based on Skew-Normal distribution. Finally, the function mmmeth() is used to find the initial values for the parameters alpha and beta using modified-moment method.
dbssn(ti, alpha=0.5, beta=1, lambda=1.5) pbssn(q, alpha=0.5, beta=1, lambda=1.5) qbssn(p, alpha=0.5, beta=1, lambda=1.5) rbssn(n, alpha=0.5, beta=1, lambda=1.5) rmixbssn(n,alpha,beta,lambda,pii) mmmeth(ti)dbssn(ti, alpha=0.5, beta=1, lambda=1.5) pbssn(q, alpha=0.5, beta=1, lambda=1.5) qbssn(p, alpha=0.5, beta=1, lambda=1.5) rbssn(n, alpha=0.5, beta=1, lambda=1.5) rmixbssn(n,alpha,beta,lambda,pii) mmmeth(ti)
ti |
vector of observations. |
q |
vector of quantiles. |
p |
vector of probabilities. |
n |
number of observations. |
alpha |
shape parameter. |
beta |
scale parameter. |
lambda |
skewness parameter. |
pii |
Are weights adding to 1. Each one of them (alpha, beta and lambda) must be a vector of length g if you want to generate a random numbers from a mixture distribution BSSN. |
If alpha, sigma or lambda are not specified they assume the default values of 0.5, 1 and 1.5, respectively, belonging to the Birnbaum-Saunders model based on Skew-Normal distribution denoted by .
As discussed in Filidor et. al (2011) we say that a random variable T is distributed as an BSSN with shape parameter , scale parameter and skewness parameter in , if its probability density function (pdf) is given by
where and are the standard normal density and cumulative distribution function respectively. Also ) and
dbssn gives the density, pbssn gives the distribution function, qbssn gives the quantile function, rbssn generates a random sample and rmixbssn genrates a mixture random sample.
The length of the result is determined by n for rbssn, and is the maximum of the lengths of the numerical arguments for the other functions dbssn, pbssn and qbssn.
Rocio Maehara [email protected] and Luis Benites [email protected]
Vilca, Filidor; Santana, L. R.; Leiva, Victor; Balakrishnan, N. (2011). Estimation of extreme percentiles in Birnbaum Saunders distributions. Computational Statistics & Data Analysis (Print), 55, 1665-1678.
Santana, Lucia; Vilca, Filidor; Leiva, Victor (2011). Influence analysis in skew-Birnbaum Saunders regression models and applications. Journal of Applied Statistics, 38, 1633-1649.
EMbssn, momentsbssn, ozone, reliabilitybssn
## Not run: ## Let's plot an Birnbaum-Saunders model based on Skew-Normal distribution! ## Density sseq <- seq(0,3,0.01) dens <- dbssn(sseq,alpha=0.2,beta=1,lambda=1.5) plot(sseq, dens,type="l", lwd=2,col="red", xlab="x", ylab="f(x)", main="BSSN Density function") # Differing densities on a graph # positive values of lambda y <- seq(0,3,0.01) f1 <- dbssn(y,0.2,1,1) f2 <- dbssn(y,0.2,1,2) f3 <- dbssn(y,0.2,1,3) f4 <- dbssn(y,0.2,1,4) den <- cbind(f1,f2,f3,f4) matplot(y,den,type="l", col=c("deepskyblue4", "firebrick1", "darkmagenta", "aquamarine4"), ylab ="Density function",xlab="y",lwd=2,sub="(a)") legend(1.5,2.8,c("BSSN(0.2,1,1)", "BSSN(0.2,1,2)", "BSSN(0.2,1,3)","BSSN(0.2,1,4)"), col = c("deepskyblue4", "firebrick1", "darkmagenta", "aquamarine4"), lty=1:4,lwd=2, seg.len=2,cex=0.8,box.lty=0,bg=NULL) #negative values of lambda y <- seq(0,3,0.01) f1 <- dbssn(y,0.2,1,-1) f2 <- dbssn(y,0.2,1,-2) f3 <- dbssn(y,0.2,1,-3) f4 <- dbssn(y,0.2,1,-4) den <- cbind(f1,f2,f3,f4) matplot(y,den,type="l", col=c("deepskyblue4", "firebrick1", "darkmagenta", "aquamarine4"), ylab ="Density function",xlab="y",lwd=2,sub="(a)") legend(1.5,2.8,c("BSSN(0.2,1,-1)", "BSSN(0.2,1,-2)","BSSN(0.2,1,-3)", "BSSN(0.2,1,-4)"), col=c("deepskyblue4","firebrick1", "darkmagenta","aquamarine4"),lty=1:4,lwd=2,seg.len=2, cex=1,box.lty=0,bg=NULL) ## Distribution Function sseq <- seq(0.1,6,0.05) df <- pbssn(q=sseq,alpha=0.75,beta=1,lambda=3) plot(sseq, df, type = "l", lwd=2, col="blue", xlab="x", ylab="F(x)", main = "BSSN Distribution function") abline(h=1,lty=2) #Inverse Distribution Function prob <- seq(0,1,length.out = 1000) idf <- qbssn(p=prob,alpha=0.75,beta=1,lambda=3) plot(prob, idf, type="l", lwd=2, col="gray30", xlab="x", ylab = expression(F^{-1}~(x)), mgp=c(2.3,1,.8)) title(main="BSSN Inverse Distribution function") abline(v=c(0,1),lty=2) #Random Sample Histogram sample <- rbssn(n=10000,alpha=0.75,beta=1,lambda=3) hist(sample,breaks = 70,freq = FALSE,main="") title(main="Histogram and True density") sseq <- seq(0,8,0.01) dens <- dbssn(sseq,alpha=0.75,beta=1,lambda=3) lines(sseq,dens,col="red",lwd=2) ##Random Sample Histogram for Mixture of BSSN alpha=c(0.55,0.25);beta=c(1,1.5);lambda=c(3,2);pii=c(0.3,0.7) sample <- rmixbssn(n=1000,alpha,beta,lambda,pii) hist(sample$y,breaks = 70,freq = FALSE,main="") title(main="Histogram and True density") temp <- seq(min(sample$y), max(sample$y), length.out=1000) lines(temp, (pii[1]*dbssn(temp, alpha[1], beta[1],lambda[1]))+(pii[2]*dbssn(temp, alpha[2] , beta[2],lambda[2])), col="red", lty=3, lwd=3) # the theoretical density lines(temp, pii[1]*dbssn(temp, alpha[1], beta[1],lambda[1]), col="blue", lty=2, lwd=3) # the first component lines(temp, pii[2]*dbssn(temp, alpha[2], beta[2],lambda[2]), col="green", lty=2, lwd=3) # the second component ## End(Not run)## Not run: ## Let's plot an Birnbaum-Saunders model based on Skew-Normal distribution! ## Density sseq <- seq(0,3,0.01) dens <- dbssn(sseq,alpha=0.2,beta=1,lambda=1.5) plot(sseq, dens,type="l", lwd=2,col="red", xlab="x", ylab="f(x)", main="BSSN Density function") # Differing densities on a graph # positive values of lambda y <- seq(0,3,0.01) f1 <- dbssn(y,0.2,1,1) f2 <- dbssn(y,0.2,1,2) f3 <- dbssn(y,0.2,1,3) f4 <- dbssn(y,0.2,1,4) den <- cbind(f1,f2,f3,f4) matplot(y,den,type="l", col=c("deepskyblue4", "firebrick1", "darkmagenta", "aquamarine4"), ylab ="Density function",xlab="y",lwd=2,sub="(a)") legend(1.5,2.8,c("BSSN(0.2,1,1)", "BSSN(0.2,1,2)", "BSSN(0.2,1,3)","BSSN(0.2,1,4)"), col = c("deepskyblue4", "firebrick1", "darkmagenta", "aquamarine4"), lty=1:4,lwd=2, seg.len=2,cex=0.8,box.lty=0,bg=NULL) #negative values of lambda y <- seq(0,3,0.01) f1 <- dbssn(y,0.2,1,-1) f2 <- dbssn(y,0.2,1,-2) f3 <- dbssn(y,0.2,1,-3) f4 <- dbssn(y,0.2,1,-4) den <- cbind(f1,f2,f3,f4) matplot(y,den,type="l", col=c("deepskyblue4", "firebrick1", "darkmagenta", "aquamarine4"), ylab ="Density function",xlab="y",lwd=2,sub="(a)") legend(1.5,2.8,c("BSSN(0.2,1,-1)", "BSSN(0.2,1,-2)","BSSN(0.2,1,-3)", "BSSN(0.2,1,-4)"), col=c("deepskyblue4","firebrick1", "darkmagenta","aquamarine4"),lty=1:4,lwd=2,seg.len=2, cex=1,box.lty=0,bg=NULL) ## Distribution Function sseq <- seq(0.1,6,0.05) df <- pbssn(q=sseq,alpha=0.75,beta=1,lambda=3) plot(sseq, df, type = "l", lwd=2, col="blue", xlab="x", ylab="F(x)", main = "BSSN Distribution function") abline(h=1,lty=2) #Inverse Distribution Function prob <- seq(0,1,length.out = 1000) idf <- qbssn(p=prob,alpha=0.75,beta=1,lambda=3) plot(prob, idf, type="l", lwd=2, col="gray30", xlab="x", ylab = expression(F^{-1}~(x)), mgp=c(2.3,1,.8)) title(main="BSSN Inverse Distribution function") abline(v=c(0,1),lty=2) #Random Sample Histogram sample <- rbssn(n=10000,alpha=0.75,beta=1,lambda=3) hist(sample,breaks = 70,freq = FALSE,main="") title(main="Histogram and True density") sseq <- seq(0,8,0.01) dens <- dbssn(sseq,alpha=0.75,beta=1,lambda=3) lines(sseq,dens,col="red",lwd=2) ##Random Sample Histogram for Mixture of BSSN alpha=c(0.55,0.25);beta=c(1,1.5);lambda=c(3,2);pii=c(0.3,0.7) sample <- rmixbssn(n=1000,alpha,beta,lambda,pii) hist(sample$y,breaks = 70,freq = FALSE,main="") title(main="Histogram and True density") temp <- seq(min(sample$y), max(sample$y), length.out=1000) lines(temp, (pii[1]*dbssn(temp, alpha[1], beta[1],lambda[1]))+(pii[2]*dbssn(temp, alpha[2] , beta[2],lambda[2])), col="red", lty=3, lwd=3) # the theoretical density lines(temp, pii[1]*dbssn(temp, alpha[1], beta[1],lambda[1]), col="blue", lty=2, lwd=3) # the first component lines(temp, pii[2]*dbssn(temp, alpha[2], beta[2],lambda[2]), col="green", lty=2, lwd=3) # the second component ## End(Not run)
Performs the EM algorithm for Birnbaum-Saunders model based on Skew-Normal distribution.
EMbssn(ti,alpha,beta,delta,initial.values=FALSE, loglik=F,accuracy=1e-8, show.envelope="FALSE",iter.max=500)EMbssn(ti,alpha,beta,delta,initial.values=FALSE, loglik=F,accuracy=1e-8, show.envelope="FALSE",iter.max=500)
ti |
Vector of observations. |
alpha, beta, delta
|
Initial values. |
initial.values |
Logical; if TRUE, get the initial values for the parameters. |
loglik |
Logical; if TRUE, showvalue of the log-likelihood. |
accuracy |
The convergence maximum error. |
show.envelope |
Logical; if TRUE, show the simulated envelope for the fitted model. |
iter.max |
The maximum number of iterations of the EM algorithm |
The function returns a list with 11 elements detailed as
iter |
Number of iterations. |
alpha |
Returns the value of the MLE of the shape parameter. |
beta |
Returns the value of the MLE of the scale parameter. |
lambda |
Returns the value of the MLE of the skewness parameter. |
SE |
Standard Errors of the ML estimates. |
table |
Table containing the ML estimates with the corresponding standard errors. |
loglik |
Log-likelihood. |
AIC |
Akaike information criterion. |
BIC |
Bayesian information criterion. |
HQC |
Hannan-Quinn information criterion. |
time |
processing time. |
Rocio Maehara [email protected] and Luis Benites [email protected]
Vilca, Filidor; Santana, L. R.; Leiva, Victor; Balakrishnan, N. (2011). Estimation of extreme percentiles in Birnbaum Saunders distributions. Computational Statistics & Data Analysis (Print), 55, 1665-1678.
Santana, Lucia; Vilca, Filidor; Leiva, Victor (2011). Influence analysis in skew-Birnbaum Saunders regression models and applications. Journal of Applied Statistics, 38, 1633-1649.
bssn, EMbssn, momentsbssn, ozone, reliabilitybssn
## Not run: #Using the ozone data data(ozone) attach(ozone) ################################# #The model ti <- dailyozonelevel #Initial values for the parameters initial <- mmmeth(ti) alpha0 <- initial$alpha0ini beta0 <- initial$beta0init lambda0 <- 0 delta0 <- lambda0/sqrt(1+lambda0^2) #Estimated parameters of the model (by default) est_param <- EMbssn(ti,alpha0,beta0,delta0,loglik=T, accuracy = 1e-8,show.envelope = "TRUE", iter.max=500) #ML estimates alpha <- est_param$res$alpha beta <- est_param$res$beta lambda <- est_param$res$lambda ######################################### #A simple output example --------------------------------------------------------- Birnbaum-Saunders model based on Skew-Normal distribution --------------------------------------------------------- Observations = 116 ----------- Estimates ----------- Estimate Std. Error z value Pr(>|z|) alpha 1.26014 0.23673 5.32311 0.00000 beta 14.65730 4.01984 3.64624 0.00027 lambda 1.06277 0.54305 1.95706 0.05034 ------------------------ Model selection criteria ------------------------ Loglik AIC BIC HQC Value -542.768 4.705 4.741 4.719 ------- Details ------- Iterations = 415 Processing time = 0.4283214 secs Convergence = TRUE ## End(Not run)## Not run: #Using the ozone data data(ozone) attach(ozone) ################################# #The model ti <- dailyozonelevel #Initial values for the parameters initial <- mmmeth(ti) alpha0 <- initial$alpha0ini beta0 <- initial$beta0init lambda0 <- 0 delta0 <- lambda0/sqrt(1+lambda0^2) #Estimated parameters of the model (by default) est_param <- EMbssn(ti,alpha0,beta0,delta0,loglik=T, accuracy = 1e-8,show.envelope = "TRUE", iter.max=500) #ML estimates alpha <- est_param$res$alpha beta <- est_param$res$beta lambda <- est_param$res$lambda ######################################### #A simple output example --------------------------------------------------------- Birnbaum-Saunders model based on Skew-Normal distribution --------------------------------------------------------- Observations = 116 ----------- Estimates ----------- Estimate Std. Error z value Pr(>|z|) alpha 1.26014 0.23673 5.32311 0.00000 beta 14.65730 4.01984 3.64624 0.00027 lambda 1.06277 0.54305 1.95706 0.05034 ------------------------ Model selection criteria ------------------------ Loglik AIC BIC HQC Value -542.768 4.705 4.741 4.719 ------- Details ------- Iterations = 415 Processing time = 0.4283214 secs Convergence = TRUE ## End(Not run)
These data correspond to representing the metabolism of carcinogenic substances among 245 unrelated individuals.
data(enzyme)data(enzyme)
enzyme is a data frame with 245 cases (rows).
For more information about dataset see Bechtel et al. (1993).
Bechtel, Y., Bonaiti-Pellie, C., Poisson, N., Magnette, J. and Bechtel, P. (1993). A population and family study of n-acetyltransferase using caffeine urinary metabolites. Clinical Pharmacology and Therapeutics, 54, 134-141.
Performs the EM-type algorithm with conditonal maximation to perform maximum likelihood inference of the parameters of the proposed model based on the assumption that the error term follows a finite mixture of Sinh-normal distributions.
FMshnReg(y, x1, alpha = NULL, Abetas = NULL, medj=NULL, pii = NULL, g = NULL, get.init = TRUE,algorithm = "K-means", accuracy = 10^-6, show.envelope="FALSE", iter.max = 100)FMshnReg(y, x1, alpha = NULL, Abetas = NULL, medj=NULL, pii = NULL, g = NULL, get.init = TRUE,algorithm = "K-means", accuracy = 10^-6, show.envelope="FALSE", iter.max = 100)
y |
the response matrix (dimension nx1). |
x1 |
Matrix or vector of covariates. |
alpha |
Value of the shape parameter for the EM algorithm. Each of them must be a vector of length g. (the algorithm considers the number of components to be adjusted based on the size of these vectors). |
Abetas |
Parameters of vector regression dimension |
medj |
a list of |
pii |
Value for the EM algorithm. Each of them must be a vector of length g. (the algorithm considers the number of components to be adjusted based on the size of these vectors). |
g |
The number of cluster to be considered in fitting. |
get.init |
if TRUE, the initial values are generated via k-means. |
algorithm |
clustering procedure of a series of vectors according to a criterion. The clustering algorithms may classified in 4 main categories: exclusive, overlapping, hierarchical and probabilistic. |
accuracy |
The convergence maximum error. |
show.envelope |
Logical; if TRUE, show the simulated envelope for the fitted model. |
iter.max |
The maximum number of iterations of the EM algorithm |
The function returns a list with 10 elements detailed as
iter |
Number of iterations. |
criteria |
Attained criteria value. |
convergence |
Convergence reached or not. |
SE |
Standard Error estimates, if the output shows |
table |
Table containing the inference for the estimated parameters. |
LK |
log-likelihood. |
AIC |
Akaike information criterion. |
BIC |
Bayesian information criterion. |
EDC |
Efficient Determination criterion. |
time |
Processing time. |
Rocio Maehara [email protected] and Luis Benites [email protected]
Maehara, R. and Benites, L. (2020). Linear regression models using finite mixture of Sinh-normal distribution. In Progress.
Bartolucci, F. and Scaccia, L. (2005). The use of mixtures for dealing with non-normal regression errors, Computational Statistics & Data Analysis 48(4): 821-834.
## Not run: #Using the AIS data library(FMsmsnReg) data(ais) ################################# #The model x1 <- cbind(1,ais$SSF,ais$Ht) y <- ais$Bfat library(ClusterR) #This library is useful for using the k-medoids algorithm. FMshnReg(y, x1, get.init = TRUE, g=2, algorithm="k-medoids", accuracy = 10^-6, show.envelope="FALSE", iter.max = 1000) ######################################### #A simple output example ------------------------------------------------------------ Finite Mixture of Sinh Normal Regression Model ------------------------------------------------------------ Observations = 202 ----------- Estimates ----------- Estimate SE alpha1 0.81346 0.10013 alpha2 3.04894 0.32140 beta0 15.08998 1.70024 beta1 0.17708 0.00242 beta2 -0.07687 0.00934 mu1 -0.25422 0.18069 mu2 0.37944 0.38802 pii1 0.59881 0.41006 ------------------------ Model selection criteria ------------------------ Loglik AIC BIC EDC Value -355.625 721.25 737.791 725.463 ------- Details ------- Convergence reached? = TRUE EM iterations = 39 / 1000 Criteria = 6.58e-07 Processing time = 0.725559 secs ## End(Not run)## Not run: #Using the AIS data library(FMsmsnReg) data(ais) ################################# #The model x1 <- cbind(1,ais$SSF,ais$Ht) y <- ais$Bfat library(ClusterR) #This library is useful for using the k-medoids algorithm. FMshnReg(y, x1, get.init = TRUE, g=2, algorithm="k-medoids", accuracy = 10^-6, show.envelope="FALSE", iter.max = 1000) ######################################### #A simple output example ------------------------------------------------------------ Finite Mixture of Sinh Normal Regression Model ------------------------------------------------------------ Observations = 202 ----------- Estimates ----------- Estimate SE alpha1 0.81346 0.10013 alpha2 3.04894 0.32140 beta0 15.08998 1.70024 beta1 0.17708 0.00242 beta2 -0.07687 0.00934 mu1 -0.25422 0.18069 mu2 0.37944 0.38802 pii1 0.59881 0.41006 ------------------------ Model selection criteria ------------------------ Loglik AIC BIC EDC Value -355.625 721.25 737.791 725.463 ------- Details ------- Convergence reached? = TRUE EM iterations = 39 / 1000 Criteria = 6.58e-07 Processing time = 0.725559 secs ## End(Not run)
Mean, variance, skewness and kurtosis for the Birnbaum-Saunders model based on Skew-Normal distribution defined in Filidor et. al (2011).
meanbssn(alpha=0.5,beta=1,lambda=1.5) varbssn(alpha=0.5,beta=1,lambda=1.5) skewbssn(alpha=0.5,beta=1,lambda=1.5) kurtbssn(alpha=0.5,beta=1,lambda=1.5)meanbssn(alpha=0.5,beta=1,lambda=1.5) varbssn(alpha=0.5,beta=1,lambda=1.5) skewbssn(alpha=0.5,beta=1,lambda=1.5) kurtbssn(alpha=0.5,beta=1,lambda=1.5)
alpha |
shape parameter |
beta |
scale parameter |
lambda |
skewness parameter |
meanbssn gives the mean, varbssn gives the variance, skewbssn gives the skewness, kurtbssn gives the kurtosis.
Rocio Maehara [email protected] and Luis Benites [email protected]
Vilca, Filidor; Santana, L. R.; Leiva, Victor; Balakrishnan, N. (2011). Estimation of extreme percentiles in Birnbaum Saunders distributions. Computational Statistics & Data Analysis (Print), 55, 1665-1678.
Santana, Lucia; Vilca, Filidor; Leiva, Victor (2011). Influence analysis in skew-Birnbaum Saunders regression models and applications. Journal of Applied Statistics, 38, 1633-1649.
bssn, EMbssn, momentsbssn, ozone, reliabilitybssn
## Let's compute some moments for a Birnbaum-Saunders model based on Skew normal Distribution. # The well known mean, variance, skewness and kurtosis meanbssn(alpha=0.5,beta=1,lambda=1.5) varbssn(alpha=0.5,beta=1,lambda=1.5) skewbssn(alpha=0.5,beta=1,lambda=1.5) kurtbssn(alpha=0.5,beta=1,lambda=1.5)## Let's compute some moments for a Birnbaum-Saunders model based on Skew normal Distribution. # The well known mean, variance, skewness and kurtosis meanbssn(alpha=0.5,beta=1,lambda=1.5) varbssn(alpha=0.5,beta=1,lambda=1.5) skewbssn(alpha=0.5,beta=1,lambda=1.5) kurtbssn(alpha=0.5,beta=1,lambda=1.5)
These data correspond to daily ozone level measurements (in ) in New York in May-September, 1973, from the New York State Department of Conservation.
data(ozone)data(ozone)
ozone is a data frame with 116 cases (rows).
For a complete description of various distributions applied to data concentration of air pollutants see Gokhale and Khare (2004).
Leiva, V., Barros, M., Paula, G. e Sanhueza, A. (2007). Generalized BirnbaumSaunders distribution applied to air pollutant concentration. Environmetrics, 19, 235-249.
Nadarajah, S. (2007). A truncated inverted beta distribution with application to air pollution data. Stoch. Environ. Res. Risk. Assess., 22, 285-289.
Gokhale, S. e Khare, M. (2004) A review of deterministic, stochastic and hybrid vehicular exhaust emission models International. J. Transp. Manag., 2, 59-74.
Two useful descriptors in reliability analysis are the reliability function (rf), and the failure rate (fr) function or hazard function. For a non-negative random variable with pdf (and cdf ), its distribution can be characterized equally in terms of the rf, or of the fr, which are respectively defined by , and , for ,and .
Rebssn(ti,alpha=0.5,beta=1,lambda=1.5) Fbssn(ti,alpha=0.5,beta=1,lambda=1.5)Rebssn(ti,alpha=0.5,beta=1,lambda=1.5) Fbssn(ti,alpha=0.5,beta=1,lambda=1.5)
ti |
dataset. |
alpha |
shape parameter |
beta |
scale parameter |
lambda |
skewness parameter |
Rbssn gives the reliability function, Fbssn gives the failure rate or hazard function.
Rocio Maehara [email protected] and Luis Benites [email protected]
Leiva, V., Vilca-Labra, F. E., Balakrishnan, N. e Sanhueza, A. (2008). A skewed sinh-normal distribution and its properties and application to air pollution. Comm. Stat. Theoret. Methods. Submetido.
Guiraud, P., Leiva, V., Fierro, R. (2009). A non-central version of the Birnbaum-Saunders distribution for reliability analysis. IEEE Transactions on Reliability 58, 152-160.
bssn, EMbssn, momentsbssn, ozone, Rebssn
## Let's compute some realiability functions for a Birnbaum-Saunders model based on ## Skew normal Distribution for different values of the shape parameter. ti <- seq(0,2,0.01) f1 <- Rebssn(ti,0.75,1,1) f2 <- Rebssn(ti,1,1,1) f3 <- Rebssn(ti,1.5,1,1) f4 <- Rebssn(ti,2,1,1) den <- cbind(f1,f2,f3,f4) matplot(ti,den,type="l", col=c("deepskyblue4","firebrick1","darkmagenta","aquamarine4"), ylab="S(t)", xlab="t",lwd=2) legend(1.5,1,c(expression(alpha==0.75), expression(alpha==1), expression(alpha==1.5), expression(alpha==2)),col= c("deepskyblue4","firebrick1","darkmagenta","aquamarine4"), lty=1:4,lwd=2,seg.len=2,cex=0.9,box.lty=0,bg=NULL) ## Let's compute some hazard functions for a Birnbaum Saunders model based on ## Skew normal Distribution for different values of the skewness parameter. ti <- seq(0,2,0.01) f1 <- Fbssn(ti,0.5,1,-1) f2 <- Fbssn(ti,0.5,1,-2) f3 <- Fbssn(ti,0.5,1,-3) f4 <- Fbssn(ti,0.5,1,-4) den <- cbind(f1,f2,f3,f4) matplot(ti,den,type = "l", col = c("deepskyblue4","firebrick1", "darkmagenta", "aquamarine4"), ylab = "h(t)" , xlab="t",lwd=2) legend(0.1,23, c(expression(lambda==-1), expression(lambda==-2), expression(lambda == -3), expression(lambda == -4)), col = c("deepskyblue4", "firebrick1", "darkmagenta","aquamarine4"), lty=1:4,lwd=2,seg.len=2,cex=0.9,box.lty=1,bg=NULL)## Let's compute some realiability functions for a Birnbaum-Saunders model based on ## Skew normal Distribution for different values of the shape parameter. ti <- seq(0,2,0.01) f1 <- Rebssn(ti,0.75,1,1) f2 <- Rebssn(ti,1,1,1) f3 <- Rebssn(ti,1.5,1,1) f4 <- Rebssn(ti,2,1,1) den <- cbind(f1,f2,f3,f4) matplot(ti,den,type="l", col=c("deepskyblue4","firebrick1","darkmagenta","aquamarine4"), ylab="S(t)", xlab="t",lwd=2) legend(1.5,1,c(expression(alpha==0.75), expression(alpha==1), expression(alpha==1.5), expression(alpha==2)),col= c("deepskyblue4","firebrick1","darkmagenta","aquamarine4"), lty=1:4,lwd=2,seg.len=2,cex=0.9,box.lty=0,bg=NULL) ## Let's compute some hazard functions for a Birnbaum Saunders model based on ## Skew normal Distribution for different values of the skewness parameter. ti <- seq(0,2,0.01) f1 <- Fbssn(ti,0.5,1,-1) f2 <- Fbssn(ti,0.5,1,-2) f3 <- Fbssn(ti,0.5,1,-3) f4 <- Fbssn(ti,0.5,1,-4) den <- cbind(f1,f2,f3,f4) matplot(ti,den,type = "l", col = c("deepskyblue4","firebrick1", "darkmagenta", "aquamarine4"), ylab = "h(t)" , xlab="t",lwd=2) legend(0.1,23, c(expression(lambda==-1), expression(lambda==-2), expression(lambda == -3), expression(lambda == -4)), col = c("deepskyblue4", "firebrick1", "darkmagenta","aquamarine4"), lty=1:4,lwd=2,seg.len=2,cex=0.9,box.lty=1,bg=NULL)