bssn {bssn} R Documentation

## Birnbaum-Saunders model based on Skew-Normal distribution

### Description

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.

### Usage

```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)
```

### Arguments

 `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.

### Details

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 BSSN(0.5,1,1.5).

As discussed in Filidor et. al (2011) we say that a random variable T is distributed as an BSSN with shape parameter α>0, scale parameter β>0 and skewness parameter λ in R, if its probability density function (pdf) is given by

f(t)=2φ(a(t;α,β))Φ(λ a(t;α,β))A(t;α,β), t>0

where φ(.) and Φ(.) are the standard normal density and cumulative distribution function respectively. Also a(t;α,β)=(1/α)(√{t/β}-√{β/t}) and A(t;α,β)=t^{-3/2}(t+β)/(2α β^{1/2})

### Value

`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`.

### Author(s)

Rocio Maehara rmaeharaa@gmail.com and Luis Benites lbenitesanchez@gmail.com

### References

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`

### Examples

```## 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*dbssn(temp, alpha, beta,lambda))+(pii*dbssn(temp, alpha
, beta,lambda)), col="red", lty=3, lwd=3) # the theoretical density
lines(temp, pii*dbssn(temp, alpha, beta,lambda), col="blue", lty=2, lwd=3)
# the first component
lines(temp, pii*dbssn(temp, alpha, beta,lambda), col="green", lty=2, lwd=3)
# the second component

## End(Not run)
```

[Package bssn version 1.0 Index]