reliaR (version 0.01)

BurrXsurvival: Survival related functions for the BurrX distribution

Description

Conditional reliability function (crf), hazard function, hazard rate average (HRA) and survival function for the BurrX distribution with shape parameter alpha and scale parameter lambda.

Usage

crf.burrX(x, t = 0, alpha, lambda)
hburrX(x, alpha, lambda)
hra.burrX(x, alpha, lambda)
sburrX(x, alpha, lambda)

Arguments

x

vector of quantiles.

alpha

shape parameter.

lambda

scale parameter.

t

age component.

Value

crf.burrX gives the conditional reliability function (crf), hburrX gives the hazard function, hra.burrX gives the hazard rate average (HRA) function, and sburrX gives the survival function for the BurrX distribution.

Details

The hazard function is defined by $$h(x) = \frac{f(x)}{1 - F(x)},\, t > 0, 0 < F(x) < 1,$$ where $f(.)$ and $F(.)$ are the pdf and cdf, respectively. The behavior of $h(x)$ allows one to characterize the aging of the units. For example, if the failure rate is increasing (IFR class), then the units age with time. If $h(x)$ is decreasing (DFR class), then the units improve in performance with time. Finally, if $h(x)$ is constant, then the lifetime distribution is necessarily exponential.

There are two more aging indicators which are the following:

The failure rate average (FRA) of X is given by $$FRA(x) = \frac{H(x)}{x} = \frac{\int^{x}_{0} h(x)\,dx}{x},\, x > 0,$$ where $H(x)$ is the cumulative hazard function. An analysis for FRA($x$) on $x$ permits to obtain the IFRA and DFRA classes.

The survival/reliability function (s.f.) and the conditional survival of X are defined by $$R(x) = 1 - F(x) \quad {\rm and} \quad R(x|t) = \frac{R(x+t)}{R(x)},\, x > 0,\, t > 0,\, R(\cdot) > 0,$$ respectively, where $F(.)$ is the cdf of X. Similarly to $h(x)$ and $FRA(x)$, the distribution of X belongs to the new better than used (NBU), exponential, or new worse than used (NWU) classes, when $R(x|t) < R(x)$, $R(x|t) = R(x)$, or $R(x|t) > R(x)$, respectively.

References

Kundu, D., and Raqab, M.Z. (2005). Generalized Rayleigh Distribution: Different Methods of Estimation, Computational Statistics and Data Analysis, 49, 187-200.

Lawless, J.F.(2003). Statistical Models and Methods for Lifetime Data, John Wiley and Sons, New York.

Marshall, A. W., Olkin, I.(2007). Life Distributions: Structure of Nonparametric, Semiparametric, and Parametric Families, Springer, New York.

See Also

dburrX for other BurrX distribution related functions;

Examples

## load data set
data(bearings)
## Maximum Likelihood(ML) Estimates of alpha & lambda for the data(bearings)
## Estimates of alpha & lambda using 'maxLik' package
## alpha.est = 1.1989515, lambda.est = 0.0130847

## Reliability indicators for data(bearings):

## Reliability function
sburrX(bearings, 1.1989515, 0.0130847)

## Hazard function
hburrX(bearings, 1.1989515, 0.0130847)

## hazard rate average(hra)
hra.burrX(bearings, 1.1989515, 0.0130847)

## Conditional reliability function (age component=0)
crf.burrX(bearings, 0.00, 1.1989515, 0.0130847)

## Conditional reliability function (age component=3.0)
crf.burrX(bearings, 3.0, 1.1989515, 0.0130847)