crfbs {bs}R Documentation

Conditional reliability function (crf) for the BSD

Description

Conditional reliability function for the BSD with shape parameter alpha and scale parameter beta.

Usage

crfbs(t, x = 0, alpha = 1, beta = 1)

Arguments

t Vector of quantiles.
x Age component in used.
alpha Shape parameter.
beta Scale parameter.

Details

One of the useful indicators in lifetime analysis is the hazard function (h.f.), which is defined by

h_T(t) = frac{f_T(t)}{1- F_T(t)},, t > 0,, 0<F_T(t)<1,

where f_T(cdot) and F_T(cdot) are the pdf and cdf, respectively; see e.g. Johnson, Kotz, and Balakrishnan (1994, p. 640). The behavior of h_T(t) 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_T(t) is decreasing (DFR class), then the units improve in performance with time. Finally, if h_T(t) is constant, then the lifetime distribution is necessarily exponential.

Another aging indicators are the following:

The failure rate average (FRA) of T is given by

FRA(t) = frac{H_T(t)}{t} = frac{int^{t}_{0} h_T(t),dt}{t},, t>0,

where H_T(t) is the cumulative hazard function. An analysis for FRA(t) on t permits to obtain the IFRA and DFRA classes.

The survival function (s.f.) and the conditional survival of T are defined by

R_T(t) = 1- F_T(t) quad {rm and} quad R_T(t|x) = frac{R_T(t+x)}{R_T(x)},, t >0,, x> 0,, R_T(cdot) > 0,

respectively, where F_T(cdot) is the cdf of T. Similarly to h_T(t) and FRA(t), the distribution of T belongs to the new better than used (NBU), exponential, or new worse than used (NWU) classes, when R_T(t|x) < R_T(t), R_T(t|x) = R_T(t), or R_T(t|x) > R_T(t), respectively.

Value

Conditional reliability function for the BSD.

Author(s)

Víctor Leiva <victor.leiva@uv.cl>, Hugo Hernández <hugo.hernande@msn.com>, and Marco Riquelme <mriquelm@ucm.cl>.

References

Hoyland, A. and Rausand, M. (1994). System Reliability Theory. John Wiley and Sons. N. Y.

Examples

## Reliability indicators:

## Reliability function
rfbs(3,0.5,1.0)

## Conditional reliability function
crfbs(3,0,0.5,1.0)

## Conditional reliability function (age component=0)
crfbs(3,x=0,0.5,1.0)

## Conditional reliability function (age component=0)
crfbs(3,x=2,0.5,1.0)

## failure rate
frbs(3,0.5,1.0)

## failure rate average
frabs(3,0.5,1.0)

[Package bs version 1.0 Index]