Type: Package
Title: Computation and Estimation of Reliability of Stress-Strength Models
Version: 1.0.2
Date: 2016-04-29
Author: Alessandro Barbiero <alessandro.barbiero@unimi.it>, Riccardo Inchingolo <dott.inchingolo_r@libero.it>
Maintainer: Alessandro Barbiero <alessandro.barbiero@unimi.it>
Description: Reliability of (normal) stress-strength models and for building two-sided or one-sided confidence intervals according to different approximate procedures.
License: GPL-2 | GPL-3 [expanded from: GPL]
LazyLoad: yes
Packaged: 2016-04-30 07:57:22 UTC; Barbiero
Repository: CRAN
Date/Publication: 2016-05-01 00:44:38
NeedsCompilation: no

Computation and Sample Estimation of Reliability of Stress-Strength Models

Description

Reliability of (normal) stress-strength models and for building two-side or one-side confidence intervals according to different approximate procedures.

Details

Package: StressStrength
Type: Package
Version: 1.0.2
Date: 2016-04-29
License: GPL
LazyLoad: yes

Author(s)

Alessandro Barbiero, Riccardo Inchingolo

Maintainer: Alessandro Barbiero <alessandro.barbiero@unimi.it>

References

Kotz S, Lumelskii Y, Pensky M (2003) The stress-strength model and its generalizations: theory and applications. World Scientific, Singapore

Guo H, Krishnamoorthy K (2004) New approximate inferential methods for the reliability parameter in a stress-strength model: The normal case. Commun Stat Theory Methods 33:1715-1731

Computation of reliability of stress-strength models

Description

For a stress-strength model, with independent r.v. X and Y representing the strength and the stress respectively, the function computes the reliability R=P(X>Y)

Usage

SSR(parx, pary, family = "normal")

Arguments

parx

parameters of X distribution (for the normal distribution, mean \mu_x and standard deviation \sigma_x)

pary

parameters of Y distribution (for the normal distribution, mean \mu_y and standard deviation \sigma_y)

family

family distribution for both X and Y (now, only "normal" available)

Details

The function computes R=P(X>Y) where X and Y are independent r.v. following the family distribution with distributional parameters parx and pary.

Value

R=P(X>Y). For normal distributions, R=\Phi(d) with d=(\mu_x-\mu_y)/\sqrt{\sigma_x^2+\sigma_y^2}.

Author(s)

Alessandro Barbiero, Riccardo Inchingolo

References

Kotz S, Lumelskii Y, Pensky M (2003) The stress-strength model and its generalizations: theory and applications. World Scientific, Singapore

See Also

estSSR

Examples

# let X be a normal r.v. with mean 1 and sd 1;
# and Y a normal r.v. with mean 0 and sd 2
# X and Y independent
parx<-c(1, 1)
pary<-c(0, 2)
# reliability of the stress-strength model (X=strength, Y=stress)
SSR(parx,pary)
# changing the parameters of Y
pary<-c(1.5, 2)
# reliability of the stress-strength model (X=strength, Y=stress)
SSR(parx,pary)

Sample estimation of reliability of stress-strength models

Description

The function provides sample estimates of reliability of stress-strength models, where stress and strength are modeled as independent r.v., whose distribution form is known except for the values of its parameters, assumed all unknown

Usage

estSSR(x, y, family="normal", twoside=TRUE, type="RG", alpha=0.05, B=2000)

Arguments

x

a random sample from r.v. X modeling strength

y

a random sample from r.v. Y modeling stress

family

the distribution of both X and Y

twoside

if TRUE, the function computes two-side confidence intervals; otherwise, one-side (a lower bound)

type

type of confidence interval (CI) to be built. For the normal family, "RG" stands for Reiser-Guttman, "AN" for large sample (asymptotically normal), "LOGIT" or "ARCSIN" for logit or arcsin variance stabilizing tranformations, "B" for percentile bootstrap, "GK" for Guo-Krishnamoorthy (one-sided only).

alpha

the complement to one of the nominal confidence level

B

number of bootstrap replicates (for type "B")

Details

For more details, please have a look at the references listed below

Value

A list comprising

ML_est

the sample value of the maximum likelihood estimator; for normal r.v. \hat{R}=\Phi[(\bar{x}-\bar{y})/\sqrt{\hat{\sigma}_x^2+\hat{\sigma}_y^2}], where \bar{x} and \bar{y} are the sample means, and \hat{\sigma}_x^2, \hat{\sigma}_y^2 the biased maximum likelihood variance estimators

Downton_est

(for normal r.v.) the sample value of one of the approximated UMVU estimators proposed by Downton \hat{R}'=\Phi[(\bar{x}-\bar{y})/\sqrt{s_x^2+s_y^2}]

CI

the confidence interval

confidence_level

the nominal confidence level 1-\alpha

Author(s)

Alessandro Barbiero, Riccardo Inchingolo

References

Barbiero A (2011) Confidence Intervals for Reliability of Stress-Strength Models in the Normal Case, Comm Stat Sim Comp 40(6):907-925

Downton F. (1973) The Estimation of Pr (Y < X) in the Normal Case, Technometrics , 15(3):551-558

Kotz S, Lumelskii Y, Pensky M (2003) The stress-strength model and its generalizations: theory and applications. World Scientific, Singapore

Guo H, Krishnamoorthy K (2004) New approximate inferential methods for the reliability parameter in a stress-strength model: The normal case. Commun Stat Theory Methods 33:1715-1731

Mukherjee SP, Maiti SS (1998) Stress-strength reliability in the Weibull case. Frontiers In Reliability 4:231-248. WorldScientific, Singapore

Reiser BJ, Guttman I (1986) Statistical inference for P(Y<X): The normal case. Technometrics 28:253-257

See Also

SSR

Examples

# distributional parameters of X and Y
parx<-c(1, 1)
pary<-c(0, 2)
# sample sizes
n<-10
m<-20
# true value of R
SSR(parx,pary)
# draw independent random samples from X and Y
x<-rnorm(n, parx[1], parx[2])
y<-rnorm(m, pary[1], pary[2])
# build two-sided confidence intervals
estSSR(x, y, type="RG")
estSSR(x, y, type="AN")
estSSR(x, y, type="LOGIT")
estSSR(x, y, type="ARCSIN")
estSSR(x, y, type="B")
estSSR(x, y, type="B",B=1000) # change number of bootstrap replicates
# and one-sided
estSSR(x, y, type="RG", twoside=FALSE)
estSSR(x, y, type="AN", twoside=FALSE)
estSSR(x, y, type="LOGIT", twoside=FALSE)
estSSR(x, y, type="ARCSIN", twoside=FALSE)
estSSR(x, y, type="B", twoside=FALSE)
estSSR(x, y, type="GK", twoside=FALSE)
# changing sample sizes
n<-20
m<-30
x<-rnorm(n, parx[1], parx[2])
y<-rnorm(m, pary[1], pary[2])
# build tow-sided confidence intervals
estSSR(x, y, type="RG")
estSSR(x, y, type="AN")
estSSR(x, y, type="LOGIT")
estSSR(x, y, type="ARCSIN")
estSSR(x, y, type="B")

Numerical solution for an equation involving noncentral T cdf

Description

It provides the solution of the equation F_t(q;df,x)=p, where F_t is the cdf (calculated in q) of a non-central Student r.v. with df degrees of freedom and unkwon noncentrality parameter x. In R code, gkf provides the solution of pt(q,df,x)=p.

Usage

gkf(p, q, df, eps = 1e-05)

Arguments

p

a probability

q

a real value

df

degrees of freedom of noncentral T

eps

tolerance

Details

This function is used for building Guo-Krishnamoorthy confidence intervals for R

Value

the noncentrality parameter x satisfying the equation F_t(q;df,x)=p

Author(s)

Alessandro Barbiero, Riccardo Inchingolo

References

Guo H, Krishnamoorthy K (2004) New approximate inferential methods for the reliability parameter in a stress-strength model: The normal case. Commun Stat Theory Methods 33:1715-1731

See Also

estSSR

Examples

p<-0.95
q<-5
df<-12
ncp<-gkf(p, q, df)
ncp
# check if the result is correct
pt(q, df, ncp)
# OK
# changing the tolerance
ncp<-gkf(p, q, df, eps=1e-10)
ncp
pt(q, df, ncp)