Title: | Computation and Estimation of Reliability of Stress-Strength Models |
---|---|
Description: | Reliability of (normal) stress-strength models and for building two-sided or one-sided confidence intervals according to different approximate procedures. |
Authors: | Alessandro Barbiero <[email protected]>, Riccardo Inchingolo <[email protected]> |
Maintainer: | Alessandro Barbiero <[email protected]> |
License: | GPL |
Version: | 1.0.2 |
Built: | 2024-12-16 06:39:01 UTC |
Source: | CRAN |
Reliability of (normal) stress-strength models and for building two-side or one-side confidence intervals according to different approximate procedures.
Package: | StressStrength |
Type: | Package |
Version: | 1.0.2 |
Date: | 2016-04-29 |
License: | GPL |
LazyLoad: | yes |
Alessandro Barbiero, Riccardo Inchingolo
Maintainer: Alessandro Barbiero <[email protected]>
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
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
estSSR(x, y, family="normal", twoside=TRUE, type="RG", alpha=0.05, B=2000)
estSSR(x, y, family="normal", twoside=TRUE, type="RG", alpha=0.05, B=2000)
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") |
For more details, please have a look at the references listed below
A list comprising
ML_est |
the sample value of the maximum likelihood estimator; for normal r.v. |
Downton_est |
(for normal r.v.) the sample value of one of the approximated UMVU estimators proposed by Downton |
CI |
the confidence interval |
confidence_level |
the nominal confidence level |
Alessandro Barbiero, Riccardo Inchingolo
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
# 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")
# 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")
It provides the solution of the equation , where
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
.
gkf(p, q, df, eps = 1e-05)
gkf(p, q, df, eps = 1e-05)
p |
a probability |
q |
a real value |
df |
degrees of freedom of noncentral T |
eps |
tolerance |
This function is used for building Guo-Krishnamoorthy confidence intervals for R
the noncentrality parameter satisfying the equation
Alessandro Barbiero, Riccardo Inchingolo
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
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)
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)
For a stress-strength model, with independent r.v. X and Y representing the strength and the stress respectively, the function computes the reliability
SSR(parx, pary, family = "normal")
SSR(parx, pary, family = "normal")
parx |
parameters of X distribution (for the normal distribution, mean |
pary |
parameters of Y distribution (for the normal distribution, mean |
family |
family distribution for both X and Y (now, only "normal" available) |
The function computes where X and Y are independent r.v. following the
family
distribution with distributional parameters parx
and pary
.
. For normal distributions,
with
.
Alessandro Barbiero, Riccardo Inchingolo
Kotz S, Lumelskii Y, Pensky M (2003) The stress-strength model and its generalizations: theory and applications. World Scientific, Singapore
# 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)
# 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)