estSSR: Sample estimation of reliability of stress-strength models
In StressStrength: Computation and Estimation of Reliability of Stress-Strength Models

Description Usage Arguments Details Value Author(s) References See Also Examples

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

1	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. \hat{R}=Φ[(\bar{x}-\bar{y})/√{\hat{σ}_x^2+\hat{σ}_y^2}], where \bar{x} and \bar{y} are the sample means, and \hat{σ}_x^2, \hat{σ}_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}'=Φ[(\bar{x}-\bar{y})/√{s_x^2+s_y^2}]
`CI`	the confidence interval
`confidence_level`	the nominal confidence level 1-α

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

SSR

# 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")

[1] 0.6726396
$ML_est
[1] 0.5549802

$Downton_est
[1] 0.5533748

$CI
[1] 0.3673573 0.7281561

$confidence_level
[1] 0.95

$ML_est
[1] 0.5549802

$Downton_est
[1] 0.5533748

$CI
[1] 0.3689178 0.7294760

$confidence_level
[1] 0.95

$ML_est
[1] 0.5549802

$Downton_est
[1] 0.5533748

$CI
[1] 0.3690406 0.7267037

$confidence_level
[1] 0.95

$ML_est
[1] 0.5549802

$Downton_est
[1] 0.5533748

$CI
[1] 0.3685168 0.7337505

$confidence_level
[1] 0.95

$ML_est
[1] 0.5549802

$Downton_est
[1] 0.5533748

$CI
     2.5%     97.5% 
0.3612767 0.7531282 

$confidence_level
[1] 0.95

$ML_est
[1] 0.5549802

$Downton_est
[1] 0.5533748

$CI
     2.5%     97.5% 
0.3638068 0.7440848 

$confidence_level
[1] 0.95

$ML_est
[1] 0.5549802

$Downton_est
[1] 0.5533748

$CI
[1] 0.3963499

$confidence_level
[1] 0.95

$ML_est
[1] 0.5549802

$Downton_est
[1] 0.5533748

$CI
[1] 0.3979414

$confidence_level
[1] 0.95

$ML_est
[1] 0.5549802

$Downton_est
[1] 0.5533748

$CI
[1] 0.3978083

$confidence_level
[1] 0.95

$ML_est
[1] 0.5549802

$Downton_est
[1] 0.5533748

$CI
[1] 0.3979228

$confidence_level
[1] 0.95

$ML_est
[1] 0.5549802

$Downton_est
[1] 0.5533748

$CI
       5% 
0.3913651 

$confidence_level
[1] 0.95

$ML_est
[1] 0.5549802

$Downton_est
[1] 0.5533748

$CI
[1] 0.3935198

$confidence_level
[1] 0.95

$ML_est
[1] 0.472842

$Downton_est
[1] 0.4733472

$CI
[1] 0.3286397 0.6217025

$confidence_level
[1] 0.95

$ML_est
[1] 0.472842

$Downton_est
[1] 0.4733472

$CI
[1] 0.3281845 0.6212163

$confidence_level
[1] 0.95

$ML_est
[1] 0.472842

$Downton_est
[1] 0.4733472

$CI
[1] 0.3294962 0.6208098

$confidence_level
[1] 0.95

$ML_est
[1] 0.472842

$Downton_est
[1] 0.4733472

$CI
[1] 0.3263167 0.6218000

$confidence_level
[1] 0.95

$ML_est
[1] 0.472842

$Downton_est
[1] 0.4733472

$CI
     2.5%     97.5% 
0.3100088 0.6311439 

$confidence_level
[1] 0.95