Reliability Estimation for StressStrength Model Based on UnitHalfNormal Distribution
Abstract
:1. Introduction
2. Entropy and Mean Residual Life
2.1. Entropy
2.2. Mean Residual Life
3. Stress–Strength Reliability Model
3.1. Maximum Likelihood Estimation of R
 1.
 ${\sum}_{i=1}^{n}{\left(\right)}^{\frac{{U}_{i}}{\eta}}2$ and ${\sum}_{j=1}^{m}{\left(\right)}^{\frac{{V}_{j}}{\lambda}}2$
 2.
 $\mathbb{E}\left({\widehat{\eta}}^{2}\right)={\eta}^{2}$ and $\mathbb{E}\left({\widehat{\lambda}}^{2}\right)={\lambda}^{2}$
 3.
 ${\left(\right)}^{\frac{\widehat{\eta}}{\widehat{\lambda}}}2\sim {F}_{(n,m)}$
 4.
 $\mathbb{E}\left(\right)open="("\; close=")">\frac{{\widehat{\eta}}^{2}}{{\widehat{\lambda}}^{2}},\phantom{\rule{1.em}{0ex}}m2$.
 1.
 ${\chi}_{\eta}=\frac{\sqrt{n}}{\eta}\widehat{\eta}\sim {\chi}_{\left(n\right)}$, then$\mathbb{E}\left({\chi}_{\eta}\right)=\sqrt{2}\frac{\Gamma \left(\right(n+1)/2)}{\Gamma (n/2)}$and$Var\left({\chi}_{\eta}\right)=n{\mathbb{E}}^{2}\left({\chi}_{\eta}\right)$.
 2.
 ${\chi}_{\lambda}=\frac{\sqrt{m}}{\lambda}\widehat{\lambda}\sim {\chi}_{\left(m\right)}$, then$\mathbb{E}\left({\chi}_{\lambda}\right)=\sqrt{2}\frac{\Gamma \left(\right(m+1)/2)}{\Gamma (m/2)}$and$Var\left({\chi}_{\lambda}\right)=m{\mathbb{E}}^{2}\left({\chi}_{\lambda}\right)$,
3.2. Confidence Intervals for $\eta $ and $\lambda $
3.3. Exact PDF for R
Algorithm 1: Algorithm to generate observations from $R\sim {f}_{R}(\eta ,\lambda ,n,m)$. 
Require Initialize the algorithm fixing $\eta $, $\lambda $, n and m

Algorithm 2: Algorithm to generate observations from $R\sim {f}_{R}(\eta ,\lambda ,n,m)$. 
Require Initialize the algorithm fixing $\eta $, $\lambda $, n and m

Algorithm 3: Algorithm to generate observations from $R\sim {f}_{R}(\eta ,\lambda ,n,m)$. 
Require Initialize the algorithm fixing $\eta $, $\lambda $, n and m

4. Interval Estimation of $\mathit{R}$
4.1. Exact Confidence Interval
4.2. Asymptotic Distribution and Confidence Interval
4.3. Bootstrap Confidence Intervals
4.3.1. Parametric Bootstrap Sampling Algorithm
 Stage 1 Compute MLE of $\eta $ and $\lambda $, say $\widehat{\eta}$ and $\widehat{\lambda}$, based on data $X={({X}_{1},\dots ,{X}_{n})}^{\prime}$ and $Y={({Y}_{1},\dots ,{Y}_{m})}^{\prime}$.
 Stage 2 Based on $\widehat{\eta}$ and $\widehat{\lambda}$, generates samples ${X}^{\u2605}={({X}_{1}^{\u2605},\dots ,{X}_{n}^{\u2605})}^{\prime}$ from $UHN\left(\widehat{\eta}\right)$ and ${Y}^{\u2605}={({Y}_{1}^{\u2605},\dots ,{Y}_{m}^{\u2605})}^{\prime}$ from $UHN\left(\widehat{\lambda}\right)$ with$$\begin{array}{cc}\hfill {X}_{i}^{\u2605}& =\frac{\widehat{\eta}\phantom{\rule{0.166667em}{0ex}}{\Phi}^{1}\left(\right)open="("\; close=")">\frac{{U}_{i1}+1}{2}}{}1+\widehat{\eta}\phantom{\rule{0.166667em}{0ex}}{\Phi}^{1}\left(\right)open="("\; close=")">\frac{{U}_{i1}+1}{2}\hfill \end{array}\hfill {Y}_{i}^{\u2605}& =\frac{\widehat{\lambda}\phantom{\rule{0.166667em}{0ex}}{\Phi}^{1}\left(\right)open="("\; close=")">\frac{{U}_{i2}+1}{2}}{}1+\widehat{\lambda}\phantom{\rule{0.166667em}{0ex}}{\Phi}^{1}\left(\right)open="("\; close=")">\frac{{U}_{i2}+1}{2}\hfill $$
 Stage 3 Compute MLE of $\eta $ and $\lambda $, say ${\widehat{\eta}}^{\u2605}$ and ${\widehat{\lambda}}^{\u2605}$, based on data ${X}^{\u2605}$ and ${Y}^{\u2605}$, respectively.
 Stage 4 Compute MLE of R, say ${\widehat{R}}^{\u2605}$, based on ${\widehat{\eta}}^{\u2605}$ and ${\widehat{\lambda}}^{\u2605}$.
 Stage 5 Repeat Steps 2 to 4 B times and generate B bootstrap estimates of $\eta $, $\lambda $ and R.
4.3.2. Nonparametric Bootstrap Sampling Algorithm
 Stage 1 Draw random samples with replacement ${X}^{\u2605}={({X}_{1}^{\u2605},\dots ,{X}_{n}^{\u2605})}^{\prime}$ and ${Y}^{\u2605}={({Y}_{1}^{\u2605},\dots ,{Y}_{m}^{\u2605})}^{\prime}$ from the original data $X={({X}_{1},\dots ,{X}_{n})}^{\prime}$ and $Y={({Y}_{1},\dots ,{Y}_{m})}^{\prime}$, respectively.
 Stage 2 Compute the bootstrap estimates $\eta $ and $\lambda $, say ${\widehat{\eta}}^{\u2605}$ and ${\widehat{\lambda}}^{\u2605}$, based on data ${X}^{\u2605}$ and ${Y}^{\u2605}$, respectively.
 Stage 3 Using ${\widehat{\eta}}^{\u2605}$ and ${\widehat{\lambda}}^{\u2605}$ and Equation (8), compute the bootstrap estimate of R, say ${\widehat{R}}^{\u2605}$.
 Stage 4 Repeat Steps 1 and 3 B times and generate B bootstrap estimates of $\eta $, $\lambda $ and R.
5. Simulation Study
6. An Illustrative Example
7. Concluding Remarks
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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$(\mathit{n},\mathit{m})$  Method  $\mathit{\eta}$  $\mathit{\lambda}$  R 

$(15,20)$  MLE  −0.0044(0.0029)  −0.0025(0.0010)  −0.0052(0.0051) 
Npar.Boot  −0.0086(0.0029)  −0.0046(0.0010)  −0.0098(0.0051)  
Par.Boot  −0.0093(0.0029)  −0.0049(0.0010)  −0.0105(0.0050)  
$(20,15)$  MLE  −0.0039(0.0022)  −0.0029(0.0013)  −0.0014(0.0049) 
Npar.Boot  −0.0072(0.0022)  −0.0057(0.0013)  −0.0022(0.0048)  
Par.Boot  −0.0076(0.0022)  −0.0062(0.0013)  −0.0021(0.0047)  
$(20,20)$  MLE  −0.0032(0.0022)  −0.0029(0.0010)  −0.0017(0.0044) 
Npar.Boot  −0.0065(0.0022)  −0.0051(0.0010)  −0.0040(0.0043)  
Par.Boot  −0.0069(0.0022)  −0.0053(0.0010)  −0.0043(0.0043)  
$(20,30)$  MLE  −0.0033(0.0022)  −0.0017(0.0007)  −0.0042(0.0038) 
Npar.Boot  −0.0065(0.0022)  −0.0032(0.0007)  −0.0081(0.0037)  
Par.Boot  −0.0070(0.0022)  −0.0034(0.0007)  −0.0088(0.0037)  
$(30,20)$  MLE  −0.0020(0.0015)  −0.0021(0.0010)  $3.6\times {10}^{5}\left(0.0036\right)$ 
Npar.Boot  −0.0043(0.0015)  −0.0043(0.0010)  $3.2\times {10}^{6}\left(0.0035\right)$  
Par.Boot  −0.0045(0.0015)  −0.0045(0.0010)  0.0002(0.0035)  
$(30,30)$  MLE  −0.0024(0.0015)  −0.0016(0.0007)  −0.0019(0.0030) 
Npar.Boot  −0.0047(0.0015)  −0.0031(0.0007)  −0.0036(0.0030)  
Par.Boot  −0.0049(0.0015)  −0.0033(0.0007)  −0.0036(0.0030)  
$(30,50)$  MLE  −0.0027(0.0015)  −0.0011(0.0004)  −0.0036(0.0024) 
Npar.Boot  −0.0050(0.0015)  −0.0021(0.0004)  −0.0066(0.0024)  
Par.Boot  −0.0052(0.0015)  −0.0021(0.0004)  −0.0069(0.0024)  
$(50,30)$  MLE  −0.0016(0.0009)  −0.0013(0.0007)  −0.0002(0.0022) 
Npar.Boot  −0.0030(0.0009)  −0.0028(0.0007)  0.0002(0.0022)  
Par.Boot  −0.0031(0.0009)  −0.0029(0.0007)  0.0003(0.0022)  
$(50,50)$  MLE  −0.0011(0.0009)  −0.0010(0.0004)  −0.0007(0.0018) 
Npar.Boot  −0.0025(0.0009)  −0.0019(0.0004)  −0.0018(0.0017)  
Par.Boot  −0.0026(0.0009)  −0.0020(0.0004)  −0.0018(0.0017)  
$(100,100)$  MLE  −0.0005(0.0004)  −0.0002(0.0002)  −0.0007(0.0009) 
Npar.Boot  −0.0013(0.0004)  −0.0007(0.0002)  −0.0013(0.0009)  
Par.Boot  −0.0013(0.0004)  −0.0007(0.0002)  −0.0013(0.0009) 
$(\mathit{n},\mathit{m})$  Method  $\mathit{\eta}$  $\mathit{\lambda}$  R  

$(15,20)$  Exact  0.2383(0.950)  0.1163(0.943)  0.2766(0.944)  
Asympt.  0.2108(0.944)  0.1225(0.934)  0.2732(0.941)  
Nonpar  t  0.2402(0.949)  0.1187(0.941)  0.2768(0.942)  
Boot  q  0.1671(0.945)  0.1101(0.942)  0.2769(0.943)  
CIs  $BCa$  0.1720(0.943)  0.1105(0.943)  0.2744(0.942)  
Par  t  0.2394(0.942)  0.1174(0.943)  0.2801(0.945)  
Boot  q  0.2384(0.951)  0.1161(0.942)  0.2765(0.943)  
CIs  $BCa$  0.2382(0.949)  0.1164(0.942)  0.2750(0.947)  
$(20,15)$  Exact  0.2018(0.947)  0.1838(0.943)  0.2719(0.951)  
Asympt.  0.1842(0.946)  0.1408(0.941)  0.2713(0.947)  
Nonpar  t  0.2201(0.950)  0.1123(0.945)  0.2718(0.948)  
Boot  q  0.1834(0.949)  0.1089(0.946)  0.2617(0.947)  
CIs  $BCa$  0.1923(0.948)  0.1166(0.947)  0.2742(0.953)  
Par  t  0.2386(0.947)  0.1812(0.947)  0.2760(0.949)  
Boot  q  0.2381(0.948)  0.1705(0.946)  0.2748(0.951)  
CIs  $BCa$  0.2385(0.949)  0.1877(0.945)  0.2742(0.958)  
$(20,20)$  Exact  0.2013(0.945)  0.1349(0.952)  0.2542(0.947)  
Asympt.  0.1838(0.942)  0.1223(0.949)  0.2527(0.942)  
Nonpar  t  0.2206(0.949)  0.1125(0.947)  0.2721(0.943)  
Boot  q  0.1832(0.949)  0.1087(0.947)  0.2611(0.948)  
CIs  $BCa$  0.1920(0.947)  0.1169(0.948)  0.2557(0.946)  
Par  t  0.2249(0.948)  0.1311(0.948)  0.2758(0.947)  
Boot  q  0.2245(0.947)  0.1201(0.949)  0.2751(0.946)  
CIs  $BCa$  0.2249(0.948)  0.1308(0.947)  0.2556(0.945)  
$(20,30)$  Exact  0.2017(0.947)  0.1271(0.943)  0.2346(0.945)  
Asympt.  0.1841(0.946)  0.1004(0.944)  0.2320(0.944)  
Nonpar  t  0.1918(0.949)  0.1169(0.945)  0.2232(0.945)  
Boot  q  0.1799(0.948)  0.1001(0.946)  0.2311(0.946)  
CIs  $BCa$  0.1801(0.949)  0.1007(0.947)  0.2301(0.951)  
Par  t  0.2116(0.946)  0.1315(0.947)  0.2351(0.946)  
Boot  q  0.2011(0.947)  0.1316(0.948)  0.2313(0.947)  
CIs  $BCa$  0.2007(0.948)  0.1318(0.949)  0.2332(0.955)  
$(30,20)$  Exact  0.1596(0.951)  0.1643(0.941)  0.2311(0.953)  
Asympt.  0.1503(0.945)  0.1224(0.942)  0.2311(0.949)  
Nonpar  t  0.1501(0.949)  0.1568(0.946)  0.2278(0.945)  
Boot  q  0.1424(0.947)  0.1502(0.947)  0.2199(0.946)  
CIs  $BCa$  0.1425(0.947)  0.1507(0.948)  0.2121(0.946)  
Par  t  0.1602(0.947)  0.1677(0.943)  0.2401(0.948)  
Boot  q  0.1599(0.948)  0.1601(0.947)  0.2397(0.946)  
CIs  $BCa$  0.1566(0.947)  0.1609(0.947)  0.2320(0.945)  
$(30,30)$  Exact  0.1597(0.949)  0.1065(0.952)  0.2086(0.948)  
Asympt.  0.1504(0.942)  0.1003(0.943)  0.2078(0.946)  
Nonpar  t  0.1495(0.951)  0.1044(0.953)  0.2084(0.951)  
Boot  q  0.1485(0.948)  0.0979(0.951)  0.1999(0.947)  
CIs  $BCa$  0.1486(0.945)  0.0977(0.949)  0.2082(0.943)  
Par  t  0.1604(0.949)  0.1071(0.950)  0.2117(0.948)  
Boot  q  0.1601(0.948)  0.1063(0.951)  0.2085(0.945)  
CIs  $BCa$  0.1598(0.948)  0.1065(0.948)  0.2080(0.944)  
$(30,50)$  Exact  0.1604(0.949)  0.1026(0.935)  0.1881(0.949)  
Asympt.  0.1510(0.934)  0.1080(0.938)  0.1865(0.945)  
Nonpar  t  0.1485(0.950)  0.0998(0.938)  0.1856(0.946)  
Boot  q  0.1449(0.948)  0.0904(0.937)  0.1855(0.947)  
CIs  $BCa$  0.1451(0.951)  0.0911(0.938)  0.1870(0.945)  
Par  t  0.1649(0.946)  0.1087(0.939)  0.1882(0.942)  
Boot  q  0.1601(0.949)  0.1023(0.940)  0.1876(0.944)  
CIs  $BCa$  0.1604(0.947)  0.1024(0.941)  0.1875(0.935)  
$(50,30)$  Exact  0.1211(0.951)  0.1372(0.937)  0.1855(0.951)  
Asympt.  0.1169(0.941)  0.1000(0.932)  0.1856(0.940)  
Nonpar  t  0.1171(0.943)  0.0989(0.942)  0.1823(0.942)  
Boot  q  0.1078(0.948)  0.0942(0.940)  0.1799(0.946)  
CIs  $BCa$  0.1081(0.947)  0.0943(0.941)  0.1854(0.946)  
Par  t  0.1285(0.946)  0.1389(0.945)  0.1899(0.948)  
Boot  q  0.1203(0.945)  0.1367(0.946)  0.1862(0.945)  
CIs  $BCa$  0.1213(0.949)  0.1369(0.945)  0.1860(0.944)  
$(50,50)$  Exact  0.1215(0.951)  0.0810(0.954)  0.1621(0.950)  
Asympt.  0.1172(0.940)  0.0781(0.942)  0.1617(0.944)  
Nonpar  t  0.1149(0.944)  0.0751(0.945)  0.1602(0.948)  
Boot  q  0.1102(0.943)  0.0733(0.943)  0.1599(0.950)  
CIs  $BCa$  0.1101(0.942)  0.0731(0.944)  0.1625(0.949)  
Par  t  0.1201(0.942)  0.0791(0.942)  0.1672(0.947)  
Boot  q  0.1172(0.941)  0.0773(0.941)  0.1624(0.943)  
CIs  $BCa$  0.1199(0.941)  0.0785(0.940)  0.1623(0.950)  
$(100,100)$  Exact  0.0845(0.948)  0.0563(0.946)  0.1149(0.943)  
Asympt.  0.0830(0.945)  0.0553(0.944)  0.1147(0.942)  
Nonpar  t  0.0865(0.942)  0.0571(0.943)  0.1153(0.944)  
Boot  q  0.0818(0.944)  0.0498(0.947)  0.1001(0.944)  
CIs  $BCa$  0.0830(0.943)  0.5341(0.941)  0.1154(0.953)  
Par  t  0.0838(0.942)  0.0862(0.951)  0.1162(0.945)  
Boot  q  0.0834(0.941)  0.0856(0.952)  0.1049(0.943)  
CIs  $BCa$  0.0836(0.943)  0.0861(0.951)  0.1150(0.954) 
0.04, 0.02, 0.06, 0.12, 0.14, 0.08, 0.22, 0.12, 0.08, 0.26, 
0.24, 0.04, 0.14, 0.16, 0.08, 0.26, 0.32, 0.28, 0.14, 0.16, 
0.24, 0.22, 0.12, 0.18, 0.24, 0.32, 0.16, 0.14, 0.08, 0.16, 
0.24, 0.16, 0.32, 0.18, 0.24, 0.22, 0.16, 0.12, 0.24, 0.06, 
0.02, 0.18, 0.22, 0.14, 0.06, 0.04, 0.14, 0.26, 0.18, 0.16 
0.06, 0.12, 0.14, 0.04, 0.14, 0.16, 0.08, 0.26, 0.32, 0.22, 
0.16, 0.12, 0.24, 0.06, 0.02, 0.18, 0.22, 0.14, 0.22, 0.16, 
0.12, 0.24, 0.06, 0.02, 0.18, 0.22, 0.14, 0.02, 0.18, 0.22, 
0.14, 0.06, 0.04, 0.14, 0.22, 0.14, 0.06, 0.04, 0.16, 0.24, 
0.16, 0.32, 0.18, 0.24, 0.22, 0.04, 0.14, 0.26, 0.18, 0.16 
$\widehat{\mathit{\eta}}$  $\widehat{\mathit{\lambda}}$  $\widehat{\mathit{R}}$  

MLE  ${0.238}_{\left(0.024\right)}$  ${0.219}_{\left(0.022\right)}$  ${0.526}_{\left(0.045\right)}$ 
Npar.Boot  ${0.237}_{\left(0.017\right)}$  ${0.219}_{\left(0.016\right)}$  ${0.526}_{\left(0.031\right)}$ 
Par.Boot  ${0.235}_{\left(0.024\right)}$  ${0.218}_{\left(0.023\right)}$  ${0.524}_{\left(0.045\right)}$ 
KS${}^{1}$:  $D\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.080$  pval = 0.726  
KS${}^{2}$:  $D\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.100$  pval = 0.607 
$\mathit{\eta}$  $\mathit{\lambda}$  R  

Exact  (0.199, 0.296)  (0.184, 0.273)  (0.437, 0.613)  
Asympt.  (0.191, 0.285)  (0.176, 0.262)  (0.438, 0.614)  
Nonpar  t  (0.205, 0.277)  (0.190, 0.257)  (0.455, 0.586) 
Boot  q  (0.204, 0.271)  (0.188, 0.250)  (0.463, 0.586) 
CIs  $BCa$  (0.207, 0.274)  (0.191, 0.254)  (0.460, 0.584) 
Par  t  (0.196, 0.302)  (0.183, 0.276)  (0.425, 0.628) 
Boot  q  (0.188, 0.287)  (0.175, 0.262)  (0.431, 0.615) 
CIs  $BCa$  (0.199, 0.296)  (0.184, 0.273)  (0.437, 0.612) 
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de la Cruz, R.; Salinas, H.S.; Meza, C. Reliability Estimation for StressStrength Model Based on UnitHalfNormal Distribution. Symmetry 2022, 14, 837. https://doi.org/10.3390/sym14040837
de la Cruz R, Salinas HS, Meza C. Reliability Estimation for StressStrength Model Based on UnitHalfNormal Distribution. Symmetry. 2022; 14(4):837. https://doi.org/10.3390/sym14040837
Chicago/Turabian Stylede la Cruz, Rolando, Hugo S. Salinas, and Cristian Meza. 2022. "Reliability Estimation for StressStrength Model Based on UnitHalfNormal Distribution" Symmetry 14, no. 4: 837. https://doi.org/10.3390/sym14040837