| Title: | Analysis of Doubly-Truncated Data |
|---|---|
| Description: | Likelihood-based inference methods with doubly-truncated data are developed under various models. Nonparametric models are based on Efron and Petrosian (1999) <doi:10.1080/01621459.1999.10474187> and Emura, Konno, and Michimae (2015) <doi:10.1007/s10985-014-9297-5>. Parametric models from the special exponential family (SEF) are based on Hu and Emura (2015) <doi:10.1007/s00180-015-0564-z> and Emura, Hu and Konno (2017) <doi:10.1007/s00362-015-0730-y>. The parametric location-scale models are based on Dorre et al. (2020) <doi:10.1007/s00180-020-01027-6>. |
| Authors: | Takeshi Emura, Ya-Hsuan Hu, Chung-Yan Huang |
| Maintainer: | Takeshi Emura <[email protected]> |
| License: | GPL-2 |
| Version: | 1.7 |
| Built: | 2024-11-16 06:44:40 UTC |
| Source: | CRAN |
Likelihood-based inference methods with doubly-truncated data are developed under various models. Nonparametric models are based on Efron and Petrosian (1999) <doi:10.1080/01621459.1999.10474187> and Emura, Konno, and Michimae (2015) <doi:10.1007/s10985-014-9297-5>. Parametric models from the special exponential family (SEF) are based on Hu and Emura (2015) <doi:10.1007/s00180-015-0564-z> and Emura, Hu and Konno (2017) <doi:10.1007/s00362-015-0730-y>. The parametric location-scale models are based on Dorre et al. (2020) <doi:10.1007/s00180-020-01027-6>.
Details are seen from the references.
Takeshi Emura, Ya-Hsuan Hu, Chung-Yan Huang Maintainer: Takeshi Emura <[email protected]>
Dorre A, Emura T (2019) Analysis of Doubly Truncated Data, An Introduction, JSS Research Series in Statistics, Springer
Dorre A, Huang CY, Tseng YK, Emura T (2020) Likelihood-based analysis of doubly-truncated data under the location-scale and AFT model, Computation Stat, DOI:10.1007/s00180-020-01027-6
Efron B, Petrosian V (1999). Nonparametric methods for doubly truncated data. J Am Stat Assoc 94: 824-834
Emura T, Konno Y, Michimae H (2015). Statistical inference based on the nonparametric maximum likelihood estimator under double-truncation. Lifetime Data Analysis 21: 397-418
Emura T, Hu YH, Konno Y (2017) Asymptotic inference for maximum likelihood estimators under the special exponential family with double-truncation, Stat Pap 58 (3): 877-909
Hu YH, Emura T (2015) Maximum likelihood estimation for a special exponential family under random double-truncation, Computation Stat 30 (4): 1199-229
Nonparametric maximum likelihood estimates are computed based on the self-consistency method (Efron and Petrosian 1999). The SE is computed from the asymptotic variance derived in Emura et al. (2015).
NPMLE(u.trunc, y.trunc, v.trunc,epsilon=1e-08)NPMLE(u.trunc, y.trunc, v.trunc,epsilon=1e-08)
u.trunc |
lower truncation limit |
y.trunc |
variable of interest |
v.trunc |
upper truncation limit |
epsilon |
error tolerance for the self-consistency algorithm |
Details are seen from the references.
f |
density |
F |
cumulative distribution |
SE |
standard error |
convergence |
Log-likelihood, and the number of iterations |
Takeshi Emura
Efron B, Petrosian V (1999). Nonparametric methods for doubly truncated data. J Am Stat Assoc 94: 824-834
Emura T, Konno Y, Michimae H (2015). Statistical inference based on the nonparametric maximum likelihood estimator under double-truncation. Lifetime Data Analysis 21: 397-418
Dorre A, Emura T (2019) Analysis of Doubly Truncated Data, An Introduction, JSS Research Series in Statistics, Springer
## A data example from Efron and Petrosian (1999) ## y.trunc=c(0.75, 1.25, 1.50, 1.05, 2.40, 2.50, 2.25) u.trunc=c(0.4, 0.8, 0.0, 0.3, 1.1, 2.3, 1.3) v.trunc=c(2.0, 1.8, 2.3, 1.4, 3.0, 3.4, 2.6) NPMLE(u.trunc,y.trunc,v.trunc)## A data example from Efron and Petrosian (1999) ## y.trunc=c(0.75, 1.25, 1.50, 1.05, 2.40, 2.50, 2.25) u.trunc=c(0.4, 0.8, 0.0, 0.3, 1.1, 2.3, 1.3) v.trunc=c(2.0, 1.8, 2.3, 1.4, 3.0, 3.4, 2.6) NPMLE(u.trunc,y.trunc,v.trunc)
Maximum likelihood estimates and their standard errors (SEs) are computed. Also computed are the likelihood value, AIC, and other qnantities.
PMLE.loglogistic(u.trunc, y.trunc, v.trunc,epsilon = 1e-5,D1=2,D2=2,d1=2,d2=2)PMLE.loglogistic(u.trunc, y.trunc, v.trunc,epsilon = 1e-5,D1=2,D2=2,d1=2,d2=2)
u.trunc |
lower truncation limit |
y.trunc |
variable of interest |
v.trunc |
upper truncation limit |
epsilon |
error tolerance for Newton-Raphson |
D1 |
Randomize the intial value if |mu_h-mu_h+1|>D1 |
D2 |
Randomize the intial value if |sigma_h-sigma_h+1|>D2 |
d1 |
U(-d1,d1) is added to the intial value of mu |
d2 |
U(-d2,d2) is added to the intial value of sigma |
Details are seen from the references.
eta |
estimates |
SE |
standard errors |
convergence |
Log-likelihood, degree of freedom, AIC, the number of iterations |
Score |
score vector at the converged value |
Hessian |
Hessian matrix at the converged value |
Takeshi Emura
Dorre A, Huang CY, Tseng YK, Emura T (2020-) Likelihood-based analysis of doubly-truncated data under the location-scale and AFT model, Computation Stat, DOI:10.1007/s00180-020-01027-6
## A data example from Efron and Petrosian (1999) ## y.trunc=c(0.75, 1.25, 1.50, 1.05, 2.40, 2.50, 2.25) u.trunc=c(0.4, 0.8, 0.0, 0.3, 1.1, 2.3, 1.3) v.trunc=c(2.0, 1.8, 2.3, 1.4, 3.0, 3.4, 2.6) PMLE.loglogistic(u.trunc,y.trunc,v.trunc)## A data example from Efron and Petrosian (1999) ## y.trunc=c(0.75, 1.25, 1.50, 1.05, 2.40, 2.50, 2.25) u.trunc=c(0.4, 0.8, 0.0, 0.3, 1.1, 2.3, 1.3) v.trunc=c(2.0, 1.8, 2.3, 1.4, 3.0, 3.4, 2.6) PMLE.loglogistic(u.trunc,y.trunc,v.trunc)
Maximum likelihood estimates and their standard errors (SEs) are computed. Also computed are the likelihood value, AIC, and other qnantities.
PMLE.lognormal(u.trunc, y.trunc, v.trunc,epsilon = 1e-5,D1=2,D2=2,d1=2,d2=2)PMLE.lognormal(u.trunc, y.trunc, v.trunc,epsilon = 1e-5,D1=2,D2=2,d1=2,d2=2)
u.trunc |
lower truncation limit |
y.trunc |
variable of interest |
v.trunc |
upper truncation limit |
epsilon |
error tolerance for Newton-Raphson |
D1 |
Randomize the intial value if |mu_h-mu_h+1|>D1 |
D2 |
Randomize the intial value if |sigma_h-sigma_h+1|>D2 |
d1 |
U(-d1,d1) is added to the intial value of mu |
d2 |
U(-d2,d2) is added to the intial value of sigma |
Details are seen from the references.
eta |
estimates |
SE |
standard errors |
convergence |
Log-likelihood, degree of freedom, AIC, the number of iterations |
Score |
score vector at the converged value |
Hessian |
Hessian matrix at the converged value |
Takeshi Emura
Dorre A, Huang CY, Tseng YK, Emura T (2020-) Likelihood-based analysis of doubly-truncated data under the location-scale and AFT model, Computation Stat, DOI:10.1007/s00180-020-01027-6
## A data example from Efron and Petrosian (1999) ## y.trunc=c(0.75, 1.25, 1.50, 1.05, 2.40, 2.50, 2.25) u.trunc=c(0.4, 0.8, 0.0, 0.3, 1.1, 2.3, 1.3) v.trunc=c(2.0, 1.8, 2.3, 1.4, 3.0, 3.4, 2.6) PMLE.lognormal(u.trunc,y.trunc,v.trunc)## A data example from Efron and Petrosian (1999) ## y.trunc=c(0.75, 1.25, 1.50, 1.05, 2.40, 2.50, 2.25) u.trunc=c(0.4, 0.8, 0.0, 0.3, 1.1, 2.3, 1.3) v.trunc=c(2.0, 1.8, 2.3, 1.4, 3.0, 3.4, 2.6) PMLE.lognormal(u.trunc,y.trunc,v.trunc)
Maximum likelihood estimates and their standard errors (SEs) are computed. Also computed are the likelihood value, AIC, and other qnantities.
PMLE.SEF1.free(u.trunc, y.trunc, v.trunc, tau1 = min(y.trunc), tau2 = max(y.trunc), epsilon = 1e-04)PMLE.SEF1.free(u.trunc, y.trunc, v.trunc, tau1 = min(y.trunc), tau2 = max(y.trunc), epsilon = 1e-04)
u.trunc |
lower truncation limit |
y.trunc |
variable of interest |
v.trunc |
upper truncation limit |
tau1 |
lower support |
tau2 |
upper support |
epsilon |
error tolerance for Newton-Raphson |
Details are seen from the references.
eta |
estimates |
SE |
standard errors |
convergence |
Log-likelihood, degree of freedom, AIC, the number of iterations |
Score |
score at the converged value |
Hessian |
Hessian at the converged value |
Takeshi Emura
Hu YH, Emura T (2015) Maximum likelihood estimation for a special exponential family under random double-truncation, Computation Stat 30 (4): 1199-229
Emura T, Hu YH, Konno Y (2017) Asymptotic inference for maximum likelihood estimators under the special exponential family with double-truncation, Stat Pap 58 (3): 877-909
Dorre A, Emura T (2019) Analysis of Doubly Truncated Data, An Introduction, JSS Research Series in Statistics, Springer
### Data generation: see Appendix of Hu and Emura (2015) ### eta_true=-3 eta_u=-9 eta_v=-1 tau=10 n=300 a=u=v=y=c() j=1 repeat{ u1=runif(1,0,1) u[j]=tau+(1/eta_u)*log(1-u1) u2=runif(1,0,1) v[j]=tau+(1/eta_v)*log(1-u2) u3=runif(1,0,1) y[j]=tau+(1/eta_true)*log(1-u3) if(u[j]<=y[j]&&y[j]<=v[j]) a[j]=1 else a[j]=0 if(sum(a)==n) break j=j+1 } mean(a) ## inclusion probability around 0.5 v.trunc=v[a==1] u.trunc=u[a==1] y.trunc=y[a==1] PMLE.SEF1.free(u.trunc,y.trunc,v.trunc)### Data generation: see Appendix of Hu and Emura (2015) ### eta_true=-3 eta_u=-9 eta_v=-1 tau=10 n=300 a=u=v=y=c() j=1 repeat{ u1=runif(1,0,1) u[j]=tau+(1/eta_u)*log(1-u1) u2=runif(1,0,1) v[j]=tau+(1/eta_v)*log(1-u2) u3=runif(1,0,1) y[j]=tau+(1/eta_true)*log(1-u3) if(u[j]<=y[j]&&y[j]<=v[j]) a[j]=1 else a[j]=0 if(sum(a)==n) break j=j+1 } mean(a) ## inclusion probability around 0.5 v.trunc=v[a==1] u.trunc=u[a==1] y.trunc=y[a==1] PMLE.SEF1.free(u.trunc,y.trunc,v.trunc)
Maximum likelihood estimates and their standard errors (SEs) are computed. Also computed are the likelihood value, AIC, and other qnantities.
PMLE.SEF1.negative(u.trunc, y.trunc, v.trunc, tau1 = min(y.trunc), epsilon = 1e-04)PMLE.SEF1.negative(u.trunc, y.trunc, v.trunc, tau1 = min(y.trunc), epsilon = 1e-04)
u.trunc |
lower truncation limit |
y.trunc |
variable of interest |
v.trunc |
upper truncation limit |
tau1 |
lower support |
epsilon |
error tolerance for Newton-Raphson |
Details are seen from the references.
eta |
estimates |
SE |
standard errors |
convergence |
Log-likelihood, degree of freedom, AIC, the number of iterations |
Score |
score at the converged value |
Hessian |
Hessian at the converged value |
Takeshi Emura, Ya-Hsuan Hu
Hu YH, Emura T (2015) Maximum likelihood estimation for a special exponential family under random double-truncation, Computation Stat 30 (4): 1199-229
Emura T, Hu YH, Konno Y (2017) Asymptotic inference for maximum likelihood estimators under the special exponential family with double-truncation, Stat Pap 58 (3): 877-909
Dorre A, Emura T (2019) Analysis of Doubly Truncated Data, An Introduction, JSS Research Series in Statistics, Springer
### Data generation: see Appendix of Hu and Emura (2015) ### eta_true=-3 eta_u=-9 eta_v=-1 tau=10 n=300 a=u=v=y=c() j=1 repeat{ u1=runif(1,0,1) u[j]=tau+(1/eta_u)*log(1-u1) u2=runif(1,0,1) v[j]=tau+(1/eta_v)*log(1-u2) u3=runif(1,0,1) y[j]=tau+(1/eta_true)*log(1-u3) if(u[j]<=y[j]&&y[j]<=v[j]) a[j]=1 else a[j]=0 if(sum(a)==n) break j=j+1 } mean(a) ## inclusion probability around 0.5 v.trunc=v[a==1] u.trunc=u[a==1] y.trunc=y[a==1] PMLE.SEF1.negative(u.trunc,y.trunc,v.trunc)### Data generation: see Appendix of Hu and Emura (2015) ### eta_true=-3 eta_u=-9 eta_v=-1 tau=10 n=300 a=u=v=y=c() j=1 repeat{ u1=runif(1,0,1) u[j]=tau+(1/eta_u)*log(1-u1) u2=runif(1,0,1) v[j]=tau+(1/eta_v)*log(1-u2) u3=runif(1,0,1) y[j]=tau+(1/eta_true)*log(1-u3) if(u[j]<=y[j]&&y[j]<=v[j]) a[j]=1 else a[j]=0 if(sum(a)==n) break j=j+1 } mean(a) ## inclusion probability around 0.5 v.trunc=v[a==1] u.trunc=u[a==1] y.trunc=y[a==1] PMLE.SEF1.negative(u.trunc,y.trunc,v.trunc)
Maximum likelihood estimates and their standard errors (SEs) are computed. Also computed are the likelihood value, AIC, and other qnantities.
PMLE.SEF1.positive(u.trunc, y.trunc, v.trunc, tau2 = max(y.trunc), epsilon = 1e-04)PMLE.SEF1.positive(u.trunc, y.trunc, v.trunc, tau2 = max(y.trunc), epsilon = 1e-04)
u.trunc |
lower truncation limit |
y.trunc |
variable of interest |
v.trunc |
upper truncation limit |
tau2 |
upper support |
epsilon |
error tolerance for Newton-Raphson |
Details are seen from the references.
eta |
estimates |
SE |
standard errors |
convergence |
Log-likelihood, degree of freedom, AIC, the number of iterations |
Score |
score at the converged value |
Hessian |
Hessian at the converged value |
Takeshi Emura, Ya-Hsuan Hu
Hu YH, Emura T (2015) Maximum likelihood estimation for a special exponential family under random double-truncation, Computation Stat 30 (4): 1199-229
Emura T, Hu YH, Konno Y (2017) Asymptotic inference for maximum likelihood estimators under the special exponential family with double-truncation, Stat Pap 58 (3): 877-909
Dorre A, Emura T (2019) Analysis of Doubly Truncated Data, An Introduction, JSS Research Series in Statistics, Springer
#### Data generation: Appendix of Hu and Emura (2015) eta_true=3 eta_u=1 eta_v=9 tau=10 n=300 a=u=v=y=c() j=1 repeat{ u1=runif(1,0,1) u[j]=tau+(1/eta_u)*log(u1) u2=runif(1,0,1) v[j]=tau+(1/eta_v)*log(u2) u3=runif(1,0,1) y[j]=tau+(1/eta_true)*log(u3) if(u[j]<=y[j]&&y[j]<=v[j]) a[j]=1 else a[j]=0 if(sum(a)==n) break j=j+1 } mean(a) ## inclusion probability around 0.5 v.trunc=v[a==1] u.trunc=u[a==1] y.trunc=y[a==1] PMLE.SEF1.positive(u.trunc,y.trunc,v.trunc)#### Data generation: Appendix of Hu and Emura (2015) eta_true=3 eta_u=1 eta_v=9 tau=10 n=300 a=u=v=y=c() j=1 repeat{ u1=runif(1,0,1) u[j]=tau+(1/eta_u)*log(u1) u2=runif(1,0,1) v[j]=tau+(1/eta_v)*log(u2) u3=runif(1,0,1) y[j]=tau+(1/eta_true)*log(u3) if(u[j]<=y[j]&&y[j]<=v[j]) a[j]=1 else a[j]=0 if(sum(a)==n) break j=j+1 } mean(a) ## inclusion probability around 0.5 v.trunc=v[a==1] u.trunc=u[a==1] y.trunc=y[a==1] PMLE.SEF1.positive(u.trunc,y.trunc,v.trunc)
Maximum likelihood estimates and their standard errors (SEs) are computed. Also computed are the likelihood value, AIC, and other qnantities. Since this is the model, estimates for the mean and SD are also computed.
PMLE.SEF2.negative(u.trunc, y.trunc, v.trunc, epsilon = 1e-04)PMLE.SEF2.negative(u.trunc, y.trunc, v.trunc, epsilon = 1e-04)
u.trunc |
lower truncation limit |
y.trunc |
variable of interest |
v.trunc |
upper truncation limit |
epsilon |
error tolerance for Newton-Raphson |
Details are seen from the references.
eta |
estimates |
SE |
standard errors |
convergence |
Log-likelihood, degree of freedom, AIC, the number of iterations |
Score |
score vector at the converged value |
Hessian |
Hessian matrix at the converged value |
Takeshi Emura, Ya-Hsuan Hu
Hu YH, Emura T (2015) Maximum likelihood estimation for a special exponential family under random double-truncation, Computation Stat 30 (4): 1199-229
Emura T, Hu YH, Konno Y (2017) Asymptotic inference for maximum likelihood estimators under the special exponential family with double-truncation, Stat Pap 58 (3): 877-909
Dorre A, Emura T (2019) Analysis of Doubly Truncated Data, An Introduction, JSS Research Series in Statistics, Springer
### Data generation: see Appendix of Hu and Emura (2015) n=300 eta1_true=30 eta2_true=-0.5 mu_true=30 mu_u=29.09 mu_v=30.91 a=u=v=y=c() ###generate n samples of (ui,yi,vi) subject to ui<=yi<=vi### j=1 repeat{ u[j]=rnorm(1,mu_u,1) v[j]=rnorm(1,mu_v,1) y[j]=rnorm(1,mu_true,1) if(u[j]<=y[j]&&y[j]<=v[j]) a[j]=1 else a[j]=0 if(sum(a)==n) break ###we need n data set### j=j+1 } mean(a) ### inclusion probability around 0.5 ### v.trunc=v[a==1] y.trunc=y[a==1] u.trunc=u[a==1] PMLE.SEF2.negative(u.trunc,y.trunc,v.trunc)### Data generation: see Appendix of Hu and Emura (2015) n=300 eta1_true=30 eta2_true=-0.5 mu_true=30 mu_u=29.09 mu_v=30.91 a=u=v=y=c() ###generate n samples of (ui,yi,vi) subject to ui<=yi<=vi### j=1 repeat{ u[j]=rnorm(1,mu_u,1) v[j]=rnorm(1,mu_v,1) y[j]=rnorm(1,mu_true,1) if(u[j]<=y[j]&&y[j]<=v[j]) a[j]=1 else a[j]=0 if(sum(a)==n) break ###we need n data set### j=j+1 } mean(a) ### inclusion probability around 0.5 ### v.trunc=v[a==1] y.trunc=y[a==1] u.trunc=u[a==1] PMLE.SEF2.negative(u.trunc,y.trunc,v.trunc)
Maximum likelihood estimates and their standard errors (SEs) are computed. Also computed are the likelihood value, AIC, and other qnantities.
PMLE.SEF3.free(u.trunc, y.trunc, v.trunc, tau1 = min(y.trunc), tau2 = max(y.trunc), epsilon = 1e-04, D1=20, D2=10, D3=1, d1=6, d2=0.5)PMLE.SEF3.free(u.trunc, y.trunc, v.trunc, tau1 = min(y.trunc), tau2 = max(y.trunc), epsilon = 1e-04, D1=20, D2=10, D3=1, d1=6, d2=0.5)
u.trunc |
lower truncation limit |
y.trunc |
variable of interest |
v.trunc |
upper truncation limit |
tau1 |
lower support |
tau2 |
upper support |
epsilon |
error tolerance for Newton-Raphson |
D1 |
Divergence condition for eta_1 |
D2 |
Divergence condition of eta_2 |
D3 |
Divergence condition of eta_3 |
d1 |
Range of randomization for eta_1 |
d2 |
Range of randomization for eta_2 |
Details are seen from the references.
eta |
estimates |
SE |
standard errors |
convergence |
Log-likelihood, degree of freedom, AIC, the number of iterations |
Score |
score vector at the converged value |
Hessian |
Hessian matrix at the converged value |
Takeshi Emura, Ya-Hsuan Hu
Hu YH, Emura T (2015) Maximum likelihood estimation for a special exponential family under random double-truncation, Computation Stat 30 (4): 1199-229
Emura T, Hu YH, Konno Y (2017) Asymptotic inference for maximum likelihood estimators under the special exponential family with double-truncation, Stat Pap 58 (3): 877-909
Dorre A, Emura T (2019) Analysis of Doubly Truncated Data, An Introduction, JSS Research Series in Statistics, Springer
## The first 10 samples of the childhood cancer data ## y.trunc=c(6,7,15,43,85,92,96,104,108,123) u.trunc=c(-1643,-24,-532,-1508,-691,-1235,-786,-261,-108,-120) v.trunc=u.trunc+1825 PMLE.SEF3.free(u.trunc,y.trunc,v.trunc)## The first 10 samples of the childhood cancer data ## y.trunc=c(6,7,15,43,85,92,96,104,108,123) u.trunc=c(-1643,-24,-532,-1508,-691,-1235,-786,-261,-108,-120) v.trunc=u.trunc+1825 PMLE.SEF3.free(u.trunc,y.trunc,v.trunc)
Maximum likelihood estimates and their standard errors (SEs) are computed. Also computed are the likelihood value, AIC, and other qnantities.
PMLE.SEF3.negative(u.trunc, y.trunc, v.trunc, tau1 = min(y.trunc), epsilon = 1e-04, D1=20, D2=10, D3=1, d1=6, d2=0.5)PMLE.SEF3.negative(u.trunc, y.trunc, v.trunc, tau1 = min(y.trunc), epsilon = 1e-04, D1=20, D2=10, D3=1, d1=6, d2=0.5)
u.trunc |
lower truncation limit |
y.trunc |
variable of interest |
v.trunc |
upper truncation limit |
tau1 |
lower support |
epsilon |
error tolerance for Newton-Raphson |
D1 |
Divergence condition for eta_1 |
D2 |
Divergence condition of eta_2 |
D3 |
Divergence condition of eta_3 |
d1 |
Range of randomization for eta_1 |
d2 |
Range of randomization for eta_2 |
Details are seen from the references.
eta |
estimates |
SE |
standard errors |
convergence |
Log-likelihood, degree of freedom, AIC, the number of iterations |
Score |
score vector at the converged value |
Hessian |
Hessian matrix at the converged value |
Takeshi Emura, Ya-Hsuan Hu
Hu YH, Emura T (2015) Maximum likelihood estimation for a special exponential family under random double-truncation, Computation Stat 30 (4): 1199-229
Emura T, Hu YH, Konno Y (2017) Asymptotic inference for maximum likelihood estimators under the special exponential family with double-truncation, Stat Pap 58 (3): 877-909
Dorre A, Emura T (2019) Analysis of Doubly Truncated Data, An Introduction, JSS Research Series in Statistics, Springer
## The first 10 samples of the childhood cancer data ## y.trunc=c(6,7,15,43,85,92,96,104,108,123) u.trunc=c(-1643,-24,-532,-1508,-691,-1235,-786,-261,-108,-120) v.trunc=u.trunc+1825 PMLE.SEF3.negative(u.trunc,y.trunc,v.trunc)## The first 10 samples of the childhood cancer data ## y.trunc=c(6,7,15,43,85,92,96,104,108,123) u.trunc=c(-1643,-24,-532,-1508,-691,-1235,-786,-261,-108,-120) v.trunc=u.trunc+1825 PMLE.SEF3.negative(u.trunc,y.trunc,v.trunc)
Maximum likelihood estimates and their standard errors (SEs) are computed. Also computed are the likelihood value, AIC, and other qnantities.
PMLE.SEF3.positive(u.trunc, y.trunc, v.trunc, tau2 = max(y.trunc), epsilon = 1e-04, D1=20, D2=10, D3=1, d1=6, d2=0.5)PMLE.SEF3.positive(u.trunc, y.trunc, v.trunc, tau2 = max(y.trunc), epsilon = 1e-04, D1=20, D2=10, D3=1, d1=6, d2=0.5)
u.trunc |
lower truncation limit |
y.trunc |
variable of interest |
v.trunc |
upper truncation limit |
tau2 |
upper support |
epsilon |
error tolerance for Newton-Raphson |
D1 |
Divergence condition for eta_1 |
D2 |
Divergence condition of eta_2 |
D3 |
Divergence condition of eta_3 |
d1 |
Range of randomization for eta_1 |
d2 |
Range of randomization for eta_2 |
Details are seen from the references.
eta |
estimates |
SE |
standard errors |
convergence |
Log-likelihood, degree of freedom, AIC, the number of iterations |
Score |
score vector at the converged value |
Hessian |
Hessian matrix at the converged value |
Takeshi Emura, Ya-Hsuan Hu
Hu YH, Emura T (2015) Maximum likelihood estimation for a special exponential family under random double-truncation, Computation Stat 30 (4): 1199-229
Emura T, Hu YH, Konno Y (2017) Asymptotic inference for maximum likelihood estimators under the special exponential family with double-truncation, Stat Pap 58 (3): 877-909
Dorre A, Emura T (2019) Analysis of Doubly Truncated Data, An Introduction, JSS Research Series in Statistics, Springer
## The first 10 samples of the childhood cancer data ## y.trunc=c(6,7,15,43,85,92,96,104,108,123) u.trunc=c(-1643,-24,-532,-1508,-691,-1235,-786,-261,-108,-120) v.trunc=u.trunc+1825 PMLE.SEF3.positive(u.trunc,y.trunc,v.trunc)## The first 10 samples of the childhood cancer data ## y.trunc=c(6,7,15,43,85,92,96,104,108,123) u.trunc=c(-1643,-24,-532,-1508,-691,-1235,-786,-261,-108,-120) v.trunc=u.trunc+1825 PMLE.SEF3.positive(u.trunc,y.trunc,v.trunc)
Maximum likelihood estimates and their standard errors (SEs) are computed. Also computed are the likelihood value, AIC, and other qnantities.
PMLE.Weibull(u.trunc, y.trunc, v.trunc,epsilon = 1e-5,D1=2,D2=2,d1=2,d2=2)PMLE.Weibull(u.trunc, y.trunc, v.trunc,epsilon = 1e-5,D1=2,D2=2,d1=2,d2=2)
u.trunc |
lower truncation limit |
y.trunc |
variable of interest |
v.trunc |
upper truncation limit |
epsilon |
error tolerance for Newton-Raphson |
D1 |
Randomize the intial value if |mu_h-mu_h+1|>D1 |
D2 |
Randomize the intial value if |sigma_h-sigma_h+1|>D2 |
d1 |
U(-d1,d1) is added to the intial value of mu |
d2 |
U(-d2,d2) is added to the intial value of sigma |
Details are seen from the references.
eta |
estimates |
SE |
standard errors |
convergence |
Log-likelihood, degree of freedom, AIC, the number of iterations |
Score |
score vector at the converged value |
Hessian |
Hessian matrix at the converged value |
Takeshi Emura
Dorre A, Huang CY, Tseng YK, Emura T (2020) Likelihood-based analysis of doubly-truncated data under the location-scale and AFT model, Computation Stat, DOI:10.1007/s00180-020-01027-6
## A data example from Efron and Petrosian (1999) ## y.trunc=c(0.75, 1.25, 1.50, 1.05, 2.40, 2.50, 2.25) u.trunc=c(0.4, 0.8, 0.0, 0.3, 1.1, 2.3, 1.3) v.trunc=c(2.0, 1.8, 2.3, 1.4, 3.0, 3.4, 2.6) PMLE.Weibull(u.trunc,y.trunc,v.trunc)## A data example from Efron and Petrosian (1999) ## y.trunc=c(0.75, 1.25, 1.50, 1.05, 2.40, 2.50, 2.25) u.trunc=c(0.4, 0.8, 0.0, 0.3, 1.1, 2.3, 1.3) v.trunc=c(2.0, 1.8, 2.3, 1.4, 3.0, 3.4, 2.6) PMLE.Weibull(u.trunc,y.trunc,v.trunc)
A data frame is generated by simulated data from the Weibull model.
simu.Weibull(n,mu,sigma,delta)simu.Weibull(n,mu,sigma,delta)
n |
sample size |
mu |
location parameter |
sigma |
scale parameter |
delta |
a positive parameter controlling the inclusion probability |
The data are generated from the random vector (U,Y,V) subject to the inclusion criterion U<=Y<=V. The random vector are defined as U=mu-delta+sigma*W, Y=mu+sigma*W, and U=mu+delta+sigma*W, where P(W>w)=exp(-exp(w)). See Section 5.1 of Dorre et al. (2020-) for details. The inclusion probability is P(U<=Y<=V).
u |
lower truncation limits |
y |
log-transformed lifetimes |
v |
upper truncation limits |
Takeshi Emura
Dorre A, Huang CY, Tseng YK, Emura T (2020-) Likelihood-based analysis of doubly-truncated data under the location-scale and AFT model, Computation Stat, DOI:10.1007/s00180-020-01027-6
## A simulation from Dorre et al.(2020) ## simu.Weibull(n=100,mu=5,sigma=2,delta=2.08) Dat=simu.Weibull(n=100,mu=5,sigma=2,delta=2.08) PMLE.Weibull(Dat$u,Dat$y,Dat$v)## A simulation from Dorre et al.(2020) ## simu.Weibull(n=100,mu=5,sigma=2,delta=2.08) Dat=simu.Weibull(n=100,mu=5,sigma=2,delta=2.08) PMLE.Weibull(Dat$u,Dat$y,Dat$v)