Marginal estimation results of the proposed nonparametric kernel Nelson-Aalen estimator (NPNA|$_{\rm marg}$|) and the Kaplan–Meier based estimator (KM)
| . | |$\widehat F_1(t)$| . | |$\widehat F_2(t)$| . | ||
|---|---|---|---|---|
| . | KM . | NPNA|$_{\rm marg}$| . | KM . | NPNA|$_{\rm marg}$| . |
| Setting 1: |$t=1$| | ||||
| bias | 0.1964 | -0.0544 | 0.1509 | -0.2704 |
| emp var | 0.0324 | 0.0332 | 0.0409 | 0.0397 |
| est var | 0.0332 | 0.0302 | 0.0358 | 0.0331 |
| 95% cov | 96.5000 | 94.0000 | 93.5000 | 92.5000 |
| MSE | 0.0336 | 0.0303 | 0.0360 | 0.0339 |
| rel eff | 100.0000 | 90.9975 | 100.0000 | 92.5993 |
| Setting 1: |$t=2$| | ||||
| bias | -0.0003 | -0.5523 | 0.1674 | -0.4324 |
| emp var | 0.0485 | 0.0492 | 0.0498 | 0.0495 |
| est var | 0.0471 | 0.0422 | 0.0439 | 0.0405 |
| 95% cov | 96.5000 | 91.5000 | 93.0000 | 89.5000 |
| MSE | 0.0471 | 0.0453 | 0.0441 | 0.0424 |
| rel eff | 100.0000 | 89.6270 | 100.0000 | 92.3248 |
| Setting 2: |$t=30$| | ||||
| bias | 0.2651 | -0.2741 | -0.0983 | -0.1784 |
| emp var | 0.0387 | 0.0318 | 0.0232 | 0.0195 |
| est var | 0.0402 | 0.0323 | 0.0207 | 0.0176 |
| 95% cov | 93.0000 | 94.0000 | 92.5000 | 93.0000 |
| MSE | 0.0409 | 0.0330 | 0.0208 | 0.0179 |
| rel eff | 100.0000 | 80.3382 | 100.0000 | 84.8547 |
| Setting 2: |$t=40$| | ||||
| bias | 0.0914 | -0.7803 | -0.1364 | -0.3885 |
| emp var | 0.0571 | 0.0439 | 0.0448 | 0.0400 |
| est var | 0.0523 | 0.0404 | 0.0400 | 0.0329 |
| 95% cov | 93.0000 | 93.0000 | 90.0000 | 90.0000 |
| MSE | 0.0524 | 0.0465 | 0.0402 | 0.0344 |
| rel eff | 100.0000 | 77.3532 | 100.0000 | 82.2426 |
| . | |$\widehat F_1(t)$| . | |$\widehat F_2(t)$| . | ||
|---|---|---|---|---|
| . | KM . | NPNA|$_{\rm marg}$| . | KM . | NPNA|$_{\rm marg}$| . |
| Setting 1: |$t=1$| | ||||
| bias | 0.1964 | -0.0544 | 0.1509 | -0.2704 |
| emp var | 0.0324 | 0.0332 | 0.0409 | 0.0397 |
| est var | 0.0332 | 0.0302 | 0.0358 | 0.0331 |
| 95% cov | 96.5000 | 94.0000 | 93.5000 | 92.5000 |
| MSE | 0.0336 | 0.0303 | 0.0360 | 0.0339 |
| rel eff | 100.0000 | 90.9975 | 100.0000 | 92.5993 |
| Setting 1: |$t=2$| | ||||
| bias | -0.0003 | -0.5523 | 0.1674 | -0.4324 |
| emp var | 0.0485 | 0.0492 | 0.0498 | 0.0495 |
| est var | 0.0471 | 0.0422 | 0.0439 | 0.0405 |
| 95% cov | 96.5000 | 91.5000 | 93.0000 | 89.5000 |
| MSE | 0.0471 | 0.0453 | 0.0441 | 0.0424 |
| rel eff | 100.0000 | 89.6270 | 100.0000 | 92.3248 |
| Setting 2: |$t=30$| | ||||
| bias | 0.2651 | -0.2741 | -0.0983 | -0.1784 |
| emp var | 0.0387 | 0.0318 | 0.0232 | 0.0195 |
| est var | 0.0402 | 0.0323 | 0.0207 | 0.0176 |
| 95% cov | 93.0000 | 94.0000 | 92.5000 | 93.0000 |
| MSE | 0.0409 | 0.0330 | 0.0208 | 0.0179 |
| rel eff | 100.0000 | 80.3382 | 100.0000 | 84.8547 |
| Setting 2: |$t=40$| | ||||
| bias | 0.0914 | -0.7803 | -0.1364 | -0.3885 |
| emp var | 0.0571 | 0.0439 | 0.0448 | 0.0400 |
| est var | 0.0523 | 0.0404 | 0.0400 | 0.0329 |
| 95% cov | 93.0000 | 93.0000 | 90.0000 | 90.0000 |
| MSE | 0.0524 | 0.0465 | 0.0402 | 0.0344 |
| rel eff | 100.0000 | 77.3532 | 100.0000 | 82.2426 |
We report bias, empirical variance (emp var), estimated bootstrap variance (est var), 95% coverage and MSE, and relative efficiency (in terms of variance relative to KM) for |$\widehat{\boldsymbol F}(t)=(\widehat F_1(t),\widehat F_2(t)) $| at specified |$t$|. Results summarized over 200 simulations with 40% censoring. Improved efficiencies from the NPNA|$_{\rm marg}$| relative to KM are boldfaced. All values are multiplied by 100.
Marginal estimation results of the proposed nonparametric kernel Nelson-Aalen estimator (NPNA|$_{\rm marg}$|) and the Kaplan–Meier based estimator (KM)
| . | |$\widehat F_1(t)$| . | |$\widehat F_2(t)$| . | ||
|---|---|---|---|---|
| . | KM . | NPNA|$_{\rm marg}$| . | KM . | NPNA|$_{\rm marg}$| . |
| Setting 1: |$t=1$| | ||||
| bias | 0.1964 | -0.0544 | 0.1509 | -0.2704 |
| emp var | 0.0324 | 0.0332 | 0.0409 | 0.0397 |
| est var | 0.0332 | 0.0302 | 0.0358 | 0.0331 |
| 95% cov | 96.5000 | 94.0000 | 93.5000 | 92.5000 |
| MSE | 0.0336 | 0.0303 | 0.0360 | 0.0339 |
| rel eff | 100.0000 | 90.9975 | 100.0000 | 92.5993 |
| Setting 1: |$t=2$| | ||||
| bias | -0.0003 | -0.5523 | 0.1674 | -0.4324 |
| emp var | 0.0485 | 0.0492 | 0.0498 | 0.0495 |
| est var | 0.0471 | 0.0422 | 0.0439 | 0.0405 |
| 95% cov | 96.5000 | 91.5000 | 93.0000 | 89.5000 |
| MSE | 0.0471 | 0.0453 | 0.0441 | 0.0424 |
| rel eff | 100.0000 | 89.6270 | 100.0000 | 92.3248 |
| Setting 2: |$t=30$| | ||||
| bias | 0.2651 | -0.2741 | -0.0983 | -0.1784 |
| emp var | 0.0387 | 0.0318 | 0.0232 | 0.0195 |
| est var | 0.0402 | 0.0323 | 0.0207 | 0.0176 |
| 95% cov | 93.0000 | 94.0000 | 92.5000 | 93.0000 |
| MSE | 0.0409 | 0.0330 | 0.0208 | 0.0179 |
| rel eff | 100.0000 | 80.3382 | 100.0000 | 84.8547 |
| Setting 2: |$t=40$| | ||||
| bias | 0.0914 | -0.7803 | -0.1364 | -0.3885 |
| emp var | 0.0571 | 0.0439 | 0.0448 | 0.0400 |
| est var | 0.0523 | 0.0404 | 0.0400 | 0.0329 |
| 95% cov | 93.0000 | 93.0000 | 90.0000 | 90.0000 |
| MSE | 0.0524 | 0.0465 | 0.0402 | 0.0344 |
| rel eff | 100.0000 | 77.3532 | 100.0000 | 82.2426 |
| . | |$\widehat F_1(t)$| . | |$\widehat F_2(t)$| . | ||
|---|---|---|---|---|
| . | KM . | NPNA|$_{\rm marg}$| . | KM . | NPNA|$_{\rm marg}$| . |
| Setting 1: |$t=1$| | ||||
| bias | 0.1964 | -0.0544 | 0.1509 | -0.2704 |
| emp var | 0.0324 | 0.0332 | 0.0409 | 0.0397 |
| est var | 0.0332 | 0.0302 | 0.0358 | 0.0331 |
| 95% cov | 96.5000 | 94.0000 | 93.5000 | 92.5000 |
| MSE | 0.0336 | 0.0303 | 0.0360 | 0.0339 |
| rel eff | 100.0000 | 90.9975 | 100.0000 | 92.5993 |
| Setting 1: |$t=2$| | ||||
| bias | -0.0003 | -0.5523 | 0.1674 | -0.4324 |
| emp var | 0.0485 | 0.0492 | 0.0498 | 0.0495 |
| est var | 0.0471 | 0.0422 | 0.0439 | 0.0405 |
| 95% cov | 96.5000 | 91.5000 | 93.0000 | 89.5000 |
| MSE | 0.0471 | 0.0453 | 0.0441 | 0.0424 |
| rel eff | 100.0000 | 89.6270 | 100.0000 | 92.3248 |
| Setting 2: |$t=30$| | ||||
| bias | 0.2651 | -0.2741 | -0.0983 | -0.1784 |
| emp var | 0.0387 | 0.0318 | 0.0232 | 0.0195 |
| est var | 0.0402 | 0.0323 | 0.0207 | 0.0176 |
| 95% cov | 93.0000 | 94.0000 | 92.5000 | 93.0000 |
| MSE | 0.0409 | 0.0330 | 0.0208 | 0.0179 |
| rel eff | 100.0000 | 80.3382 | 100.0000 | 84.8547 |
| Setting 2: |$t=40$| | ||||
| bias | 0.0914 | -0.7803 | -0.1364 | -0.3885 |
| emp var | 0.0571 | 0.0439 | 0.0448 | 0.0400 |
| est var | 0.0523 | 0.0404 | 0.0400 | 0.0329 |
| 95% cov | 93.0000 | 93.0000 | 90.0000 | 90.0000 |
| MSE | 0.0524 | 0.0465 | 0.0402 | 0.0344 |
| rel eff | 100.0000 | 77.3532 | 100.0000 | 82.2426 |
We report bias, empirical variance (emp var), estimated bootstrap variance (est var), 95% coverage and MSE, and relative efficiency (in terms of variance relative to KM) for |$\widehat{\boldsymbol F}(t)=(\widehat F_1(t),\widehat F_2(t)) $| at specified |$t$|. Results summarized over 200 simulations with 40% censoring. Improved efficiencies from the NPNA|$_{\rm marg}$| relative to KM are boldfaced. All values are multiplied by 100.
This PDF is available to Subscribers Only
View Article Abstract & Purchase OptionsFor full access to this pdf, sign in to an existing account, or purchase an annual subscription.
