**2. Bandwidth Selector**

Let *r Ih* <sup>⊂</sup> <sup>R</sup> be an appropriate pilot bandwidth. The bootstrap resampling algorithm consists of generating *Ui* ∼ *U*(0, 1) and *Vi* ∼ *K* and obtaining

$$X\_i^\* = X\_{[n\underline{U}i]+1} + rV\_{i\prime}$$

$$Z\_i^\* = Z\_{[n\underline{U}i]+1\prime}$$

$$\delta\_i^\* = \delta\_{[n\underline{U}i]+1\prime}$$

for each *i* = 1, . . . , *n*. The bootstrap sample is formed as {(*X*<sup>∗</sup> *<sup>i</sup>* , *Z*<sup>∗</sup> *<sup>i</sup>* , *δ*<sup>∗</sup> *<sup>i</sup>* )}*<sup>n</sup> <sup>i</sup>*=1.

**Citation:** Suárez, R.P.; Abad, R.C.; Fernández, J.M.V. Bootstrap Selector for the Smoothing Parameter of Beran's Estimator. *Eng. Proc.* **2021**, *7*, 28. https://doi.org/10.3390/ engproc2021007028


Academic Editors: Joaquim de Moura, Marco A. González, Javier Pereira and Manuel G. Penedo

Published: 14 October 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The optimal smoothing parameter is the bandwidth that minimizes the mean integrated squared error given by:

$$MSE\_x(h) = E\left(\int \left(\widehat{S}\_h(t|x) - S(t|x)\right)^2 dt\right).$$

Then, the bootstrap bandwidth is obtained by minimizing the Monte Carlo approximation of the bootstrap MISE defined as follows

$$MISE^\*\_{\mathfrak{X}}(h) \quad \simeq \quad \frac{1}{B} \sum\_{j=1}^{B} \left( \int \left( \widehat{S}^{\*(j)}\_{h}(t|\mathbf{x}) - \widehat{S}\_{r}(t|\mathbf{x}) \right)^2 dt \right),$$

where *<sup>S</sup><sup>r</sup>*(*t*|*x*) is the Beran survival estimation with pilot bandwidth *r* using the original sample {(*Xi*, *Zi*, *<sup>δ</sup>i*)}*<sup>n</sup> <sup>i</sup>*=1, *<sup>S</sup>*-∗(*j*) *<sup>h</sup>* (*t*|*x*) is the Beran survival estimation with bandwidth *h* using the bootstrap resample {(*X*∗(*j*) *<sup>i</sup>* , *<sup>Z</sup>*∗(*j*) *<sup>i</sup>* , *δ* ∗(*j*) *<sup>i</sup>* )}*<sup>n</sup> <sup>i</sup>*=1, and *B* the number of bootstrap resamples.

#### **3. Bootstrap Confidence Intervals**

Let *<sup>h</sup>* <sup>∈</sup> *Ih* <sup>⊂</sup> <sup>R</sup> be an appropriate smoothing parameter and fixed values (*t*, *<sup>x</sup>*) <sup>∈</sup> [*a*, *b*] × *I*, the bootstrap confidence interval for a confidence level of 1 − *α* is given by

$$\left(\widehat{S}\_r(t|\mathbf{x}) - \frac{\rho\_{1-\alpha/2}}{\sqrt{nh}}, \widehat{S}\_r(t|\mathbf{x}) - \frac{\rho\_{\alpha/2}}{\sqrt{nh}}\right),$$

where *<sup>S</sup><sup>r</sup>*(*t*|*x*) is the Beran estimation with the pilot bandwidth *r* that is used in the bootstrap resampling, and *ρα*/2 and *<sup>ρ</sup>*1−*α*/2 are the 100*α*/2 and 100(1 − *<sup>α</sup>*/2) percentiles of the resampling distribution of <sup>√</sup>*nh S*-∗ *<sup>h</sup>*(*t*|*x*) <sup>−</sup> *<sup>S</sup><sup>r</sup>*(*t*|*x*) , being *<sup>S</sup>*-∗ *<sup>h</sup>*(*t*|*x*) the Beran survival estimation of the bootstrap resample.

#### **4. Simulation Study**

A simulation study is carried out to analyse the behaviour of the bootstrap algorithm previously described. Several models with different conditional probabilities of censoring were considered. Figure 1 shows the bootstrap estimations of the conditional survival function in two of these scenarios: Model 1 considers the Weibull distribution for life and censoring times and Model 2 considers exponential life and censoring times. Both models have a conditional probability of censoring equal to 0.5. Figure 2 shows the bootstrap confidence intervals in one sample from Models 1 and 2.

**Figure 1.** Theoretical survival function *S*(*t*|*x*) (solid line), Beran's estimation with optimal bandwidth (dotted line) and Beran's estimation with bootstrap bandwidth (dashed line) for Model 1 (**left**) and Model 2 (**right**).

**Figure 2.** Theoretical survival function (solid line), Beran's estimator with bootstrap bandwidth (dashed line) and the bootstrap confidence intervals (dotted line) for each *t* in a grid of size *nt* = 100 in Model 1 (**left**) and Model 2 (**right**).

#### **5. Conclusions**

The results of the simulations show that this bootstrap algorithm provides adequate smoothing parameters to estimate the survival function in this context. The bootstrap bandwidths obtained are similar to the optimal ones and the estimation errors of both are quite similar. Bootstrap confidence intervals have a reasonable behaviour.

Future lines of work focus on developing a method for choosing the bidimensional smoothing parameters involved in the doubly smoothed Beran estimator presented in [2]. In addition, we deal with the construction of confidence intervals for the conditional survival function based on the doubly smoothed Beran estimator.

**Funding:** This research has been supported by MINECO Grant MTM2017-82724-R, and by the Xunta de Galicia (Grupos de Referencia Competitiva ED431C-2016-015 and Centro Singular de Investigación de Galicia ED431G/01), all of them through the ERDF.

