Computationally Efficient Bootstrap Expressions for Bandwidth Selection in Nonparametric Curve Estimation †

Bootstrap methods are used for bandwidth selection in: (1) nonparametric kernel density estimation with dependent data (smoothed stationary bootstrap and smoothed moving blocks bootstrap), and (2) nonparametric kernel hazard rate estimation (smoothed bootstrap). In these contexts, four new bandwidth parameter selectors are proposed based on closed bootstrap expressions of the MISE of the kernel density estimator (case 1) and two approximations of the kernel hazard rate estimation (case 2). These expressions turn out to be very useful since Monte Carlo approximation is no longer needed. Finally, these smoothing parameter selectors are empirically compared with the already existing ones via a simulation study.


Introduction
This work deals with the well known problem of data-driven choice of smoothing parameters in nonparametric density and hazard rate estimation (see [1][2][3][4]).Our aim is also to propose new bootstrap procedures for nonparametric density estimation considering dependent data.On the other hand, hazard rate estimation is considered and two bootstrap bandwidth selectors based on some approximation of the kernel hazard rate estimator are proposed.

Nonparametric Density Estimation
Let us consider a random sample, (X 1 , . . ., X n ), coming from a population with density f and the kernel density estimator (see [5,6]), which strongly depends on a bandwidth selector, h.In fact, its choice is really important since it regulates the degree of smoothing applied to the data.
In this context, the smoothed stationary bootstrap (SSB) resampling plan has been proposed (see the Appendix for a detailed description of the algorithm and [7]), as well as a bandwidth selector, namely h * SSB .It is the result of minimizing the SSB version of the MISE.A closed expression for the bootstrap MISE is also obtained by [7].On the other hand, smoothed moving blocks bootstrap (SMBB) has been proposed (see the Appendix for a complete description of the method), as well as a bandwidth selector, h * SMBB , which is the minimizer in h of the closed expression for the MISE * SMBB (see [8] for a deeper insight on the topic).It is worth mentioning that the exact expressions for the MISE * SSB (h) and MISE * SMBB (h) are really useful since Monte Carlo approximation is no longer necessary.

Nonparametric Hazard Rate Estimation
Let us consider (X 1 , X 2 , . . ., X n ), a simple random sample coming from a population with continuous density f and cumulative distribution function F. Consider, additionally, the nonparametric hazard rate estimator (see [3,4]), the kernel density estimator fh and the kernel distribution estimator Fh .In order to establish a bootstrap bandwidth selector for the hazard rate estimator, two approximations of the hazard rate estimator are considered.The two hazard rate approximated versions are given by: rh,1 (x) = fh (x) Closed-form expressions of the MISE of rh,1 and rh,2 , as well as their bootstrap versions can be found in [9].Moreover, two bootstrap bandwidth selectors, namely h BOOT1 and h BOOT2 , are defined as the minimizers of MISE * rh,1 ,w (h) and MISE * rh,2 ,w (h), respectively (see [9] for a deeper insight on the approach).It is worth mentioning that Monte Carlo approximation is not required.

Simulation Results
A simulation study is now carried out in order to check the good empirical behaviour of the new smoothing parameter selectors in both contexts.These are the models considered: 1.

Discussion
Figure 1 shows that h * SSB and h * SMBB display a similar performance, actually the best one.

Conflicts of Interest:
The authors declare no conflict of interest.The founding sponsors had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the decision to publish the results.

Abbreviations
The following abbreviations are used in this manuscript:

MISE
Mean integrated squared error ISE Integrated squared error SSB Smoothed stationary bootstrap SMBB Smoothed moving blocks bootstrap iid Independent and identically distributed h DO DO-validation bandwidth selector for hazard rate estimation (see [10]) h * GCM González-Manteiga, Cao, Marron bandwidth selector for hazard rate estimation (see [11]) h PI Plug-in bandwidth selector for bandwidth selection with dependent data (see [12]) h CV l Leave-(2l + 1)-out cross-validation for density estimation (see [13]) h SMCV Modified cross validation for density estimation with dependent data (see [8]) h PCV Penalized cross validation for density estimation with dependent data (see [8]) h CV Cross validation bandwidth selector for hazard rate estimation (see [14] , for all i = 1, 2, . . ., n

Table 1 .
According to Table1, h BOOT1 and h BOOT2 display the overall best performance.Boxplot of log MISE( ĥ)/MISE(h MISE ) , n = 100, where ĥ = h CV l (first box), h SMCV (second box), h PCV (third box), h * SSB (fourth box), h * SMBB (fifth box) and h PI (sixth box).Mean and median of ISE( ĥ), n = 100, where ĥ = h CV (third column), h DO (fourth column), h BOOT1 (fifth column), h BOOT2 (sixth column) and h * GCM (seventh column).The authors acknowledge partial support by MINECO grants MTM2014-52876-R and MTM2017-82724-R (EU ERDF support included).Additionally, financial support from the Xunta de Galicia (Centro Singular de Investigación de Galicia accreditation ED431G/01 2016-2019 and Grupos de Referencia Competitiva ED431C2016-015) and the European Union (European Regional Development Fund -ERDF), is gratefully acknowledged.The first author aknowledges financial support from the Xunta de Galicia and the European Union (European Social Fund -ESF), the reference of which is ED481A-2017/215.Additionally, the work of the first author has been partially carried out during a visit at the University of California, San Diego, financed by INDITEX, with reference INDITEX-UDC 2017.