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.

THE MOVING BLOCKS BOOTSTRAP FOR PANEL LINEAR REGRESSION MODELS WITH INDIVIDUAL FIXED EFFECTS

In this paper we propose a bootstrap method for panel data linear regression models with individual fixed effects. The method consists of applying the standard moving blocks bootstrap of Künsch (1989, Annals of Statistics 17, 1217–1241) and Liu and Singh (1992, in R. LePage & L. Billiard (eds.), Exploring the Limits of the Bootstrap) to the vector containing all the individual observations at each point in time. We show that this bootstrap is robust to serial and cross-sectional dependence of unknown form under the assumption that n (the cross-sectional dimension) is an arbitrary nondecreasing function of T (the time series dimension), where T → ∞, thus allowing for the possibility that both n and T diverge to infinity. The time series dependence is assumed to be weak (of the mixing type), but we allow the cross-sectional dependence to be either strong or weak (including the case where it is absent). Under appropriate conditions, we show that the fixed effects estimator (and also its bootstrap analogue) has a convergence rate that depends on the degree of cross-section dependence in the panel. Despite this, the same studentized test statistics can be computed without reference to the degree of cross-section dependence. Our simulation results show that the moving blocks bootstrap percentile-t intervals have very good coverage properties even when the degree of serial and cross-sectional correlation is large, provided the block size is appropriately chosen.

Field Significance Revisited: Spatial Bias Errors in Forecasts as Applied to the Eta Model

The spatial structure of bias errors in numerical model output is valuable to both model developers and operational forecasters, especially if the field containing the structure itself has statistical significance in the face of naturally occurring spatial correlation. A semiparametric Monte Carlo method, along with a moving blocks bootstrap method is used to determine the field significance of spatial bias errors within spatially correlated error fields. This process can be completely automated, making it an attractive addition to the verification tools already in use. The process demonstrated here results in statistically significant spatial bias error fields at any arbitrary significance level. To demonstrate the technique, 0000 and 1200 UTC runs of the operational Eta Model and the operational Eta Model using the Kain–Fritsch convective parameterization scheme are examined. The resulting fields for forecast errors for geopotential heights and winds at 850, 700, 500, and 250 hPa over a period of 14 months (26 January 2001–31 March 2002) are examined and compared using the verifying initial analysis. Specific examples are shown, and some plausible causes for the resulting significant bias errors are proposed.

THE BOOTSTRAP OF THE MEAN FOR DEPENDENT HETEROGENEOUS ARRAYS

Presently, conditions ensuring the validity of bootstrap methods for the sample mean of (possibly heterogeneous) near epoch dependent (NED) functions of mixing processes are unknown. Here we establish the validity of the bootstrap in this context, extending the applicability of bootstrap methods to a class of processes broadly relevant for applications in economics and finance. Our results apply to two block bootstrap methods: the moving blocks bootstrap of Künsch (1989, Annals of Statistics 17, 1217–1241) and Liu and Singh (1992, in R. LePage & L. Billiard (eds.), Exploring the Limits of the Bootstrap, 224–248) and the stationary bootstrap of Politis and Romano (1994a, Journal of the American Statistical Association 89, 1303–1313). In particular, the consistency of the bootstrap variance estimator for the sample mean is shown to be robust against heteroskedasticity and dependence of unknown form. The first-order asymptotic validity of the bootstrap approximation to the actual distribution of the sample mean is also established in this heterogeneous NED context.