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Recently, we proposed a method for estimating the distribution of a kernel spectral density estimator using the bootstrap. In this contribution, we address the issue of accuracy of the estimation. It is well known that the bootstrap is second order accurate. This holds whenever a bootstrap version of the population-sample relationship is constructed in such a way that each component that depends on the population is replaced by its sample version, and each part that depends on the sample is replaced by its resample counterpart. Our contribution consists of a proof that the accuracy of the distribution estimate of the kernel spectral density estimator, which includes kernel bandwidth estimation, is of second order, even when we do not follow the above rule. As a result, the computational burden is greatly reduced while maintaining statistical accuracy.