FRM Part II · FRM Exam Part II · Non-parametric Approaches

A risk team applies a kernel with bandwidth h to estimate the density of daily losses from n independent observations. They then double the sample size to 2n, keeping the data-generating process unchanged, and choose the bandwidth following the standard principle that the optimal bandwidth shrinks as the sample grows. Which combination of consequences is most consistent with this practice?

With a larger sample, the optimal bandwidth shrinks: more data per local window keeps variance low, while the smaller bandwidth reduces bias. Both errors fall, so the kernel estimate becomes more accurate as the sample grows, which is the idea behind consistency.

AVariance falls due to more data, and the bandwidth can be reduced so bias also falls, giving a more accurate estimate overallCorrect
BBandwidth must be increased to keep the variance constant, so bias rises
CBias and variance both stay the same because the kernel function is unchanged
DVariance rises because more observations are being smoothed together

Explanation

With more observations, each local window contains enough data to keep variance low even with a smaller bandwidth. Reducing h lowers bias, so both components improve and the estimator is consistent. Increasing h would add bias unnecessarily, and the sample size clearly affects variance.

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