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

A analyst fits a kernel density estimator to a sample of 1,000 daily returns using a Gaussian kernel. She first uses a very small bandwidth, then a very large one. Which outcome is most likely?

A very small bandwidth produces a spiky, noisy density with high variance and low bias, while a very large bandwidth oversmooths and produces a flat density with high bias but low variance. Bandwidth selection is therefore a bias-variance trade-off.

  1. ASmall bandwidth: oversmoothed density with high bias; large bandwidth: spiky density with high variance
  2. BSmall bandwidth: spiky density with high variance; large bandwidth: oversmoothed density with high biasCorrect
  3. CBoth bandwidths give identical density estimates because the sample size is large
  4. DSmall bandwidth: biased tail estimates; large bandwidth: exact reproduction of the empirical distribution

Explanation

A small bandwidth lets each observation dominate locally, giving a noisy, spiky estimate with low bias but high variance. A large bandwidth averages over many points, flattening features and adding bias. The first option reverses this trade-off.

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