Supplementary PDF.pdf

The proof of Lemma 1 involves showing that the derivative of the population risk with respect to the parameter θ is equal to the negative expectation of the gradient of the loss function. The authors then use this result to derive an expression for the derivative of the minimizer of the population risk with respect to the contamination proportion ε. They also show that the operator norm of the Hessian of the population risk is bounded, which is used to establish a lower bound on the norm of the derivative of the minimizer.

The proof of Lemma 2 involves showing that the subset risk minimization (SRM) estimator is equivalent to the minimizer of the population risk with respect to a mixture distribution. The authors then use this result to derive an expression for the bias of the SRM estimator and show that it is bounded by a term that depends on the contamination proportion ε and the trace of the covariance matrices of the true and contaminated distributions.

The proof of Lemma 3 involves showing that the interval estimator ˆI obtained using the algorithm contains at least a certain proportion of points from the true distribution and that the length of ˆI is bounded. The authors then use this result to derive an expression for the final error of the estimator and show that it is bounded by a term that depends on the contamination proportion ε and the standard deviation of the true distribution.

The key results of the proofs are:

The derivative of the population risk with respect to the parameter θ is equal to the negative expectation of the gradient of the loss function.
The operator norm of the Hessian of the population risk is bounded.
The SRM estimator is equivalent to the minimizer of the population risk with respect to a mixture distribution.
The bias of the SRM estimator is bounded by a term that depends on the contamination proportion ε and the trace of the covariance matrices of the true and contaminated distributions.
The interval estimator ˆI obtained using the algorithm contains at least a certain proportion of points from the true distribution and has a bounded length.
The final error of the estimator is bounded by a term that depends on the contamination proportion ε and the standard deviation of the true distribution.