I am following a book: Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization by Svetlozar T. Rachev, Stoyan V. Stoyanov, Frank J. Fabozzi

I'm learning about average value at risk. In particular from a sample as in equation 7.7 on page 215. I think they have made a mistake since in the Rockafellar, Uryasev 2002 paper (https://pdfs.semanticscholar.org/8863/790b149edfb586db318363e28182a6fedc80.pdf) they have a $\frac{1}{1-\epsilon}$ instead of a $\frac{1}{\epsilon}$.

In any case, I would like to show some properties about this point estimator. It seems like the book just took a natural choice for an estimator, but didn't discuss anything like the bias or consistency.

I know that the avar is given by

$$\min_{\theta \in \mathbb{R}} \bigg( \theta + \frac{1}{(1-\epsilon)} \mathbb{E}[\max (-X - \theta, 0)] \bigg)$$

and would like some info about the statistic

$$\min_{\theta \in \mathbb{R}} \bigg( \theta + \frac{1}{n(1-\epsilon)} \sum_{i=1}^n \max (-X_i - \theta, 0)\bigg)$$ like for example, is it an unbiased estimator?

I was wondering if anyone was knowledgeable about this estimator and could discuss with me. I think in general we cannot pass the expectation from the outside of a $\min$ to the inside as outlined in my other question here: https://math.stackexchange.com/questions/3335421/passing-an-integral-to-the-inside-of-a-min/3335444?noredirect=1#comment6867762_3335444

Does anyone know about the bias or other properties of this estimator?


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