Bias and Consistency of the Sample Average Value-at-Risk Estimator
Summary
The document asks about statistical properties of a sample estimator for average value at risk, also called conditional value at risk. It presents the population objective as a minimization over a threshold parameter, with expected tail loss beyond that threshold, and compares it with the empirical version formed from observed losses. The central questions are whether a coefficient in a cited book is a typo and whether the resulting estimator is unbiased or consistent.
No answer or proof is included. The prompt highlights a key estimation issue: taking a minimum over a random empirical objective need not preserve unbiasedness, even when the sample average estimates the expectation at each fixed threshold. It provides no distributional assumptions, finite-sample analysis, or conclusion about consistency, so those properties remain open in the document.
Key ideas
- Average value at risk can be represented as a minimization over a threshold plus a scaled expected tail-loss term.
- The sample counterpart replaces the expectation with an average over observed losses.
- Unbiasedness cannot be inferred solely from unbiased sample averages at fixed thresholds because the threshold is also optimized.
- The document raises but does not resolve questions about the estimator’s bias and consistency.
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Full text
# Estimator for Conditional value at risk (average value at risk)
# Estimator for Conditional value at risk (average value at risk)
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?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.