We present an approach for determining sample sizes required to detect underestimations of the expected shortfall with a prescribed power when applying the recently proposed e-backtesting procedure. We consider scenarios in which the value-at-risk at level $p$ is always estimated correctly, while the difference between the true expected shortfall and the value-at-risk is underestimated by a given factor $r$. We show that exploiting the structure of the backtest e-statistic proposed for backtesting the expected shortfall at level $p$ enables the derivation of approximate lower bounds for the required sample sizes by considering a sequence of independent and identically distributed Bernoulli random variables. We also discuss potential limitations of this approximation and compare the resulting sample size requirements with those obtained in practical applications using Monte Carlo simulations. Furthermore, we present generalizations of the e-backtesting procedure, in particular to risk measures which constitute Bayes pairs.
Expected Shortfall (ES) measures the average size of tail losses. Its sequential backtests typically also require a Value-at-Risk (VaR) forecast. We develop two e-backtesting methods using only observed losses and ES forecasts: one for iid losses only and the other for time series under an AR-GARCH conditional VaR stru...
A simple decomposition of a neural-network-based normalizing flow that naturally uncovers a pivotal statistic in the presence of nuisance parameters, based only on a sample generator from the distribution of interest is presented.
E-values provide a flexible framework for statistical inference, but the universal threshold $1/\alpha$ can be conservative when the null distribution has additional structure. We study how structural restrictions limit the worst-case concentration underlying Markov's inequality, using a nondecreasing density model to...
Jia-Yi Shen, Hou-Zhen Liu, Qi-Run Zeng et al.· 0 citations
We investigate Empirical Bayes (EB) methods in the context of compound adaptive experiments, where the arm distribution in each experiment follows a normal distribution with an unknown mean that we seek to estimate. There are two main EB strategies: $g$-modeling, which estimates the prior by maximizing the marginal lik...
Karun Adusumilli, Jia-Ying Gu, J. Tao· 0 citations
Infinite-order
U
-statistics play a vital role in modern statistics. However, given their degree
k
n
$k_n$
k Subscript n
depends on the sample size
n
, quantifying their variability is quite challenging, as fixed-degree theories cannot be directly adopted. In this article, we propose a ja...
On two real Bayesian posteriors scored against long NUTS references, uniform pooling improved on the sample-count-matched single run under marginal JS, energy distance, and MMD2 in every recorded run, while the aggregation’s deficit is confined to the biased marginal metric.
Ming-Yu Shi, Fan Zhang· Mathematics· 0 citations
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