For two independent non-negative random variables X and Y , we treat X as the initial variable of major importance and Y as a modifier (such as the interest rate of a portfolio). Stability in the tail behaviors of the product compared with that of the original variable X is of practical interests. In this paper, we study the tail behaviors of the product XY when the distribution of X belongs to the classes L and S, respectively. Under appropriate conditions, we show that the distribution of the product XY is in the same class as X when X belongs to class L or S, in other words, classes L and S are stable under some mild conditions on the distribution of Y . We also show that if the distribution of X is in class L(γ) (γ > 0) and continuous, then the product XY is in L if and only if Y is unbounded.

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Assessing conditional tail risk at very high or low levels is of great interest in numerous applications. Due to data sparsity in high tails, the widely used quantile regression method can suffer from high variability at the tails, especially for heavy-tailed distributions. As an alternative to quantile regression, expectile regression, which relies on the minimization of the asymmetric l2-norm and is more sensitive to the magnitudes of extreme losses than quantile regression, is considered. In this article, we develop a new estimation method for high conditional tail risk by first estimating the intermediate conditional expectiles in regression framework, and then estimating the underlying tail index via weighted combinations of the top order conditional expectiles. The resulting conditional tail index estimators are then used as the basis for extrapolating these intermediate conditional expectiles to high tails based on reasonable assumptions on tail behaviors. Finally, we use these high conditional tail expectiles to estimate alternative risk measures such as the Value at Risk (VaR) and Expected Shortfall (ES), both in high tails. The asymptotic properties of the proposed estimators are investigated. Simulation studies and real data analysis show that the proposed method outperforms alternative approaches.

The uterine endometrium, which lines the mammalian uterus, is essential for embryo implantation. This lining undergoes significant changes during sexual and menstrual cycles. The endometrium is also associated with hormone-related diseases such as endometriosis and endometrial cancer. Circular RNAs (circRNAs) play a role in various biological processes. Recent studies have determined that circRNAs function in both normal and pathological endometrial environments. Here, we review high-throughput studies pertaining to circRNAs as well as individual circRNAs active in the endometrium, in order to explore the myriad functions of circRNAs in the endometrium and mechanisms underlying these functions, from panoramic and individual perspectives. Owing to their abundant expression, stability, and small size, circRNAs have displayed potential usefulness as diagnostic markers and treatment targets for endometrial-related diseases. Therefore, the specific role of circRNAs in the endometrium warrants systematic investigation in the future.