Abstract This paper generalizes the results in Lo (2002) by allowing the distributions of asset returns exhibiting fat tails with the tail index 0 < κ < 4. We show that the asymptotic behavior of the estimated Sharpe ratio is biased and converges at a slower rate as 2< κ < 4. Therefore, the risk-return payoff information provided by the estimated Sharpe ratio can be misleading. It is also shown that the asymptotic unbiasedness and normality of the estimated Sharpe ratio can only be established as asset returns have a finite (4 + ε ) th moment with ε ≥ 0, i.e., κ > 4. Of particular interest are the cases as the tail index 2 < κ < 4, where the ex ante Sharpe ratio is well defined, and an inference method is developed. Empirical examples are used to illustrate the derived results, and an empirical study on major ETFs and mutual funds suggests caution in using the Sharpe ratio for comparing investment performance.

ABSTRACT In this paper, we establish a matrix Freedman inequality for martingales with sub‐Weibull tails. Under conditional control of the increments, the top eigenvalue admits a non‐asymptotic tail bound with explicit, dimension‐aware constants. Via Hermitian dilation, our result extends to rectangular matrices, recovers the sub‐Gaussian case at and admits a time‐uniform (supremum‐over‐time) form. Relative to recent Bennett/Bernstein bounds for sub‐Weibull matrix martingales, our thresholds depend only on a variance proxy and a radius. Concretely, in high‐confidence regimes with , these new thresholds match or improve the corresponding modern envelopes at the same confidence level. We illustrate the utility of our bound in two applications: (i) self‐normalized confidence sets for stochastic linear bandits with heavy‐tailed noise and (ii) operator‐norm error bounds for covariance estimation. We corroborate the theory and highlight constant‐level effects through simulations over a range of tail indices and variance levels.

Accurate modeling of daily rainfall, encompassing both dry and wet days as well as extreme precipitation events, is critical for robust hydrological and climatological analyses. This study proposes a zero-inflated extended generalized Pareto distribution (ZIEGPD) model that unifies the modeling of dry days, low, moderate, and extreme rainfall within a single framework. Unlike traditional approaches that rely on prespecified threshold selection to identify extremes, our proposed model captures tail behavior intrinsically through a tail index that aligns with the GPD. The model also accommodates covariate effects via generalized additive modeling, allowing for the representation of complex climatic variability. The current implementation is limited to a univariate setting, modeling daily rainfall independently of covariates. Model estimation is carried out using both maximum likelihood and Bayesian approaches. Simulation studies and empirical applications demonstrate the model’s flexibility in capturing zero inflation and heavy-tailed behavior characteristics of daily rainfall distributions.

In this paper, we use a tail index estimator to examine how the likelihood of extreme natural gas prices in the German market has evolved between 2011 and 2024. After controlling for the extraordinary supply disruptions associated with the 2022 energy crisis, we find no evidence of a sustained increase in the fatness of price tails over time. While the crisis period clearly heightened the probability of extreme price outcomes, this effect appears to be transient rather than indicative of a permanent structural change. Furthermore, the influence of natural gas demand on tail fatness is less clear-cut once the crisis conditions are accounted for, with storage levels emerging as a key factor in mitigating extreme price risks. Overall, our results suggest that the German natural gas market, despite facing significant short-term shocks, does not exhibit a long-term trend toward fatter tails in its price distributions. JEL Classification: C13 - Estimation: General, C14 - Semiparametric and Nonparametric Methods: General, Q42 - Alternative Energy Sources, Q47 - Energy Forecasting, G11 - Portfolio Choice; Investment Decisions, L16 - Industrial Organization and Macroeconomics: Industrial Structure and Structural Change; Industrial Price Indices

ABSTRACTWe consider a borderline case: The central limit theorem for a strictly stationary time series with infinite variance but a Gaussian limit. In the i.i.d. case, a well‐known sufficient condition for this central limit theorem is regular variation of the marginal distribution with tail index . In the dependent case, we assume the stronger condition of sequential regular variation of the time series with tail index . We assume that a sample of size from this time series can be split into blocks of size such that as and that the block sums are asymptotically independent. Then we apply classical central limit theory for row‐wise i.i.d. triangular arrays. The necessary and sufficient conditions for such independent block sums will be verified by using large deviation results for the time series. We derive the central limit theorem for ‐dependent sequences, linear processes, stochastic volatility processes and solutions to affine stochastic recurrence equations whose marginal distributions have infinite variance and are regularly varying with tail index .

A new estimator for the tail index of a heavy-tailed distribution is proposed by combining the block maxima and peaks-over-threshold methods, tailored for block data where only a few large values are observed within each block. The weak consistency of the estimator is established, and its asymptotic expansion as well as asymptotic normality are derived under the second order regular variation condition. A small simulation study and a case study are conducted to evaluate the performance of the proposed estimator and compare the new estimator with closely related estimators, indicating that our new estimator may have better performance in some cases, in terms of the estimated mean bias and mean squared error.

Abstract Extreme value regression offers a convenient framework to assess the effect of market variables on hedge funds tail risks, proxied by the tail index of the cross-section of hedge funds returns. However, its major limitation lies in the need to select a threshold below which data are discarded, leading to significant estimation inefficiencies. In this article, our main contribution consists in introducing a method to estimate simultaneously the tail index and the threshold parameter from the entire sample at hand, improving estimation efficiency. To do so, we extend the tail regression model to non-tail observations with an auxiliary splicing density, enabling the threshold to be internally determined without truncating the data. We then apply an artificial censoring mechanism to decrease specification issues at the estimation stage. Empirically, we investigate the determinants of hedge funds tail risks over time, and find a significant link with funding liquidity indicators. We also find that our tail risk measure has a significant predictive ability for the returns of around 25% of the funds. In addition, sorting funds along a tail risk sensitivity measure, we are able to discriminate between high- and low-alpha funds under some asset pricing models.

The Pareto distribution has been widely used to model income distribution and inequality. The tail index and the Gini index are typically computed by iteration using Maximum Likelihood and are usually interpreted in terms of the Lorenz curve. We derive an alternative method by considering a truncated Pareto distribution and deriving a simple closed-form approximation for the tail index and the Gini coefficient in terms of the mean absolute deviation and weighted quartile differences. The obtained expressions can be used for any Pareto distribution, even without a finite mean or variance. These expressions are resistant to outliers and have a simple geometric and “economic” interpretation in terms of the quantile function and quartiles. Extensive simulations demonstrate that the proposed approximate values for the tail index and the Gini coefficient are within a few percent relative error of the exact values, even for a moderate number of data points. Our paper offers practical and computationally simple methods to analyze a class of models with Pareto distributions. The proposed methodology can be extended to many other distributions used in econometrics and related fields.

Confidence intervals for the tail index of a heavy-tailed distribution with right-censored data: empirical likelihood and Bayesian composite likelihood methods

ABSTRACTWe are interested in the relationship between the large values of a real random variable and its associated multidimensional covariate, in the context where the conditional distribution is heavy‐tailed. Estimating the positive conditional tail index of a heavy‐tailed conditional distribution is a crucial step for statistical inference, but the task becomes increasingly challenging as the covariate dimension increases. In this work, we assume the existence of a lower‐dimensional linear subspace such that the conditional tail index depends on the covariate only through its projection onto this subspace. We propose a method to estimate this dimension reduction subspace and establish its consistency. Additionally, we introduce an estimator of the conditional tail index that leverages this dimension reduction and prove its consistency. We illustrate the benefits of this dimension reduction approach for estimating the conditional tail index through simulations and an application to real‐world data.

This article studies estimation and inference in the autoregressive (AR) models with unspecified and heavy-tailed heteroskedastic noises. A piece-wise locally stationary structure of the noise is constructed to capture various forms of heterogeneity, without imposing any restrictions on the tail index. The new nonstationary AR model allows for not only time-varying conditional features but also unconditional variance and tail index. This makes it appealing in practice, with wide applications in economics and finance. To obtain a feasible inference, we investigate the self-weighted least absolute deviation estimator and derive its asymptotic normality. Since the asymptotic variance relies on an unobserved density, a bootstrap method is proposed to approximate the limiting distribution. Based on the conditional moment condition, a portmanteau test from residuals is further proposed to detect misspecifications in the proposed model. A simulation study and two applications to time series illustrate our inference procedures.

Abstract Extremes occur in stationary regularly varying time series as short periods with several large observations, known as extremal blocks. We study cluster statistics summarizing the behavior of functions acting on these extremal blocks. Examples of cluster statistics are the extremal index, cluster size probabilities, and other cluster indices. The purpose of our work is twofold. First, we state the asymptotic normality of block estimators for cluster inference based on consecutive observations with large $$\ell ^p-$$ ℓ p - norms, for $$p > 0$$ p > 0 . The case $$p=\alpha $$ p = α , where $$\alpha > 0$$ α > 0 is the tail index of the time series, has specific nice properties thus we analyze the asymptotics of block estimators when approximating $${\alpha }$$ α using the Hill estimator. Second, we verify the conditions we require on classical models such as linear models and solutions of stochastic recurrence equations. Regarding linear models, we prove that the asymptotic variance of classical index cluster-based estimators is null as first conjectured in Hsing (Probab. Theory Related Fields 95, 331–356 1993). We illustrate our findings on simulations.

Abstract It is the purpose of this paper to investigate the issue of estimating the regularity index $$\beta >0$$ β > 0 of a discrete heavy-tailed r.v. S, i.e. a r.v. S valued in $$\mathbb {N}^*$$ N ∗ such that $$\mathbb {P}(S>n)=L(n)\cdot n^{-\beta }$$ P ( S > n ) = L ( n ) · n - β for all $$n\ge 1$$ n ≥ 1 , where $$L:\mathbb {R}^*_+\rightarrow \mathbb {R}_+$$ L : R + ∗ → R + is a slowly varying function. Such discrete probability laws, referred to as generalized Zipf’s laws sometimes, are commonly used to model rank-size distributions after a preliminary range segmentation in a wide variety of areas such as e.g. quantitative linguistics, social sciences or information theory. As a first go, we consider the situation where inference is based on independent copies $$S_1,\; \ldots ,\; S_n$$ S 1 , … , S n of the generic variable S. The estimator $$\widehat{\beta }$$ β ^ we propose can be derived by means of a suitable reformulation of the regularly varying condition, replacing S’s survivor function by its empirical counterpart. Under mild assumptions, a non-asymptotic bound for the deviation between $$\widehat{\beta }$$ β ^ and $$\beta $$ β is established, as well as limit results (consistency and asymptotic normality). Beyond the i.i.d. case, the inference method proposed is extended to the estimation of the regularity index of a regenerative $$\beta $$ β -null-recurrent Markov chain. Since the parameter $$\beta $$ β can be then viewed as the tail index of the (regularly varying) distribution of the return time of the chain X to any (pseudo-) regenerative set, in this case, the estimator is constructed from the successive regeneration times. Because the durations between consecutive regeneration times are asymptotically independent, we can prove that the consistency of the estimator promoted is preserved. In addition to the theoretical analysis carried out, simulation results provide empirical evidence of the relevance of the inference technique proposed.

In this paper, we introduce a class of semi-parametric estimators of the distortion risk premiums for dependent insurance losses with heavy-tailed marginals. Our approach is based on the kernel estimation of the tail index and extreme quantiles under the first and second orders regularly varying assumptions for stationary insured risks with heavy-tailed distribution under dependence serials. Moreover, we illustrate the behaviour of our proposed estimator and give a comparison between this estimator and the classical one in terms of the absolute bias and the root median squared error.

It was observed that the number of cases and deaths for infectious diseases were associated with heavy-tailed power law distributions such as the Pareto distribution. While Pareto distribution was widely used to model the cases and deaths of infectious diseases, a major limitation of Pareto distribution is that it can only fit a given data set beyond a certain threshold. Thus, it can only model part of the data set. Thus, we proposed some novel discrete composite distributions with Pareto tails to fit the real infectious disease data. To provide necessary statistical inference for the tail behavior of the data, we developed a hypothesis testing procedure to test the tail index parameter. COVID-19 reported cases in Singapore and monkeypox reported cases in France were analyzed to evaluate the performance of the new distributions. The results from the analysis suggested that the discrete composite distributions could demonstrate competitive performance compared to the commonly used discrete distributions. Furthermore, the analysis of the tail index parameter can provide great insights into preventing and controlling infectious diseases.

A new compound extension of the Fréchet distribution is introduced and studied. Some of its properties including moments, incomplete moments, probability weighted moments, moment generating function, stress strength reliability model, residual life and reversed residual life functions are derived. The mean squared errors (MSEs) for some estimation methods including maximum likelihood estimation (MLE), Cram\'{e}r--von Mises (CVM) estimation, Bootstrapping (Boot.) estimation and Kolmogorov estimates (KE) method are used to estimate the unknown parameter via a simulation study. Two real applications are presented for comparing the estimation methods. Another two real applications are presented for comparing the competitive models. The nonparametric Hill estimator under the breaking stress of carbon fibers is estimated using the tail index (TIx) of the new model. Finally, a case study on reliability analysis of composite materials for aerospace applications is presented.

Abstract Considering a double-indexed array $(Y_{n,i:\,n\ge 1,i\ge 1})$ of non-negative regularly varying random variables, we study the random-length weighted sums and maxima from its ‘row’ sequences. These sums and maxima may have the same tail and extremal indices (Markovich and Rodionov 2020). The main constraints of the latter results are that there exists a unique series in a scheme of series with the minimum tail index and the tail of the term number is lighter than the tail of the terms. Here, a bounded random number of series are allowed to have the minimum tail index and the tail of the term number may be heavier than the tail of the terms. We derive the tail and extremal indices of the weighted non-stationary random-length sequences under a broader set of conditions than in Markovich and Rodionov (2020). We provide examples of random sequences for which the assumptions are valid. Perspectives in adopting the results in different application areas are formulated.

On a new family of composite regression models with covariate dependent threshold via tail index parameter

Abstract We show theoretically and empirically that the cross-section of stock return idiosyncratic volatilities contains useful information about the ICAPM. We construct a proxy cross-sectional bivariate idiosyncratic volatility (CBIV) for the covariance risk between the market and the unobserved hedge portfolio under the ICAPM. Consistent with the ICAPM pricing relation, CBIV is a robust and significant predictor of the equity risk premium. We further show that the return predictability of the tail index in Kelly and Jiang (2014) can be explained by the ICAPM covariance risk.

ABSTRACTData analysis derives statistical inference from the result of data‐driven model (variable) selection or averaging. One puzzle however is that inference after model selection may not be guaranteed to satisfy tests and confidence intervals provided by classical statistical theory. This paper proposes a valid post‐averaging confidence interval in an AR model driven by a general GARCH (G/GARCH) model, in which the innovations exhibit a heavy‐tailed structure with a tail index . To achieve this, we investigate the asymptotic inference of the nested least squares averaging estimator under model uncertainty with a fixed coefficient setup. Interestingly, based on a Mallows‐type model averaging (MTMA) criterion, the weights of under‐fitted models decay to zero whereas asymptotically random weights are assigned only to just‐fitted and over‐fitted models. Utilizing the asymptotic behavior of model weights, we derive the asymptotic distributions of the MTMA estimator and show that the proposed confidence interval is valid for any . Monte Carlo simulations show that the proposed method achieves the nominal level.