arXiv subjects

Nawaf Mohammed

Publications and source records attributed to Nawaf Mohammed.

5 recordsLinked to original sources

arXiv2026

The Elliptically Optimal Confidence Interval: A Bivariate Extension of Wilson's Score Method

Constructing a confidence interval for the difference between two independent binomial proportions involves a nuisance direction that is not identified by the estimand. The one-sample Wilson score interval inverts a scalar score test, but has no direct bivariate analogue isolating the difference: inverting the joint normal approximation yields an elliptical region in the unit square, whereas the estimand $p_1-p_2$ is one-dimensional. We define the Elliptically Optimal (EO) confidence interval as the range of $p_1-p_2$ over this region and solve the resulting optimization problem in closed form, obtaining explicit bounds in six mutually exclusive and exhaustive cases. The solution admits a compact characterization: the EO interval is the score interval obtained by maximizing over the nuisance variance rather than estimating it. It is therefore the shortest interval obtained by projecting the elliptical region, and inherits its coverage guarantee. We derive the exact coverage excess, $2[Φ(z\mathcal{R})-Φ(z)]$, where $\mathcal{R}$ is the ratio of the least-favourable to the true standard deviation. The excess vanishes on an explicit line through the parameter space, is bounded by $α$, and is invariant under proportional scaling of the sample sizes. Exact enumeration of the binomial coverage shows that the Wald interval, whose variance estimator is downward biased by a factor $1-1/n$ under balanced allocation, falls below nominal coverage almost everywhere. The EO interval never under-covers under the normal approximation and always yields admissible, non-degenerate bounds. Its price is over-coverage when both proportions are extreme, which we quantify exactly.

math.ST↗

arXiv2026

Joint Exclusivity

We introduce \emph{joint exclusivity} (JE), a flexible extension of mutual exclusivity for non-negative random vectors that prohibits only the simultaneous positivity of all $n$ components. Unlike mutual exclusivity, JE admits a broad class of marginals while retaining a sharp existence criterion. We prove that a JE random vector with prescribed marginals $F_1,\ldots,F_n$ exists if and only if \[ \sum_{i=1}^n \overline{F}_i(0)\le n-1. \] We also provide a canonical face-based construction with freely specified copulas within each face, and show that the trivariate case can elegantly be reduced to a one-parameter family. Finally, we extend the framework through Wang-type distortions and an $m$-exclusivity hierarchy interpolating between mutual exclusivity and JE.

math.ST↗

arXiv2026

VaR at Its Extremes: Impossibilities and Conditions for One-Sided Random Variables

We investigate the extremal aggregation behavior of Value-at-Risk (VaR) -- that is, its additivity properties across all probability levels -- for sums of one-sided random variables. For risks supported on $[0,\infty)$, we show that VaR sub-additivity is impossible except in the degenerate case of exact additivity, which holds only under co-monotonicity. To characterize when VaR is instead fully super-additive, we introduce two structural conditions: negative simplex dependence (NSD) for the joint distribution and simplex dominance (SD) for a margin-dependent functional. Together, these conditions provide a unified and easily verifiable framework that accommodates non-identical margins, heavy-tailed laws, and a wide spectrum of negative dependence structures. All results extend to random variables with arbitrary finite lower or upper endpoints, yielding sharp constraints on when strict sub- or super-additivity can occur.

q-fin.RM↗

arXiv2026

Tail Structure and the Ordering of the Standard Deviation and Gini Mean Difference

We investigate the ordering between two fundamental measures of dispersion for real-valued risks: the standard deviation (SD) and the Gini mean difference (GMD). Our analysis is driven by a single structural object, namely the mean excess function of the pairwise difference $|X - X'|$. We show that its monotonicity is determined by the tail behavior of the underlying distribution, giving rise to two distinct dispersion regimes. In a heavy-tailed regime, characterized by decreasing hazard rates or increasing reverse hazard rates, the SD dominates the GMD. Conversely, when both tails of the distribution are light, the GMD dominates the SD. These dominance regimes are shown to be stable under truncation, convolution, and mixtures. Discrete analogues of the main results are also developed. Overall, the results provide an intuitive interpretation of the dispersion ordering phenomena that goes beyond the existing general comparisons, with direct relevance for risk modeling and actuarial applications.

q-fin.RM↗

arXiv2021

Can a regulatory risk measure induce profit-maximizing risk capital allocations? The case of Conditional Tail Expectation

Risk capital allocations (RCAs) are an important tool in quantitative risk management, where they are utilized to, e.g., gauge the profitability of distinct business units, determine the price of a new product, and conduct the marginal economic capital analysis. Nevertheless, the notion of RCA has been living in the shadow of another, closely related notion, of risk measure (RM) in the sense that the latter notion often shapes the fashion in which the former notion is implemented. In fact, as the majority of the RCAs known nowadays are induced by RMs, the popularity of the two are apparently very much correlated. As a result, it is the RCA that is induced by the Conditional Tail Expectation (CTE) RM that has arguably prevailed in scholarly literature and applications. Admittedly, the CTE RM is a sound mathematical object and an important regulatory RM, but its appropriateness is controversial in, e.g., profitability analysis and pricing. In this paper, we address the question as to whether or not the RCA induced by the CTE RM may concur with alternatives that arise from the context of profit maximization. More specifically, we provide exhaustive description of all those probabilistic model settings, in which the mathematical and regulatory CTE RM may also reflect the risk perception of a profit-maximizing insurer.

q-fin.RM↗