arXiv subjects

Jong Sung Kim

Publications and source records attributed to Jong Sung Kim.

4 recordsLinked to original sources

arXiv2026

A Laplace equation approach to the Behrens--Fisher problem

We develop a partial differential equation formulation of the Behrens-Fisher problem for two independent normal samples with unknown and unequal variances. An orthogonal decomposition separates mean and residual components (corresponding to the centered within-sample variation left after removal of the mean directions) and recasts the studentized difference of sample means as a scale-invariant geometric constraint. This reduction transforms the distributional problem into the evaluation of spherical wedge probabilities, which are identified with harmonic measure and with the value at the origin of a Laplace-Dirichlet boundary value problem. From this framework, we derive exact finite-sample representations for the cumulative distribution function and the probability density function in terms of beta functions, with dependence only on the sample sizes and the variance ratio. These representations place the Behrens-Fisher law in a standard special-function form that is directly accessible in widely available commercial software -- including Microsoft Excel -- thereby facilitating distributional evaluation and quantile computation. We also obtain a Gegenbauer separation-of-variables expansion for the associated harmonic extension and its threshold derivative, with coefficients in closed Beta-Gamma form, and derive sharp tail expansions with explicit leading constants and higher-order corrections.

math.ST↗

arXiv2026

Statistical two-round search for one excellent element

We formulate and study a statistical version of Katona's two-round search problem of finding at least one excellent element in a set. A population of $n$ elements is considered, where each element is independently excellent with probability $λ/n$, $λ> 0$. A subset test is noiseless: it returns positive exactly when the queried subset contains at least one excellent element. The goal is to minimize the expected number of tests subject to finding one excellent element with probability at least $1-α$, where $0<α<1$, under the restriction that testing is performed in two rounds. Unlike classical group testing, the objective is not to recover the full set of excellent elements, but only to identify one of them. We first show that success is fundamentally limited by the possibility that no excellent element exists. In the sparse Poisson regime, this imposes the necessary feasibility condition $α\ge e^{-λ}$. When the target success probability is feasible, we prove that the optimal expected number of tests grows logarithmically with the population size. The upper bound is obtained by combining an initial existence test with a second-round separating design; the lower bound follows from an information-counting argument. Numerical illustrations show the feasibility boundary and the resulting logarithmic scaling.

cs.IT↗

arXiv2026

A unified approach to the Behrens-Fisher problem

A unified framework is presented to study the two-sample Behrens--Fisher problem -- testing equality of means when two normal populations have unequal, unknown variances -- and a compact expression is derived for the null distribution of the classical test statistic. Our new approach involves a Mellin--Barnes factorization that decouples the square root of a weighted sum of independent chi-square variates, thereby collapsing a challenging two-dimensional integral to a tractable single-contour integral. Closing the contour yields a residue series that terminates whenever either sample's degrees of freedom is odd. A complementary Euler--Beta reduction identifies the density as a Gauss hypergeometric function with explicit parameters, yielding a numerically stable form that recovers Student's $t$ under equal variances. Ramanujan's master theorem supplies exact inverse-power tail coefficients, which bound Lugannani--Rice saddle-point approximation errors and support reliable tail analyses. The proposed framework reveals why hypergeometric structure appears, why certain finite-sum cases arise, and how one can pass from the bulk of the distribution to its tails without altering the analytic framework. Finally, it lets us tabulate exact two-sided critical values over a broad grid of sample sizes and variance ratios that reveal the parameter surface on which the well-known Welch's approximation switches from conservative to liberal, quantifying its maximum size distortion.

math.ST↗

arXiv2025

Cassini-Catalan Determinants via Ramanujan's Theta Identity

In this paper, we show that the classical Cassini and Catalan identities for Fibonacci numbers arise naturally from a single quadratic theta-function identity of Ramanujan. Expanding the identity $ψ(q)ψ(q^{3})=ψ(q^{4})φ(q^{6})+q\,φ(q^{2})ψ(q^{12})$ via the Jacobi triple product and equating coefficients yields the unified $q$-determinant $F_{n+r}(q)F_{n-r}(q)-F_{n}(q)^{2}=(-q)^{\,n-r}F_{r}(q)^{2}$, $n\ge r\ge 1$, where $ψ(q)$ and $φ(q)$ are Ramanujan's theta functions with $q$ a complex parameter in the unit disc $(\lvert q \rvert < 1)$ and $F_n(q)$ denotes the Carlitz $q$-Fibonacci polynomials. The radial limit $q\to1^{-}$ recovers Cassini's formula ($r=1$) and Catalan's one-parameter extension, while the same derivation with an auxiliary weight produces new partition-refined versions. The argument uses only standard $q$-series algebra (triple-product expansions, $q$-Pochhammer cancellations, and coefficient extraction), providing a transparent modular explanation of the alternating sign $(-1)^{\,n-r}$ in Catalan's identity through the level-6 provenance of $φ$ and $ψ$. Beyond unifying Cassini\textendash Catalan in a single framework, the method lifts seamlessly to higher-order recurrences, giving a template for Tribonacci-type determinants and suggesting congruence phenomena obtained from modular dissections and root-of-unity limits. The results place familiar Fibonacci determinants within Ramanujan's analytic landscape, indicate routes to combinatorial bijections that mirror the analytic cancellations, and connect with themes in modern $q$-series\textemdash ranging from colored partition identities to quantum-modular and exactly solvable models\textemdash thereby highlighting both the explanatory power and the ongoing relevance of Ramanujan's theta identities.

math.CO↗