arXiv subjects

Nagananda K G

Publications and source records attributed to Nagananda K G.

7 recordsLinked to original sources

arXiv2026

Regret-weighted Bayes Fusion for Distributed Experimental Design

We study distributed experimental design with multiple candidate experiments, where local sites possess only partial information and transmit design recommendations to a fusion center. Unlike centralized design, in which the experiment that maximizes expected information gain can be selected directly, distributed design requires combining heterogeneous and potentially conflicting local recommendations. Formulating as a multi-class Bayes fusion problem, centralized oracle design is treated as an unknown label and each site is characterized by a local recommendation mechanism. The proposed fusion rule minimizes posterior expected information regret, rather than merely maximizing the number of local votes or the posterior probability (MAP) of the oracle label. This distinction is essential because different incorrect experimental choices may incur different losses in information gain. We show that majority vote is optimal only under restrictive symmetry assumptions and can otherwise be strictly suboptimal. Regret-weighted multi-class Chernoff bounds are derived to identify the pairwise separations governing distributed design performance. Numerical studies identify two operational regimes: MAP is effective when oracle-label accuracy and information regret are aligned, while regret-weighted Bayes fusion reduces information loss when the most probable oracle label is not the lowest-regret decision.

math.ST↗

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

Distributed Experimental Design: Bayes-optimal Fusion of Local Designs

We develop a decision-theoretic framework for distributed Bayesian experimental design in which local agents evaluate candidate experiments using expected information gain and transmit their local design decisions to a fusion center. Unlike centralized Bayesian design, where all likelihood components and information-gain values are available to a single planner, the fusion center in the distributed setting chooses a global experiment from compressed local recommendations. We derive the Bayes-optimal fusion rule, which selects the experiment with largest conditional expected centralized information gain given the observed local design decisions. This rule is analogous in spirit to optimal fusion rules in distributed detection, but differs fundamentally because the underlying utility is expected information gain and the resulting loss is information-gain regret rather than classification error. We also establish information-loss bounds and identify conditions under which the decision-only fusion rule is asymptotically equivalent to the centralized design. Numerical experiments show that Bayes-optimal fusion closely approximates the centralized oracle, whereas majority voting can be highly suboptimal when a minority of sites carry disproportionate information.

stat.AP↗

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↗

arXiv2026

A projector-rank partition theorem for exact degrees of freedom in experimental design

In many experimental designs -- split-plots, blocked or nested layouts, fractional factorials, and studies with missing or unequal replication -- standard ANOVA procedures no longer tell us exactly how many independent pieces of information each effect truly contributes. We provide a general degrees of freedom $(\mathrm{df})$ partition theorem that resolves this ambiguity. For $N$ observations, we show that the total information in the data (i.e., $N-1$ $\mathrm{df}$) can be split exactly across experimental effects and randomization strata by projecting the data onto each stratum and counting the $\mathrm{df}$ each effect contributes there. This yields integer $\mathrm{df}$ -- not approximations -- for any mix of fixed and random effects, blocking structures, fractionation, or imbalance. This result yields closed-form $\mathrm{df}$ tables for unbalanced split-plot, row-column, lattice, and crossed-nested designs. We introduce practical diagnostics -- the $\mathrm{df}$-retention ratio $ρ$, df deficiency $δ$, and variance-inflation index $α$ -- that measure exactly how many $\mathrm{df}$ an effect retains under blocking or fractionation and the resulting loss of precision, thereby extending Box-Hunter's resolution idea to multi-stratum and incomplete designs. Classical results emerge as corollaries: Cochran's one-stratum identity; Yates's split-plot $\mathrm{df}$; resolution-$R$ identified when an effect retains no $\mathrm{df}$. Empirical studies on split-plot and nested designs, a blocked fractional-factorial design-selection experiment, and timing benchmarks show that our approach delivers calibrated error rates, recovers information to raise power by up to 60% without additional runs, and is orders of magnitude faster than bootstrap-based $\mathrm{df}$ approximations.

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↗