Source-linked AI summary
Asymptotic results for linear combinations of spacings generated by i.i.d. exponential random variables
Camilla Calì, Maria Longobardi, Claudio Macci, Barbara Pacchiarotti
TL;DR
The paper studies large and moderate deviations for linear combinations of spacings from i.i.d. exponential variables, extending results for empirical cumulative entropies. It uses coefficients defined by suitable continuous functions and establishes deviation principles under stated conditions, while noting difficulties beyond exponential sampling.
Problem
Existing results treated particular spacing combinations, including empirical cumulative entropies, while a broader class of continuous-function coefficients remained to be addressed.
Source: §1 · pp. 1–2Method
The paper studies spacing combinations C_n(w) with continuous-function coefficients under suitable conditions, using properties of exponential spacings to establish deviation results.
Source: §1 · pp. 1–2, §3 · p. 11Results
For exponentially distributed samples, C_n(w) satisfies an LDP with speed n and good rate function Λ*, and the paper also establishes a moderate-deviation LDP with speed 1/a_n and good rate function ˜Λ*.
Source: §3 · pp. 7, 9Takeaways & Limitations
The results generalize Giuliano et al. (2015) from empirical cumulative entropies to a wider class of spacing combinations defined by suitable continuous functions.
Source: §1 · p. 2, §3 · p. 6For non-exponential variables, the authors report difficulty applying Gärtner–Ellis and leave possible extensions under suitable technical conditions for future work.
Source: §3 · p. 7, §4 · p. 13
Abstract
from arXiv · showhide
We prove large (and moderate) deviations for a class of linear combinations of spacings generated by i.i.d. exponentially distributed random variables. We allow a wide class of coefficients which can be expressed in terms of continuous functions defined on [0, 1] which satisfy some suitable conditions. In this way we generalize some recent results by Giuliano et al. (2015) which concern the empirical cumulative entropies defined in Di Crescenzo and Longobardi (2009a).
1 Introduction
The paper extends large-deviation work on linear combinations of spacings to a broad class of continuous-function coefficients. Its setting generalizes results for empirical cumulative entropies under exponential sampling.
1 Introduction
- Linear combinations of order statistics, including spacings, are an important part of empirical-process research and have been studied for large deviations.A spacing is the difference between consecutive order statistics.Source: §1 · p. 1
- The paper generalizes Giuliano et al. (2015) results for empirical cumulative entropies when the underlying variables are exponentially distributed.Those entropies were defined for i.i.d. positive variables with a common absolutely continuous distribution.Source: §1 · p. 1
- The coefficients are drawn from a wide class expressible through continuous functions on [0, 1] that satisfy suitable conditions.Source: §1 · p. 2
- The paper also relates its moderate-deviation result to Gao and Zhao (2011) and discusses function choices based on empirical entropies.Source: §1 · p. 2
2 Preliminaries
The preliminaries define the deviation framework and the spacing-based statistic, then give its convergence and normality context. They also connect the statistic to L-statistics and empirical cumulative entropies.
2.1 Preliminaries on large deviations
- The paper defines a good rate function as one with compact level sets and invokes Gärtner–Ellis to obtain LDPs with good rate functions.The theorem uses the Legendre transform Λ*(z) = supθ∈R{θz−Λ(θ)} when Λ meets the stated regularity conditions.Source: §2.1 · p. 2
2.2 Preliminaries on the sequence {Cn(w) : n ≥1}
- For exponentially distributed samples, spacings are independent, with X_{k+1:n}−X_{k:n} distributed as EXP(λ(n−k)).This yields explicit moment-generating-function, mean, and variance formulas for C_n(w).Source: §2.2 · p. 3
- The preliminaries discuss almost-sure convergence and asymptotic normality of C_n(w), including convergence of its centered, scaled form to a centered Normal distribution.The normality discussion follows Di Crescenzo and Longobardi (2009a).Source: §2.2 · pp. 3, 5
- Condition 1 requires continuous w with |w(x)|≤c(1−x)^β near 1, for β∈(0,1], and this ensures w(1)=0 for a finite almost-sure limit.The paper assumes the common distribution is EXP(λ), λ>0.Source: §2.2 · p. 3
2.3 Connections with some literature
- With a suitable score function, C_n(w) coincides with L-statistics in Gao and Zhao (2011), empirical cumulative entropies in Di Crescenzo and Longobardi (2009a), and the sequence in Giuliano et al. (2015).Gao and Zhao (2011) do not require exponentially distributed variables.Source: §2.3 · pp. 5–6
3 Results
The results extend large- and moderate-deviation analysis to spacing combinations with admissible continuous weight functions. Under the stated conditions, the paper establishes an LDP, derives a rate-function bound, and connects selected weights to empirical means.
Moderate deviations
- The moderate-deviation result assumes Λw is finite in a neighbourhood of the origin θ = 0; the paper remarks it would not be finite under Condition 1 with β ∈ (0, 1).Source: §3 · p. 6
Large deviations
- The sequence C_n(w) satisfies an LDP with speed n and good rate function Λ* when the exponential-input, weight, and regularity conditions hold.Proposition 3.1 also assumes Λ_w is essentially smooth and lower semi-continuous.Source: §3 · pp. 6–7
Examples
- For w_1(x)=1−x, C_n(w_1) is equally distributed with a sum of n exponential variables, yielding a Gamma law and the corresponding Cramér rate function.The cited rate function is λy−1−log(λy) for y>0 and infinity otherwise.Source: §3 · p. 8
Large deviations
- For positive h_w, Proposition 3.2 gives an upper bound on the rate function through M_w and identifies its minimizer as ȳ_w.M_w(y)=0 if and only if y=ȳ_w.Source: §3 · pp. 8–9
Moderate deviations
- Under Proposition 3.3’s sequence conditions, centered and rescaled C_n(w) satisfies an LDP with speed 1/a_n and a good rate function.The paper describes this moderate-deviation class as filling the gap between two asymptotic regimes.Source: §3 · pp. 9, 11
Examples
- For ℓ_w(x)=γ_w(1−x), C_n(ℓ_w) is a sequence of empirical means of exponentially distributed variables.Its rate function is the Cramér rate function for EXP(λγ_w^-1) variables.Source: §3 · p. 11
4 Some analogies with a moderate deviation result for L-statistics
The section relates the paper’s moderate-deviation result for a score-function choice of w to Gao and Zhao’s L-statistic result, comparing their centering and rate-function behavior under suitable conditions.
4 Some analogies with a moderate deviation result for L-statistics
- Both moderate-deviation rate functions are quadratic and uniquely vanish at the origin, while (a(n))^-2 plays the role of a_n.The comparison concerns Proposition 3.3 and Theorem 4.8 in Gao and Zhao (2011).Source: §4 · p. 12
- The computations establish the expected equality m(J, F) = μ_w(J;·) for the score-function choice w(J;·).The derivation uses symmetry of the integrand, a change of variables, and integration by parts.Source: §4 · pp. 12–13
- For w(J;·), the stated Condition 2 applies with β = 1, and the score function is bounded, continuous, and trimmed.Trimmed means it is zero near r = 0 and r = 1.Source: §4 · pp. 12–13
- For non-exponential variables, applying the Gärtner–Ellis theorem is difficult; adapting Gao and Zhao’s proof to general w is left for future work.The cited Gao and Zhao result treats w(J;·) under suitable hypotheses.Source: §4 · p. 13
5 Applications to some empirical entropies
The section connects linear combinations of spacings to empirical entropy functionals, including cumulative entropy and generalized or residual-entropy examples, and describes conditions their weight functions satisfy.
5 Applications to some empirical entropies
- The section defines cumulative entropy for an absolutely continuous distribution and its empirical counterpart using the empirical distribution function.The empirical distribution function is formed from an i.i.d. sample with distribution function F.Source: §5 · p. 13
- The empirical cumulative entropy CE(F̂_n) converges almost surely to CE(F), and C_n(w(1)) coincides with CE(F̂_n).The weight w(1) satisfies both Conditions 1 and 2.Source: §5 · pp. 13–14
- The fractional generalized cumulative entropy extends the cumulative-entropy example, which it recovers at α = 1.For α ≥ 1, its weight satisfies Conditions 1 and 2; for α ∈ (0, 1), Condition 1 holds but Condition 2 fails.Source: §5 · p. 14
- For q = 1, the cumulative residual entropy is recovered; for q > 1, its weight satisfies Condition 1 but generally not Condition 2.The stated exception to failure of Condition 2 is q = 0.Source: §5 · p. 14
Funding
The acknowledgements identify institutional and project support for the authors.
Funding
- CC and ML list Indam-GNAMPA and MIUR-PRIN 2017 support; CM and BP list Tor Vergata, Indam-GNAMPA, and related project support.The listed support includes university and department research programs.Source: Funding · p. 14