Probability Inequalities for the Sum of Independent Random Variables

Bennett, G. G.

doi:10.1080/01621459.1962.10482149

Cited by 527 publications

(113 citation statements)

References 8 publications

Supporting

Mentioning

100

Contrasting

Unclassified

Order By: Relevance

“…For simplicity, we use the uniform knots, s 1 = 0, s 2 = 1/(k n + 2µ − 1), · · · , s kn+2µ = 1, which are sufficient when the function g(·) does not exhibit dramatic changes in its derivatives. By equation (2) in Chapter IX of [12] B knj has support [s j , s j+µ ]. Then g(·) can be approximated by a linear combination α B kn (·) of the basis, where α ∈ R kn+µ and B kn (·) = B kn1 (·), · · · , B kn,kn+µ (·) .…”

Section: The Main Resultsmentioning

confidence: 99%

Uniform Convergence Rate of Estimators of Autocovariances in Partly Linear Regression Models with Correlated Errors

You

Chen

et al. 2003

Acta Mathematicae Applicatae Sinica, English Series, English Se

View full text Add to dashboard Cite

Consider the partly linear regression modelx ip ) and t i ∈ T are known and nonrandom design points, T is a compact set in the real line R, β = (β 1 , · · · , βp) is an unknown parameter vector, g(·) is an unknown function andrandom variables with zero mean and variance σ 2 e . Drawing upon B-spline estimation of g(·) and least squares estimation of β, we construct estimators of the autocovariances of {ε i }. The uniform strong convergence rate of these estimators to their true values is then established. These results not only are a compensation for those of [23], but also have some application in modeling error structure. When the errors {ε i } are an ARMA process, our result can be used to develop a consistent procedure for determining the order of the ARMA process and identifying the non-zero coefficients of the process. Moreover, our result can be used to construct the asymptotically efficient estimators for parameters in the ARMA error process.

show abstract

Section: The Main Resultsmentioning

confidence: 99%

Uniform Convergence Rate of Estimators of Autocovariances in Partly Linear Regression Models with Correlated Errors

You

Chen

et al. 2003

Acta Mathematicae Applicatae Sinica, English Series, English Se

View full text Add to dashboard Cite

show abstract

“…Elementary algebra shows that (4) is minimized with the choice t := log (1 + ηM), which yields first a Bennett exponential bound (see [3]) and because of (…”

Section: Decoupling Inequality and Coupling Exponential Boundsmentioning

confidence: 99%

Optimal calibration for multiple testing against local inhomogeneity in higher dimension

Rohde

2010

Probab. Theory Relat. Fields

View full text Add to dashboard Cite

Based on two independent samples X 1 , . . . , X m and X m+1 , . . . , X n drawn from multivariate distributions with unknown Lebesgue densities p and q respectively, we propose an exact multiple test in order to identify simultaneously regions of significant deviations between p and q. The construction is built from randomized nearest-neighbor statistics. It does not require any preliminary information about the multivariate densities such as compact support, strict positivity or smoothness and shape properties. The properly adjusted multiple testing procedure is shown to be sharp-optimal for typical arrangements of the observation values which appear with probability close to one. The proof relies on a new coupling Bernstein type exponential inequality, reflecting the non-subgaussian tail behavior of a combinatorial process. For power investigation of the proposed method a reparametrized minimax set-up is introduced, reducing the composite hypothesis " p = q" to a simple one with the multivariate mixed density (m/n) p + (1 − m/n)q as infinite dimensional nuisance parameter. Within this framework, the test is shown to be spatially and sharply asymptotically adaptive with respect to uniform loss on isotropic Hölder classes. The exact minimax risk asymptotics are obtained in terms of solutions of the optimal recovery.

show abstract

“…[2]) that says that if {X j } j =1,2,... is a collection of centered IID variables such that |X 1 | ≤ 1, then for every t ≥ 0…”

Section: Probability Lower Bounds: Quenched (And Annealed) Estimatesmentioning

confidence: 99%

Enhanced interface repulsion from quenched hard–wall randomness

Bertacchi¹,

Giacomin²

2002

Probab Theory Relat Fields

View full text Add to dashboard Cite

We consider the harmonic crystal, or massless free field, ϕ = {ϕ x } x∈Z d , d ≥ 3, that is the centered Gaussian field with covariance given by the Green function of the simple random walk on Z d . Our main aim is to obtain quantitative information on the repulsion phenomenon that arises when we condition ϕ x to be larger than σ x , σ = {σ x } x∈Z d is an IID field (which is also independent of ϕ), for every x in a large region D N = ND ∩ Z d , with N a positive integer and D a bounded subset of R d . We are mostly motivated by results for given typical realizations of σ (quenched set-up), since the conditioned harmonic crystal may be seen as a model for an equilibrium interface, living in a (d + 1)-dimensional space, constrained not to go below an inhomogeneous substrate that acts as a hard wall. We consider various types of substrate and we observe that the interface is pushed away from the wall much more than in the case of a flat wall as soon as the upward tail of σ 0 is heavier than Gaussian, while essentially no effect is observed if the tail is sub-Gaussian. In the critical case, that is the one of approximately Gaussian tail, the interplay of the two sources of randomness, ϕ and σ , leads to an enhanced repulsion effect of additive type. This generalizes work done in the case of a flat wall and also in our case the crucial estimates are optimal Large Deviation type asymptotics as N ∞ of the probability that ϕ lies above σ in D N .

show abstract

Probability Inequalities for the Sum of Independent Random Variables

Cited by 527 publications

References 8 publications

Uniform Convergence Rate of Estimators of Autocovariances in Partly Linear Regression Models with Correlated Errors

Uniform Convergence Rate of Estimators of Autocovariances in Partly Linear Regression Models with Correlated Errors

Optimal calibration for multiple testing against local inhomogeneity in higher dimension

Enhanced interface repulsion from quenched hard–wall randomness

Contact Info

Product

Resources

About