On the gaussian approximation of convolutions under multidimensional analogues of S.N. Bernstein's inequality conditions

Summary Model organisms and human studies have led to increasing empirical evidence that interactions among genes contribute broadly to genetic variation of complex traits. In the presence of gene-by-gene interactions, the dimensionality of the feature space becomes extremely high relative to the sample size. This imposes a significant methodological challenge in identifying gene-by-gene interactions. In the present paper, through a Gaussian graphical model framework, we translate the problem of identifying gene-by-gene interactions associated with a binary trait D into an inference problem on the difference of two high-dimensional precision matrices, which summarize the conditional dependence network structures of the genes. We propose a procedure for testing the differential network globally that is particularly powerful against sparse alternatives. In addition, a multiple testing procedure with false discovery rate control is developed to infer the specific structure of the differential network. Theoretical justification is provided to ensure the validity of the proposed tests and optimality results are derived under sparsity assumptions. A simulation study demonstrates that the proposed tests maintain the desired error rates under the null and have good power under the alternative. The methods are applied to a breast cancer gene expression study.

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“…Define

{false| a false|}_{min} = {min}_{1 \leq i \leq d} | a_{i} |

for any vector a ∈ R d . Then we have

pr (\cap_{j = 1}^{d} F_{m_{j}}) = pr ({| {n_{2}}^{- \frac{1}{2}} \sum_{k = 1}^{n_{1} + n_{2}} W_{k} |}_{min} \geq y_{p}^{\frac{1}{2}}) .

Then it follows from Theorem 1 in Zaïtsev (1987) that

\begin{matrix} left \\ pr ({| {n_{2}}^{- 1 / 2} true \sum_{k = 1}^{n_{1} + n_{2}} W_{k} |}_{min} \geq y_{p}^{1 / 2}) \leq pr {{| N_{d} |}_{min} \geq y_{p}^{1 / 2} - ε_{n} {(\log p)}^{- 1 / 2}} \\ + c_{1} d^{\frac{5}{2}} \exp {- \frac{n^{1 / 2} ε_{n}}{c_{2} d^{3} τ_{n} {(normallog p)}^{1 / 2}}}, \end{matrix}

where c 1 > 0 and c 2 > 0 are constants, ε n → 0 which will be specified later and

N_{d} = (N_{m_{1}}, \dots N_{m_{d}})

is a normal random vector with E ( N d ) = 0 and

normalcov false(N_{d} false) = n_{2} / n_{1} normalcov false(W_{1} false) + no...

…”

Section: A·1 Technical Lemmasmentioning

confidence: 84%

Testing differential networks with applications to the detection of gene-gene interactions

2015

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“…[20] used a finite approximation of a (possibly uncountably) infinite class of functions and apply a coupling inequality of [60] to the discretized empirical process (more precisely, [20] used a version of Yurinskii's inequality proved by [18]). [1] and [53], on the other hand, used a coupling inequality of [61] instead of Yurinskii's and some recent empirical process techniques such as Talagrand's [56] concentration inequality, which leads to refinements of Dudley and Philipp's results in some cases. However, the rates that [18], [1] and [53] established do not lead to tight conditions for the Gaussian approximation in non-Donsker cases, with important examples being the suprema of empirical processes arising in nonparametric estimation, namely the suprema of local and series empirical processes (see Section 3 for detailed treatment).…”

mentioning

confidence: 99%

Gaussian approximation of suprema of empirical processes

Chernozhukov¹,

Chetverikov²,

Kato³

2013

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Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. This paper develops a new direct approach to approximating suprema of general empirical processes by a sequence of suprema of Gaussian processes, without taking the route of approximating whole empirical processes in the sup-norm. We prove an abstract approximation theorem applicable to a wide variety of statistical problems, such as construction of uniform confidence bands for functions. Notably, the bound in the main approximation theorem is nonasymptotic and the theorem does not require uniform boundedness of the class of functions. The proof of the approximation theorem builds on a new coupling inequality for maxima of sums of random vectors, the proof of which depends on an e↵ective use of Stein's method for normal approximation, and some new empirical process techniques. We study applications of this approximation theorem to local and series empirical processes arising in nonparametric estimation via kernel and series methods, where the classes of functions change with the sample size and are non-Donsker. Importantly, our new technique is able to prove the Gaussian approximation for the supremum type statistics under weak regularity conditions, especially concerning the bandwidth and the number of series functions, in those examples. Terms of use: Documents in

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“…Our method of proof is somewhat similar to the one employed by Dudley and Philipp (1983). We ÿrst establish a coupling inequality for multidimensional random vectors, where we use a result of Zaitsev (1987) on the rate of convergence in the multidimensional central limit theorem in combination with the Strassen-Dudley theorem. We then employ a recent inequality of Talagrand (1994) to approximate the empirical process by a suitable ÿnite dimensional process.…”

Section: Andmentioning

confidence: 99%

“…The following inequality follows from the work of Zaitsev (1987). where c 1 and c 2 are positive constants depending only on d.…”

Section: A Useful Coupling Inequalitymentioning

confidence: 99%

Gaussian approximation of local empirical processes indexed by functions

Einmahl

Mason

1997

Probab Theory Relat Fields

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An extended notion of a local empirical process indexed by functions is introduced, which includes kernel density and regression function estimators and the conditional empirical process as special cases. Under suitable regularity conditions a central limit theorem and a strong approximation by a sequence of Gaussian processes are established for such processes. A compact law of the iterated logarithm (LIL) is then inferred from the corresponding LIL for the approximating sequence of Gaussian processes. A number of statistical applications of our results are indicated.Mathematics Subject Classiÿcation (1991): 60F15, 62G05

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On the gaussian approximation of convolutions under multidimensional analogues of S.N. Bernstein's inequality conditions

Cited by 79 publications

References 9 publications

Testing differential networks with applications to the detection of gene-gene interactions

Testing differential networks with applications to the detection of gene-gene interactions

Gaussian approximation of suprema of empirical processes

Gaussian approximation of local empirical processes indexed by functions

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