Multidimensional Integral Limit Theorems for Large Deviation Probabilities

Aleshkyavichene, A. K.

doi:10.1137/1128004

Cited by 8 publications

(5 citation statements)

References 7 publications

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“…is valid as n → +∞, where d is the dimension of the random vectors X i . The proof is a small variation of a classical argument developed, e.g., in Chapter VIII of Petrov (1975), in von Bahr (1967, or in Aleshkyavichene (1983).…”

Section: B3 Proof Of (16)mentioning

confidence: 88%

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The power of private likelihood-ratio tests for goodness-of-fit in frequency tables

Dolera¹,

Favaro²

2021

Preprint

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Privacy-protecting data analysis investigates statistical methods under privacy constraints. This is a rising challenge in modern statistics, as the achievement of confidentiality guarantees, which typically occurs through suitable perturbations of the data, may determine a loss in the statistical utility of the data. In this paper, we consider privacy-protecting tests for goodness-of-fit in frequency tables, this being arguably the most common form of releasing data. Under the popular framework of (ε, δ)-differential privacy for perturbed data, we introduce a private likelihood-ratio test for goodnessof-fit and we study its large sample properties, showing the importance of taking the perturbation into account to avoid a loss in the statistical significance of the test. Our main contribution provides a quantitative characterization of the trade-off between confidentiality, measured via differential privacy parameters ε and δ, and utility, measured via the power of the test. In particular, we establish a precise Bahadur-Rao type large deviation expansion for the power of the private likelihood-ratio test, which leads to: i) identify a critical quantity, as a function of the sample size and (ε, δ), which determines a loss in the power of the private likelihood-ratio test; ii) quantify the sample cost of (ε, δ)-differential privacy in the private likelihood-ratio test, namely the additional sample size that is required to recover the power of the likelihood-ratio test in the absence of perturbation. Such a result relies on a novel multidimensional large deviation principle for sum of i.i.d. random vectors, which is of independent interest. Our work presents the first rigorous treatment of privacy-protecting likelihood-ratio tests for goodness-of-fit in frequency tables, making use of the power of the test to quantify the trade-off between confidentiality and utility.

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Section: B3 Proof Of (16)mentioning

confidence: 88%

“…See Appendix B for the proof of ( 16). Analogous multidimensional large deviation principles, though not useful in our context, can be found in Aleshkyavichene (1983), Osipov (1982), Saulis (1983), von Bahr (1967. See also Saulis and Statulevicius (1991), and references therein, for a detailed account.…”

Section: The Power Of the "True" Lr Testmentioning

confidence: 88%

The power of private likelihood-ratio tests for goodness-of-fit in frequency tables

Dolera¹,

Favaro²

2021

Preprint

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“…be independent identically distributed random vectors taking values in R d , d 1. Assume that their common distribution P 0 is essentially d-dimensional and set S n = ξ (1) + · · · + ξ (n) , n = 1, 2, . .…”

Section: Introduction Let ξ ξmentioning

confidence: 99%

“…It is worth noting that in all the above-mentioned works, as well as in [1], [2], [13], [15], [23], [28], [31], [32], the order of deviations is bounded by O(n 1/2 …”

Section: Introduction Let ξ ξmentioning

confidence: 99%

Abelian Theorems, Limit Properties of Conjugate Distributions, and Large Deviations for Sums of Independent Random Vectors

Zaigraev¹,

Nagaev$^dag$²

2004

Theory Probab. Appl.

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Abstract.A class of multidimensional absolutely continuous distributions is considered. Each distribution has a moment generating function, which is finite in a bounded convex set S and generates a family of the so-called conjugate distributions. We focus our attention on the limit distributions for this family when the conjugate parameter tends to the boundary of S. As in the one-dimensional case, each limit distribution is obtained as a corollary of the Abel-type theorem. The results obtained are utilized for establishing a local limit theorem for large deviations of arbitrarily high order.

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“…To complete the proof, observe that, for any fixed ⑀ ) 0, the right-hand side of 14 1 is less than n log n for large enough n. As the probability of positive drift 2 1 increases monotonically with batch size m a batch size of n log n will result in a 2 1 probability of positive drift at least as large as q ⑀ for large enough n. B Ž . from w with high probability.…”

mentioning

confidence: 97%