×
Albertus, Mickael; Berthet, Philippe

Auxiliary information: the raking-ratio empirical process. (English) Zbl 1411.62028

Electron. J. Stat. 13, No. 1, 120-165 (2019).
Authors’ abstract: We study the empirical measure associated to a sample of size \(n\) and modified by \(N\) iterations of the raking-ratio method. This empirical measure is adjusted to match the true probability of sets in a finite partition which changes each step. We establish asymptotic properties of the raking-ratio empirical process indexed by functions as \(n\rightarrow +\infty \), for \(N\) fixed. We study nonasymptotic properties by using a Gaussian approximation which yields uniform Berry-Esseen type bounds depending on \(n,N\) and provides estimates of the uniform quadratic risk reduction. A closed-form expression of the limiting covariance matrices is derived as \(N\rightarrow +\infty \). In the two-way contingency table case the limiting process has a simple explicit formula.
Reviewer: Wiesław Dziubdziela (Miedziana Gora)

Cited in 2 Documents

MSC:

62D05 Sampling theory, sample surveys
62G07 Density estimation
62H17 Contingency tables
62J10 Analysis of variance and covariance (ANOVA)
60F17 Functional limit theorems; invariance principles

Keywords:

raking-ratio method; empirical processes; strong approximation; nonparametric statistics; auxiliary information; Sinkhorn algorithm; contingency table
Cite Review PDF
Full Text: DOI arXiv Euclid

References:

[1] Alexander, K. S. (1984). Probability inequalities for empirical processes and a law of the iterated logarithm., The Annals of Probability12 1041-1067. · Zbl 0549.60024 · doi:10.1214/aop/1176993141
[2] Bankier, M.D. (1986). Estimators based in several stratified samples with applications to multiple frame surveys., Journal of the American Statistical Association81 1074-1079. · Zbl 0614.62007 · doi:10.1080/01621459.1986.10478376
[3] Berthet, P. and Mason, D. M. (2006). Revisiting two strong approximation results of Dudley and Philipp., IMS Lecture Notes-Monograph Series High Dimensional Probability51 155-172. · Zbl 1122.60031
[4] Binder, D. A. and Théberge, A (1988). Estimating the variance of raking-ratio estimators., The Canadian Journal of Statistics16 47-55. · Zbl 0709.62600 · doi:10.2307/3315215
[5] Birgé, L. and Massart, P. (1998). Minimum contrast estimators on sieves: exponential bounds and rates of convergence., Bernoulli. Official Journal of the Bernoulli Society for Mathematical Statistics and Probability4 329-375. · Zbl 0954.62033 · doi:10.2307/3318720
[6] Brackstone, G. J. and Rao, J. N. K. (1979). An investigation of raking ratio estimators., The Indian journal of Statistics41 97-114. · Zbl 0484.62025
[7] Brown, D. T. (1959). A note on approximations to discrete probability distributions., Information and control2 386-392. · Zbl 0117.14804 · doi:10.1016/S0019-9958(59)80016-4
[8] Choudhry, G.H. and Lee, H. (1987). Variance estimation for the Canadian Labour Force Survey., Survey Methodology13 147-161.
[9] Cover, T. M. and Thomas, J. A. (2012). Elements of information theory., John Wiley & Sons · Zbl 0762.94001
[10] Deming, W. E. and Stephan, F. F. (1940). On a least squares adjustment of a sampled frequency table when the expected marginal totals are known., The Annals of Mathematical Statistics11 427-444. · Zbl 0024.05502 · doi:10.1214/aoms/1177731829
[11] Deville, J-C. and Särndal, C-E (1992). Calibration estimators in survey sampling., Journal of the American Statistical Association87 376-382. · Zbl 0760.62010
[12] Deville, J-C. and Särndal, C-E (1993). Generalized raking procedures in survey sampling., Journal of the American Statistical Association88 1013-1020. · Zbl 0794.62005 · doi:10.1080/01621459.1993.10476369
[13] Dudley, R. M. (1989). Real analysis and probability., Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA · Zbl 0686.60001
[14] Dudley, R. M. (2014). Uniform central limit theorems., Cambridge university press142 · Zbl 1317.60030
[15] Franklin, J. and Lorenz, J. (1989). On the scaling of multidimensional matrices., Linear Algebra and its Applications114-115 717-735. · Zbl 0674.15001 · doi:10.1016/0024-3795(89)90490-4
[16] Ireland, C. T. and Kullback, S. (1968). Contingency tables with given marginals., Biometrika55 179-188. · Zbl 0155.26701 · doi:10.1093/biomet/55.1.179
[17] Konijn, H. S. (1981). Biases, variances and covariances of raking ratio estimators for marginal and cell totals and averages of observed characteristics., Metrika28 109-121. · Zbl 0461.62012 · doi:10.1007/BF01902883
[18] Lewis, P. M. (1959). Approximating probability distributions to reduce storage requirements., Information and control2 214-225. · Zbl 0095.32602 · doi:10.1016/S0019-9958(59)90207-4
[19] Pollard, D. (1984). Convergence of stochastic processes., Springer Science & Business Media · Zbl 0544.60045
[20] Pollard, D. (1990). Empirical processes: theory and applications., NSF-CBMS regional conference series in probability and statistics · Zbl 0741.60001
[21] Shorack, G. R. and Wellner, J. A. (1986). Empirical processes with applications to statistics., Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics · Zbl 1170.62365
[22] Sinkhorn, R. (1964). A relationship between arbitrary positive matrices and doubly stochastic matrices., The Annals of Mathematical Statistics35 876-879. · Zbl 0134.25302 · doi:10.1214/aoms/1177703591
[23] Sinkhorn, R. (1967). Diagonal equivalence to matrices with prescribed row and column sums., The American Mathematical Monthly74 402-405. · Zbl 0166.03702 · doi:10.2307/2314570
[24] Sinkhorn, R. and Knopp, P. (1967). Concerning nonnegative matrices and doubly stochastic matrices., Pacific Journal of Mathematics21 343-348. · Zbl 0152.01403 · doi:10.2140/pjm.1967.21.343
[25] Stephan, F. F. (1942). An iterative method of adjusting sample frequency tables when expected marginal totals are known., The Annals of Mathematical Statistics13 166-178. · Zbl 0060.31505 · doi:10.1214/aoms/1177731604
[26] Talagrand, M. (1994). Sharper bounds for Gaussian and empirical processes., The Annals of Probability22 28-76. · Zbl 0798.60051 · doi:10.1214/aop/1176988847
[27] Van der Vaart, A. W. and Wellner, J. A. (1996). Weak convergence and empirical processes with applications to statistics., Springer series in statistics · Zbl 0862.60002
[28] Van der Vaart, A. W. (2000). Asymptotic statistics., Cambridge university press · Zbl 0910.62001
[29] Wellner, J. A. (1992). Empirical processes in action: a review., International Statistical Review/Revue Internationale de Statistique60 247-269. · Zbl 0757.62028 · doi:10.2307/1403678
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.