Abstract
The main goal of this article is to study how an auxiliary information can be used to improve the efficiency of two famous statistical tests: the Z-test and the chi-square test. Many definitions of auxiliary information can be found in the statistical literature. In this article, the notion of auxiliary information is discussed from a very general point of view and depends on the relevant test. These two statistical tests are modified so that this information is taken into account. It is shown in particular that the efficiency of these new tests is improved in the sense of Pitman’s ARE. Some statistical examples illustrate the use of this method.
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References
Albertus M (2020) Raking-ratio empirical process with auxiliary information learning. ESAIM: Probab Stat. https://doi.org/10.1051/ps/2020017
Albertus M, Berthet P (2019) Auxiliary information: the raking-ratio empirical process. Electronic Jo Stat 13(1):120–165. https://doi.org/10.1214/18-EJS1526
Benhamou E, Melot V (2018) Seven proofs of the Pearson Chi-squared independence test and its graphical interpretation. SSRN Electronic J. https://doi.org/10.2139/ssrn.3239829
Binder DA, Théberge A (1988) Estimating the variance of raking-ratio estimators. Can J Stat/ La Revue Canadienne de Statistique 16:47. https://doi.org/10.2307/3315215
Brackstone GJ, Rao JNK (1979) An investigation of raking-ratio estimators. Sankhya. Indian J Stat 41:97–114. https://doi.org/10.1093/ia/13.2.271a
Cochran WG (1952) The \(\chi ^2\) Test of Goodness of Fit. Ann Math Stat 23(3):315–345. https://doi.org/10.1214/aoms/1177729380
Deming WE, Stephan FF (1940) On a least squares adjustment of a sampled frequency table when the expected marginal totals are known. Ann Math Stat 11(4):427–444. https://doi.org/10.1214/aoms/1177731829
Deville JC (2002) La correction de la non-réponse par calage généralisé. Paris. INSEE, Journées de Méthodologie Statistique
Dmitriev YG, Tarasenko PF (1992) The use of a priori information in the statistical processing of experimental data. Russ Phys J 35(9):888–893. https://doi.org/10.1007/BF00560063
Ireland CT, Kullback S (1968) Contingency tables with given marginals. Biometrika 55(1):179–188. https://doi.org/10.1093/biomet/55.1.179
Konijn HS (1981) Biases, variances and covariances of raking ratio estimators for marginal and cell totals and averages of observed characteristics. Metrika 28(1):109–121. https://doi.org/10.1007/BF01902883
Patnaik PB (1949) The Non-Central \(\chi ^2\)-and F-Distribution and their Applications. Tech. Rep. 1
Sinkhorn R (1964) A relationship between arbitrary positive matrices and doubly stochastic matrices. Ann Math Stat 35:876–879
Stephan FF (1942) An iterative method of adjusting sample frequency tables when expected marginal totals are known. Ann Math Stat 13(2):166–178. https://doi.org/10.1214/aoms/1177731604
Tarima S, Pavlov D (2006) Using auxiliary information in statistical function estimation. ESAIM - Probab Stat 10:11–23. https://doi.org/10.1051/ps:2005019
van den Vaart AW (1998) Asymptotic Statistics. Cambridge University Press, Cambridge. https://doi.org/10.1017/cbo9780511802256
Zhang B (1995) M-estimation and quantile estimation in the presence of auxiliary information. J Stat Plan Inference 44(1):77–94. https://doi.org/10.1016/0378-3758(94)00040-3
Zhang B (1997) Estimating a distribution function in the presence of auxiliary information. Metrika 46(3):221–244. https://doi.org/10.1007/bf02717176
Zhang B (1997) Quantile processes in the presence of auxiliary information. Ann Inst Stat Math 49(1):35–55. https://doi.org/10.1023/A:1003158521261
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I would like to sincerely thank the reviewers for their help which was invaluable to me. They gave me leads that I had not thought, for their advice which gave more consistency to this paper, made it easier to read and for the time they took to underline the mistakes.
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Albertus, M. Asymptotic Z and chi-squared tests with auxiliary information. Metrika 85, 859–883 (2022). https://doi.org/10.1007/s00184-021-00853-y
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DOI: https://doi.org/10.1007/s00184-021-00853-y