Lepage test

In statistics, the Lepage test is an exactly distribution-free test (nonparametric test) for jointly monitoring the location (central tendency) and scale (variability) in two-sample treatment versus control comparisons. This is one of the most famous rank tests for the two-sample location-scale problem. The Lepage test statistic is the squared Euclidean distance of standardized Wilcoxon rank-sum test for location and the standardized Ansari–Bradley test for scale. The Lepage test was first introduced by Yves Lepage in 1971 in a paper in Biometrika.[1] A large number of Lepage-type tests exists in statistical literature for simultaneously testing location and scale shifts in case-control studies. The details may be found in the book: Nonparametric statistical tests: A computational approach.[2] Kössler, W.[3] in 2006 also introduced various Lepage type tests using some alternative score functions optimal for various distributions. Dr. Amitava Mukherjee and Dr. Marco Marozzi introduced a class of percentile modified version of the Lepage test.[4] An alternative to the Lepage-type tests is known as the Cucconi test proposed by Odoardo Cucconi in 1968.[5]

Conducting the Lepage test with R, an open-source software

Practitioners can apply the Lepage test using the pLepage function of the contributory package NSM3,[6] built under R software. Andreas Schulz and Markus Neuhäuser also provided detailed R code for computation of test statistic and p-value of the Lepage test[7] for the users.

Application in statistical process monitoring

In recent years, Lepage statistic is widely used statistical process monitoring and quality control. In a classical development, in 2012, Amitava Mukherjee, an Indian statistician, and Subhabrata Chakraborti, an American statistician of Indian origin, introduced a distribution-free Shewhart-type Phase-II monitoring scheme[8] (control chart) for simultaneously monitoring of location and scale parameter of a process using a test sample of fixed size, when a reference sample of sufficiently large size is available from an in-control population. Later in 2015, the same statisticians along with Shovan Chowdhury, proposed a distribution-free CUSUM-type Phase-II monitoring scheme[9] based on the Lepage statistic. In 2017, Mukherjee further designed an EWMA-type distribution-free Phase-II monitoring scheme[10] for joint monitoring of location and scale. In the same year, Mukherjee, with Marco Marozzi, an Italian statistician known for promoting the Cucconi test, came together to design Circular-Grid Lepage chart – a new type of joint monitoring scheme.[11]

Multisample version of the Lepage test

In 2005, František Rublìk introduced the multisample version of the original two-sample Lepage test.[12] This work recently emerge as the motivation behind the proposal of the Phase-I distribution-free Shewhart-type control chart for joint monitoring of location and scale.

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See also

References
  1. Lepage, Yves (April 1971). "A Combination of Wilcoxon's and Ansari-Bradley's Statistics". Biometrika. 58 (1): 213–217. doi:10.2307/2334333. ISSN 0006-3444. JSTOR 2334333.
  2. Neuhäuser, Markus (2011-12-19). Nonparametric Statistical Tests. Chapman and Hall/CRC. doi:10.1201/b11427. ISBN 9781439867037.
  3. Kössler, W. (Wolfgang) (2006). Asymptotic power and efficiency of lepage-type tests for the treatment of combined location-scale alternatives. Humboldt-Universität zu Berlin. doi:10.18452/2462. hdl:18452/3114. OCLC 243600853.
  4. Mukherjee, Amitava; Marozzi, Marco (2019-08-01). "A class of percentile modified Lepage-type tests". Metrika. 82 (6): 657–689. doi:10.1007/s00184-018-0700-1. ISSN 1435-926X.
  5. Cucconi, Odoardo (1968). "Un Nuovo Test non Parametrico per Il Confronto Fra Due Gruppi di Valori Campionari". Giornale Degli Economisti e Annali di Economia. 27 (3/4): 225–248. JSTOR 23241361.
  6. Schneider, Grant; Chicken, Eric; Becvarik, Rachel (2018-05-16), NSM3: Functions and Datasets to Accompany Hollander, Wolfe, and Chicken – Nonparametric Statistical Methods, Third Edition, retrieved 2019-09-17
  7. Schulz, Andreas. "R Programme for Lepage Test" (PDF).
  8. Mukherjee, A.; Chakraborti, S. (2011-09-26). "A Distribution-free Control Chart for the Joint Monitoring of Location and Scale". Quality and Reliability Engineering International. 28 (3): 335–352. doi:10.1002/qre.1249. ISSN 0748-8017.
  9. Chowdhury, Shovan; Mukherjee, Amitava; Chakraborti, Subhabrata (2014-11-07). "Distribution-free Phase II CUSUM Control Chart for Joint Monitoring of Location and Scale" (PDF). Quality and Reliability Engineering International. 31 (1): 135–151. doi:10.1002/qre.1677. hdl:2263/50153. ISSN 0748-8017.
  10. Mukherjee, Amitava (2017-02-18). "Distribution-free phase-II exponentially weighted moving average schemes for joint monitoring of location and scale based on subgroup samples". The International Journal of Advanced Manufacturing Technology. 92 (1–4): 101–116. doi:10.1007/s00170-016-9977-2. ISSN 0268-3768.
  11. Mukherjee, Amitava; Marozzi, Marco (2016-05-17). "Distribution-free Lepage Type Circular-grid Charts for Joint Monitoring of Location and Scale Parameters of a Process". Quality and Reliability Engineering International. 33 (2): 241–274. doi:10.1002/qre.2002. ISSN 0748-8017.
  12. Rublík, František (2005). "The multisample version of the Lepage test". Kybernetika. 41 (6): [713]–733. hdl:10338.dmlcz/135688. ISSN 0023-5954.
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