The Chi-Squared Test and Modifications: From 1900 to 2010

• Activity Number: 580
• Type: Contributed
• Date/Time: Wednesday, August 4, 2010 : 2:00 PM to 3:50 PM
• Sponsor: Section on Statistical Education
• Author(s): Vassilly Voinov*+
• Companies: KIMEP
• Keywords: Pearson-Fisher test ; Nikulin's test ; Decompositions of chi-squared tests

Abstract

The chi-squared test was discovered by Karl Pearson in 1900. Pearson believed that the limit distribution of that test will unchange if parameters would be replaced by estimates. In 1924 Fisher showed that the number of degrees of freedom of Pearson's test with parameters replaced by estimates based on grouped data must be reduced by the number of parameters. In 1954 Chernoff and Lehman showed that using raw data for estimating parameters will change the limit distribution of Pearson's sum making impossible to use it for testing. Erroneous using of Pearson's test under such settings is reproduced up to date in many textbooks. In 1973 Nikulin modified Pearson's test such that using estimates based on raw data will lead to a distribution free test. New modifications based on decompositions of chi-squared quadratic forms and vector-valued goodness-of-fit tests are presented.