A simple derivation on the chi-square approximation of Pearson's statistic

Abstract

One of the most popular test statistics is Pearson's c2 goodness-of-fit statistic which is known to have an approximate chi-square distribution for large sample size. All the proofs so far seem to make use of the chisquaredness of quadratic forms in normal variables, manipulations of quadratic forms and singular covariance matrix. An elegant simple derivation of this result is given here without making use of chisquaredness of quadratic forms or other such results. Two numerical examples are also given to caution about the use of this statistic.