Tests for normality using estimated score function

The score function, defined as the negative logarithmic derivative of the probability density function, plays an ubiquitous role in statistics. Since the score function of the normal distribution is linear, testing normality amounts to checking the linearity of the empirical score function. Using the score function, we present a graphical alternative to the Q-Q plot for detecting departures from normality. Even though graphical approaches are informative, they lack the objectivity of formal testing procedures. We, therefore, supplement our graphical approach with a formal correlation coefficient test and a large sample chi-square test. The finite sample size and power performances of the chi square test and correlation coefficient test are investigated through a Monte Carlo study.