  This paper proposes a new deep-learning method to construct test statistics
by computer vision and metrics learning. The application highlighted in this
paper is applying computer vision on Q-Q plot to construct a new test statistic
for normality test. To the best of our knowledge, there is no similar
application documented in the literature. Traditionally, there are two families
of approaches for verifying the probability distribution of a random variable.
Researchers either subjectively assess the Q-Q plot or objectively use a
mathematical formula, such as Kolmogorov-Smirnov test, to formally conduct a
normality test. Graphical assessment by human beings is not rigorous whereas
normality test statistics may not be accurate enough when the uniformly most
powerful test does not exist. It may take tens of years for statistician to
develop a new test statistic that is more powerful statistically. Our proposed
method integrates four components based on deep learning: an image
representation learning component of a Q-Q plot, a dimension reduction
component, a metrics learning component that best quantifies the differences
between two Q-Q plots for normality test, and a new normality hypothesis
testing process. Our experimentation results show that the
machine-learning-based test statistics can outperform several widely-used
traditional normality tests. This study provides convincing evidence that the
proposed method could objectively create a powerful test statistic based on Q-Q
plots and this method could be modified to construct many more powerful test
statistics for other applications in the future.
