On the Distribution of Weighted Sum of Two Chi-squares with Applications to Shape Analysis

In certain cutting-edge applications, it is found that a weighted sum of two χ2-distributions plays an important role. It is well-known that the work on both central and non-central χ2-distributions is classical and the next obvious step for extension is the weighted sum of two χ2-distributions. Although there has been considerable theoretical work on the distribution of general linear combinations of χ2, there have been no dedicated work on either getting deep insight into the distribution even in the particular case of weighted sum of two χ2-distributions or its applications. We first derive the most general distribution of the weighted sum of two non-central χ2 and give some properties. Particular cases are considered, and one important case arises when one of the χ2 has 2 degrees of freedom, so that it has an exponential distribution. We refer to the resulted weighted sum as the exponentially modified χ2-distribution. Another important case is when one of the χ2 has large degrees of freedom, hence approximates a normal distribution. The resulted weighted sum is known as the exponentially modified Gaussian distribution in the literature. We give further insight into these skew distributions and we also consider some inference problems for these distributions. This work is motivated by new challenges in shape analysis on how to deal with asymmetry of bilateral shapes and we illustrate our methodology by applying it to a shape analysis problem involving a smile data on the cleft lip patients.