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

Abstract In certain cutting-edge applications, it is found that a weighted sum of two $$\chi ^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>χ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> -distributions plays an important role. It is well-known that the work on both central and non-central $$\chi ^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>χ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> -distributions is classical and the next obvious step for extension is the weighted sum of two $$\chi ^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>χ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> -distributions. Although there has been considerable theoretical work on the distribution of general linear combinations of $$\chi ^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>χ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> , there have been no dedicated work on either getting deep insight into the distribution even in the particular case of weighted sum of two $$\chi ^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>χ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> -distributions or its applications. We first derive the most general distribution of the weighted sum of two non-central $$\chi ^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>χ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> and give some properties. Particular cases are considered, and one important case arises when one of the $$\chi ^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>χ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> has 2 degrees of freedom, so that it has an exponential distribution. We refer to the resulted weighted sum as the exponentially modified $$\chi ^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>χ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> -distribution. Another important case is when one of the $$\chi ^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>χ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> 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.