Fitting the Distribution of Linear Combinations of <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <mi>t</mi> <mo>−</mo> </math> Variables with more than 2 Degrees of Freedom

The linear combination of Student’s <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M2"> <a:mi>t</a:mi> </a:math> random variables (RVs) appears in many statistical applications. Unfortunately, the Student’s <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" id="M3"> <c:mi>t</c:mi> </c:math> distribution is not closed under convolution, thus, deriving an exact and general distribution for the linear combination of <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" id="M4"> <e:mi>K</e:mi> </e:math> Student’s <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" id="M5"> <g:mi>t</g:mi> </g:math> RVs is infeasible, which motivates a fitting/approximation approach. Here, we focus on the scenario where the only constraint is that the number of degrees of freedom of each <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" id="M6"> <i:mi>t</i:mi> <i:mo>−</i:mo> </i:math> RV is greater than two. Notice that since the odd moments/cumulants of the Student’s <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" id="M7"> <k:mi>t</k:mi> </k:math> distribution are zero and the even moments/cumulants do not exist when their order is greater than the number of degrees of freedom, it becomes impossible to use conventional approaches based on moments/cumulants of order one or higher than two. To circumvent this issue, herein we propose fitting such a distribution to that of a scaled Student’s <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" id="M8"> <m:mi>t</m:mi> </m:math> RV by exploiting the second moment together with either the first absolute moment or the characteristic function (CF). For the fitting based on the absolute moment, we depart from the case of the linear combination of <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" id="M9"> <o:mi>K</o:mi> <o:mo>=</o:mo> <o:mn>2</o:mn> </o:math> Student’s <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" id="M10"> <q:mi>t</q:mi> </q:math> RVs and then generalize to <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" id="M11"> <s:mi>K</s:mi> <s:mo>≥</s:mo> <s:mn>2</s:mn> </s:math> through a simple iterative procedure. Meanwhile, the CF-based fitting is direct, but its accuracy (measured in terms of the Bhattacharyya distance metric) depends on the CF parameter configuration, for which we propose a simple but accurate approach. We numerically show that the CF-based fitting usually outperforms the absolute moment-based fitting and that both the scale and number of degrees of freedom of the fitting distribution increase almost linearly with <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" id="M12"> <u:mi>K</u:mi> </u:math> .