The Calculation of the Density and Distribution Functions of Strictly Stable Laws

• Main Author: Viacheslav Saenko
• Format: Article
• Language: English
• Published: MDPI AG 2020-05-01
• Series: Mathematics
• Subjects: stable distribution; probability density function; distribution function
• Online Access: https://www.mdpi.com/2227-7390/8/5/775

Integral representations for the probability density and distribution function of a strictly stable law with the characteristic function in the Zolotarev’s “C” parametrization were obtained in the paper. The obtained integral representations express the probability density and distribution function of standard strictly stable laws through a definite integral. Using the methods of numerical integration, the obtained integral representations allow us to calculate the probability density and distribution function of a strictly stable law for a wide range of admissible values of parameters <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>α</mi> <mo>,</mo> <mi>θ</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>. A number of cases were given when numerical algorithms had difficulty in calculating the density. Formulas were given to calculate the density and distribution function with an arbitrary value of the scale parameter <inline-formula> <math display="inline"> <semantics> <mi>λ</mi> </semantics> </math> </inline-formula>.