Dimensionless Equations, Asymptotic Analyses, and Closed-Form Relations for Parameter Identification and Optimization

2011 ◽  
pp. 131-150
Keyword(s):  
Parameter Identification ◽  
Closed Form ◽  
Asymptotic Analyses ◽  
Dimensionless Equations

scholarly journals On closed-form tight bounds and approximations for the median of a gamma distribution

PLoS ONE ◽  
2021 ◽  
Vol 16 (5) ◽  
pp. e0251626
Author(s):  
Richard F. Lyon
Keyword(s):  
Lower Bound ◽  
Closed Form ◽  
Lower Bounds ◽  
Gamma Distribution ◽  
Upper Bound ◽  
Entire Range ◽  
Shape Parameter ◽  
Upper And Lower Bounds ◽  
Elementary Functions ◽  
Asymptotic Analyses

The median of a gamma distribution, as a function of its shape parameter k, has no known representation in terms of elementary functions. In this work we use numerical simulations and asymptotic analyses to bound the median, finding bounds of the form 2−1/k(A + Bk), including an upper bound that is tight for low k and a lower bound that is tight for high k. These bounds have closed-form expressions for the constant parameters A and B, and are valid over the entire range of k > 0, staying between 48 and 55 percentile. Furthermore, an interpolation between these bounds yields closed-form expressions that more tightly bound the median, with absolute and relative margins to both upper and lower bounds approaching zero at both low and high values of k. These bound results are not supported with analytical proofs, and hence should be regarded as conjectures. Simple approximation expressions between the bounds are also found, including one in closed form that is exact at k = 1 and stays between 49.97 and 50.03 percentile.

Sign in / Sign up

Export Citation Format

Share Document