On bounds for the mode and median of the generalized hyperbolic and related distributions

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

“…That is, for k > 3.021, it's a looser lower bound and a worse approximation than k − 1/3, even though it is the tightest lower bound of the form we're considering. On the other hand, ν U (k) is a much tighter upper bound than k is, for all k, with an asymptotic margin near 1 3 À ðlog 2 À e À g Þ � 0:202; for k � 1, the recent straight-line bound k − 1 + log 2 [4] is tighter still, with an asymptotic margin 1 3 À ð1 À log 2Þ � 0:026.…”

“… Comparison of several median bounds and asymptotes. B&P refers to Berg and Pedersen [ 3 ], and G&M refers to Gaunt and Merkle [ 4 ]. Conjectured bounds that have not been proved, but are supported by asymptotic and numerical results, are marked with an asterisk(*).…”

“…The Chen and Rubin bounds are tight [ 1 ]; the upper bound in the low- k limit and the lower bound in the high- k limit. Recently, Gaunt and Merkle [ 4 ] proved that the line of slope 1 that intersects ν ( k ) at the known value ν (1) = log 2 is an upper bound for k ≥ 1; that is, ν ( k )< k − 1 + log 2, much tighter than the ν ( k )< k bound, leveraging the prior result at integers by Choi [ 2 ] and the result by Berg and Pedersen that the slope of the median ν ′( k ) is everywhere less than 1 [ 3 ]. As shown in Fig 1 , this new upper bound can be combined with a chord for 0 ≤ k ≤ 1, based on convexity shown by Berg and Pedersen [ 5 ].…”

“… The bounds and ν ( k )>0 (solid lines) are shown along with the true value (solid curve), the piecewise-linear bound that combines the recent linear bound for k > 1 [ 4 ] with a chord segment (dashed lines), and the linear lower bound that is tangent at k = 1 (dash-dot line). The region k < 1 is not very usefully bounded.…”

Stein factors for variance-gamma approximation in the Wasserstein and Kolmogorov distances

“…The proof of inequality (2.7) in Proposition 2.1 makes use of the two-sided inequality (2.6). An alternative lower bound for the mode of the VG(r, θ, σ, µ) distribution is given in [28,Corollary 2.6], which improves on the lower bound of (2.6) for r > 4.…”