Closed-form Tight Bounds and Approximations for the Median of a Gamma Distribution

Abstract

We show how to find upper and lower bounds to the median of a gamma distribution, over the entire range of shape parameter $k > 0$, that are the tightest possible bounds of the form $2^{-1/k} (A + Bk)$, with closed-form parameters $A$ and $B$. The lower bound of this form that is best at high $k$ stays between 48 and 50 percentile, while the uniquely best upper bound stays between 50 and 55 percentile. We show how to form even tighter bounds by interpolating between these bounds, yielding closed-form expressions that more tightly bound the median. Good closed-form approximations between the bounds are also found, including one that is exact at $k = 1$ and stays between 49.97 and 50.03 percentile.

Figures (9)

• Figure 1: Previously published bounds (red; lower bounds solid, upper bounds dashed) for the median of a gamma distribution (black dotted), are good at high $k$ or low $k$, but not both. Their margins (errors) are shown in blue (corresponding line styles). At the left, at $k = 0.01$, the median is near $10^{-30}$, and at $k = 0.001$ near $10^{-300}$.
• Figure 2: The $A$--$B$ parameter space is shaded with dash-dot lines where $\nu(k) 2^{1/k} = A + Bk$, for a set of very small to very large $k$ values in geometric progression (using numerically computed $\nu(k)$ values). Key values of $A$ and $B$ are indicated. Points outside (or on the edge) of the shaded region represent bounds, while points inside the shaded region represent functions that cross the median function. There is an obvious uniquely tight upper bound $\nu_U$, and a curved locus of tight lower bounds from $\nu_{L0}$, which is tightest near $k=0$, to $\nu_{L\infty}$, which is tightest for high $k$. One point (pentagram) on the curved locus represents a lower bound $\nu_{L1}$ that is tight at $k=1$, for which $A + B = 2 \log 2$ (which is the equation of the dashed line). The point $\nu_1$ represents a good asymptotic approximation close to $\nu_{L0}$, but not a bound; see the next figure. The dotted line from $\nu_U$ to $\nu_{L\infty}$ at $B = 1$ intersects the lines for all $k$ in monotonic order, with $A$ decreasing while $k$ increases.
• Figure 3: Zooming in to $\nu_1$ and $\nu_{L0}$, note that the point with $B = e^{-\gamma}\pi^2/12$, which we got from the Taylor series of the power of the gamma function, is actually inside the shaded area, so does not represent a bound; but a point at slightly lower $B = 0.45965$ is on the edge, so represents a lower bound. These points give zero error at approximately $k = 0.1003$ and $k = 0.0708$, respectively (see the next figure). We do not have analytic formulations for these numeric and graphical observations.
• Figure 4: The lower bounds $\nu_{L0}$, $\nu_{L1}$, and $\nu_{L\infty}$ (solid), and upper bound $\nu_U$ (dashed) are shown in red over the ideal median (black heavy dots), with their absolute errors in blue, all premultiplied by $2^{1/k}$ to reduce the required plot range. The approximation $\nu_1$, which is not a bound, is also shown; note that its error curve changes from solid to dashed at the cusp, while the errors for $\nu_{L0}$ and $\nu_{L1}$ have $\log(0)$ cusps where the error grazes zero but does not change sign. The $k$ parameters at these cusps correspond to the sloped lines indicated in the previous figures.
• Figure 5: The percentiles achieved by four new median bounds of the form $2^{-1/k}(A + Bk)$ (solid curves) are plotted, along with the linear bounds $k$ and $k - \frac{1}{3}$ (dotted), upper and lower bounds from Berg and Pedersen berg2006chen (dash-dot), and a pair of closer bounds formed by interpolation between $\nu_U$ and $\nu_{L\infty}$ using a one-parameter rational function (dashed). The bounds $2^{1/k}k$, $\nu_U$, $\nu_{L\infty}$, and the interpolated bounds converge on 50th percentile at both low and high $k$, while the other six do not. Both the upper and lower interpolated bounds are close to $\nu_U$ at low $k$ and close to $\nu_{L\infty}$ at high $k$; tighter such interpolated bounds, developed in a later section, would crowd the center of the graph.