A Half-Integer Inequality Between the Negative Binomial and Gamma Laws, With Binomial Tail Bounds

For a binomial count with $n$ trials and success probability $p$, the signed log-likelihood ratio at $x$ is the square root of twice the log-likelihood ratio of $x/n$ against $p$, with the sign of $x/n-p$. Zubkov and Serov proved that, for $0<j<n-1$, the binomial distribution function at $j$ lies between the standard Gaussian distribution function at the signed log-likelihood ratios of $j$ and $j+1$. If the distribution function of the Gamma law with shape $k$ and scale $\theta$ at $x$ equals the probability of a negative binomial count with these parameters below an integer $m$, Harremoës proved that the Gamma signed log-likelihood ratio at $x$ lies between the negative binomial ones at $m-1$ and $m$, and conjectured that $m-1$ can be raised to $m-1/2$. We prove this conjecture for all $k,\theta>0$. The ratio of the derivatives of the two distribution functions in $q=1/(1+\theta)$ is not monotone at $m-1/2$, unlike at $m-1$, and we show that it starts below one and first falls and then rises. Our theorem bounds the binomial distribution function at $j$ by the standard Gaussian distribution function at the signed log-likelihood ratios of $j+1/2$ and $j+1$, with $n+1/2$ in place of $n$, through a tail identity and a known Gaussian bound for the Gamma law. For $0<j<n-1$, each bound is strictly tighter than the corresponding Zubkov--Serov bound. On a numerical grid, the three-term expressions of Alfers and Dinges, which are bounds only under an unproved convexity condition, are tighter in the ranges for which they recommend them.