Related papers: Convexity of the median in the gamma distribution
This note contains sufficient conditions for the probability density function of an arbitrary continuous univariate distribution, supported on $(0,\infty),$ such that the corresponding Mills ratio to be reciprocally convex (concave). To…
The logarithmic convexity of restrictions of the Beta functions to rays parallel to the main diagonal and the functional equation \[ \phi\left( x+1\right) =\frac{x\left( x+k\right) }{\left( 2x+k+1\right) \left( 2x+k\right) }\phi\left(…
We study the median in the Gamma distribution as a function of the parameter. Using methods from complex analysis we show that the derivative of the median is less than 1 and we thereby solve a conjecture of Chen and Rubin from 1986. We…
In this paper, we obtain some new inequalities for ({\alpha},m)-convex functions. The analysis used in the proofs is fairly elementary and based on the use of Power-mean inequality.
The gamma density function is usually defined in interval between zero and infinity. This paper introduces an upper and a lower boundary to this distribution. The parameters which characterize the truncated gamma distribution are evaluated.…
The gamma difference distribution is defined as the difference of two gamma distributions, with in general different shape and rate parameters. Starting with knowledge of the corresponding characteristic function, a second order linear…
The full width at half maximum (FWHM) is a useful quantity for characterizing the bandwidth of unimodal functions. However, a closed-form expression for the FWHM of gamma-shaped functions-i.e. functions that are shaped like the gamma…
Polynomial sequence ${P_m}_{m\geq0}$ is $q$-logarithmically concave if $P_{m}^2-P_{m+1}P_{m-1}$ is a polynomial with nonnegative coefficients for any $m\geq{1}$. We introduce an analogue of this notion for formal power series whose…
This work includes a new characterization of the multivariate normal distribution. In particular, it is shown that a positive density function $f$ is Gaussian if and only if the $f(x+ y)/f(x)$ is convex in $x$ for every $y$. This result has…
The beta distribution is a two-parameter family of probability distributions whose distribution function is the (regularised) incomplete beta function. In this paper, the inverse incomplete beta function is studied analytically as…
The generalized Marcum functions $Q_{\mu}(x,y)$ and $P_{\mu}(x,y)$ have as particular cases the non-central $\chi^2$ and gamma cumulative distributions, which become central distributions (incomplete gamma function ratios) when the…
Several representations of the exact cdf of the sum of squares of n independent gamma-distributed random variables Xi are given, in particular by a series of gamma distribution functions. Using a characterization of the gamma distribution…
We give a statement on extension with estimates of convex functions defined on a linear subspace, inspired by similar extension results concerning metrics on positive line bundles
In this paper, we present a very accurate approximation for gamma function: \begin{equation*} \Gamma \left( x+1\right) \thicksim \sqrt{2\pi x}\left( \dfrac{x}{e}\right) ^{x}\left( x\sinh \frac{1}{x}\right) ^{x/2}\exp \left(…
This paper is concerned with the study of Green's functions for one dimensional diffusions with constant diffusion coefficient and linear time inhomogeneous drift. It is well know that the whole line Green's function is given by a Gaussian.…
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…
The gamma distribution arises frequently in Bayesian models, but there is not an easy-to-use conjugate prior for the shape parameter of a gamma. This inconvenience is usually dealt with by using either Metropolis-Hastings moves, rejection…
In this paper, we study some properties such as the monotonicity, logarithmically complete monotonicity, logarithmic convexity, and geometric convexity, of the combinations of gamma function and power function. The results we obtain…
Typically, when we are given the section (or projection) function of a convex body, it means that in each direction we know the size of the central section (or projection) perpendicular to this direction. Suppose now that we can only get…
We study the geometry of $m$-regular domains within the Caffarelli-Nirenberg-Spruck model in terms of barrier functions, envelopes, exhaustion functions, and Jensen measures. We prove among other things that every $m$-hyperconvex domain…