The Chen-Rubin Conjecture in a Continuous Setting

@article{Berg2004TheCC,
  title={The Chen-Rubin Conjecture in a Continuous Setting},
  author={Henrik L. Pedersen Christian Berg},
  journal={Methods and applications of analysis},
  year={2004},
  volume={13},
  pages={63-88}
}
We study the median m(x) in the gamma distribution with parameter x and show that 0 0. This proves a generalization of a conjecture of Chen and Rubin from 1986: The sequence m(n) n decreases for n � 1. We also describe the asymptotic behaviour of m and m 0 at zero and at infinity. 

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References

SHOWING 1-10 OF 26 REFERENCES

On the medians of gamma distributions and an equation of Ramanujan

For n > 0, let k(n) denote the median of the T(n +1,1) distribution. We prove that n + \ j. We show that the bounds on X(n) imply s log2 < median(Z^) < p. + ±. This proves a conjecture of Chen and

Sharp estimates for the median of the Γ(n+1,1) distribution☆

Proof of the Chen–Rubin conjecture

  • H. Alzer
  • Mathematics
    Proceedings of the Royal Society of Edinburgh: Section A Mathematics
  • 2005
Let n ≥ 0 be an integer and let λ(n) be the median of the Gamma distribution of order n + 1 with parameter 1. In 1986, Chen and Rubin conjectured that n ↦ λ (n) − n (n = 0, 1, 2, …) is decreasing. We

C249. The incomplete gamma function and ramanujan’s rational approximation to e x

(1986). C249. The incomplete gamma function and ramanujan’s rational approximation to e x. Journal of Statistical Computation and Simulation: Vol. 24, No. 2, pp. 163-168.

Monotonicity of the difference between median and mean of gamma distributions and of a related Ramanujan sequence

For n 0, let n be the median of the ( n + 1,1) distribution. We prove that the sequence { n = n n} decreases from log2 to 2/3 as n increases from 0 to 1. The difference, 1 n, between the mean and the

The Logarithmic Integral

Preface Introduction 1. Jensen's formula 2. Szego's theorem 3. Entire functions of exponential type 4. Quasianalyticity 5. The moment problem on the real line 6. Weighted approximation on the real

Bounds for the difference between median and mean of gamma and poisson distributions

Infinite Divisibility of Probability Distributions on the Real Line

infinitely divisible distributions on the nonnegative integers infinitely divisible distributions on the nonnegative reals infinitely divisible distributions on the real line self-decomposability and

On Ramanujan's Q-function

Monotone Matrix Functions and Analytic Continuation

I. Preliminaries.- II. Pick Functions.- III. Pick Matrices and Loewner Determinants.- IV. Fatou Theorems.- V. The Spectral Theorem.- VI. One-Dimensional Perturbations.- VII. Monotone Matrix