# The Mean, Median, and Mode of Unimodal Distributions:A Characterization

@article{Basu1997TheMM,
title={The Mean, Median, and Mode of Unimodal Distributions:A Characterization},
author={Sunanda Basu and A. Dasgupta},
journal={Theory of Probability and Its Applications},
year={1997},
volume={41},
pages={210-223}
}
• Published 1997
• Mathematics
• Theory of Probability and Its Applications
For a unimodal distribution on the real line, the celebrated mean-median-mode inequality states that they often occur in an alphabetical (or its reverse) order. Various sufficient conditions for th...
51 Citations

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## References

SHOWING 1-10 OF 13 REFERENCES

### The Mode, Median, and Mean Inequality

• Mathematics
• 1977
Abstract An elementary method of proof of the mode, median, and mean inequality is given for skewed, unimodal distributions of continuous random variables. A proof of the inequality for the gamma, F,

### The Mean, Median, Mode Inequality and Skewness for a Class of Densities

Summary Recent work on the mean, median, mode inequality and skewness for unimodal continuous distributions is discussed with reference to the properties of totally positive kernels. These

### Unimodality, convexity, and applications

• Mathematics
• 1989
Properties of Univariate Unimodal Distributions. Concepts of Multivariate Unimodality. Some More Notions of Unimodality. Unimodality for Discrete Distributions. Unimodality of Infinitely Divisible

### Mean, median, mode

Summary This note is an attempt to avoid doing the same search for the third time. It happened twice in my life that I wished to prove that the median is located between mean and mode for certain

### A generalized unimodality

• Mathematics
Journal of Applied Probability
• 1970
A definition -- more exactly, a one parameter family of definitions -- of unimodality for random objects taking values in a finite dimensional vector space and an extension of Khintchine's theorem to alpha-unimodality are given.

### A New General Method for Constructing Confidence Sets in Arbitrary Dimensions: With Applications

• Mathematics
• 1995
Let $\mathbf{X}$ have a star unimodal distribution $P_0$ on $\mathbb{R}^p$. We describe a general method for constructing a star-shaped set $S$ with the property \$P_0(\mathbf{X} \in S) \geq 1 -

### Robust Bayesian Experimental Designs in Normal Linear Models

• Mathematics
• 1991
We address the problem of finding a design that minimizes the Bayes risk with respect to a fixed prior subject to being robust with respect to misspecification of the prior. Uncertainty in the prior