Having a distribution for example and the asymptotic behavior is satisfied i.e. the limit as x tend to 0 and infinity is equal to zero. They conclude that the distribution is a unimodal
If distribution is unimodal, this implies that the frequency (mode) of the variable you are observing peaks at only one point. This would be indicative of a single value being particularly dominant among others. Analyses of unimodal distributions make use of inequalities (valid only for unimodal distributions), which allows for important results regarding the graph.
Dear colleagues,
I would like to add to the previous answers the following condition for a distribution to be unequivocally unimodal:
K>=0.0428*A^4+1.7948*A^2+2.9943 for 0
@george and Samson;
I have indeed read some of that book; i think it can be extended to functions and not just distributions as can symmetry, schur convexity and all of these other concepts that sometimes mean slightly different things.
It can apply to even linear like functions; say strictly increasing functions which are convex, say at some value at 0.5, and concave beyond that pt (rather then necessarily increasing or decreasing from that point). In that cases the function F, of interest, is a cumulative distribution function, whose density function function , f is uni-modal in the traditional sense; at some pt x, c>x, f(x) and for c
here seem to be a lot of similar albeit distinct uses of the terms; uni-modal strongly unimodal, centrally symmetric, monotone unimodal, symmetric, mirror symmetric unimodal; and centrally convex unimodal, linear uni-modal. You might also want to look at these two related books:https://www.elsevier.com/books/unimodality-convexity-and-applications/dharmadhikari/978-0-08-091766-5 and https://statistics.stanford.edu/sites/default/files/OLK%20NSF%20147.pdf as well as M. Kuczma, An introduction to the theory of functional equations and inequalities. Cauchy’s equation and Jensen’s inequality. 2nd ed., Birkh¨auser, 2009 [1st ed. PWN, Warszawa, 1985]>
It means the distribution has only one mode or the shape is inverted bathtub.
The asymptotic behaviour does not tell you anything of the modlity (number of modes) of a distribution, at least not when this is the only information given.
If, for instance, it was given that the distribution is from the gamma family, theny it must be unimodal (for the gamma-distribution when f(x=0)=0 and f(x->inf)=0, fitting to your given asymptotic behaviour). Howere, you can easily imagine a gamma mixture distribution that has the same asymptotic behaviour but is not unimodal. The attached picture shows the density of a mixture of two gammas: 0.4*gamma(5,2) + 0.6*gamma(18,3). The individual (scaled) gamma densities are shown as dottes lines.
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