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The Asymmetric Laplace distribution is commonly used with an alternative parameterization for performing quantile regression in a Bayesian inference context. [4] Under this approach, the κ {\displaystyle \kappa } parameter describing asymmetry is replaced with a p {\displaystyle p} parameter indicating the percentile or quantile desired.
The asymmetric generalized normal distribution is a family of continuous probability distributions in which the shape parameter can be used to introduce asymmetry or skewness. [ 15 ] [ 16 ] When the shape parameter is zero, the normal distribution results.
The non-central t-distribution is asymmetric unless μ is zero, i.e., a central t-distribution. In addition, the asymmetry becomes smaller the larger degree of freedom. The right tail will be heavier than the left when μ > 0, and vice versa.
The exponentially modified normal distribution is another 3-parameter distribution that is a generalization of the normal distribution to skewed cases. The skew normal still has a normal-like tail in the direction of the skew, with a shorter tail in the other direction; that is, its density is asymptotically proportional to for some positive .
In probability theory and directional statistics, a wrapped asymmetric Laplace distribution is a wrapped probability distribution that results from the "wrapping" of the asymmetric Laplace distribution around the unit circle. For the symmetric case (asymmetry parameter κ = 1), the distribution becomes a wrapped Laplace distribution.
Thus, a d-variate distribution is defined to be mirror symmetric when its chiral index is null. The distribution can be discrete or continuous, and the existence of a density is not required, but the inertia must be finite and non null. In the univariate case, this index was proposed as a non parametric test of symmetry. [2]
Example distribution with positive skewness. These data are from experiments on wheat grass growth. In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean.
A subset A of positive integers has natural density α if the proportion of elements of A among all natural numbers from 1 to n converges to α as n tends to infinity.. More explicitly, if one defines for any natural number n the counting function a(n) as the number of elements of A less than or equal to n, then the natural density of A being α exactly means that [1]