Smoothness (probability theory)

In probability theory and statistics, smoothness of a density function is a measure which determines how many times the density function can be differentiated, or equivalently the limiting behavior of distribution’s characteristic function. Formally, we call the distribution of a random variable X ordinary smooth of order β ^[1] if its characteristic function satisfies

[math]\displaystyle{ d_0 |t|^{-\beta} \leq |\varphi_X(t)| \leq d_1 |t|^{-\beta} \quad \text{as } t\to\infty }[/math]

for some positive constants d₀, d₁, β. The examples of such distributions are gamma, exponential, uniform, etc.

The distribution is called supersmooth of order β ^[1] if its characteristic function satisfies

[math]\displaystyle{ d_0 |t|^{\beta_0}\exp\big(-|t|^\beta/\gamma\big) \leq |\varphi_X(t)| \leq d_1 |t|^{\beta_1}\exp\big(-|t|^\beta/\gamma\big) \quad \text{as } t\to\infty }[/math]

for some positive constants d₀, d₁, β, γ and constants β₀, β₁. Such supersmooth distributions have derivatives of all orders. Examples: normal, Cauchy, mixture normal.

References

• Lighthill, M. J. (1962). Introduction to Fourier analysis and generalized functions. London: Cambridge University Press. 

0.00 (0 votes) Original source: https://en.wikipedia.org/wiki/Smoothness (probability theory). Read more