“Skewed generalized t distribution”的意思、由来-开放百科全书

In probability and statistics, the skewed generalized “t” distribution is a family of continuous probability distributions. The distribution was first introduced by Panayiotis Theodossiou^[1] in 1998. The distribution has since been used in different applications.^[2]^[3]^[4]^[5]^[6]^[7] There are different parameterizations for the skewed generalized t distribution,^[1]^[5] which we account for in this article.

Definition

Probability density function

where

is the beta function,

is the location parameter,

is the scale parameter,

is the skewness parameter, and

and

are the parameters that control the kurtosis. Note that

and

are not parameters, but functions of the other parameters that are used here to scale or shift the distribution appropriately to match the various parameterizations of this distribution.

These values for

and

yield a distribution with mean of

and a variance of

. In order for

to take on this value however, it must be the case that

. Similarly, for

to equal the above value,

The parameterization that yields the simplest functional form of the probability density function sets

and

. This gives a mean of

The

parameter controls the skewness of the distribution. To see this, let

denote the mode of the distribution, and note that

Since

, the probability left of the mode, and therefore right of the mode as well, can equal any value in (0,1) depending on the value of

. Thus the skewed generalized t distribution can be highly skewed as well as symmetric. If

, then the distribution is negatively skewed. If

, then the distribution is positively skewed. If

, then the distribution is symmetric.

Finally,

and

control the kurtosis of the distribution. As

and

get smaller, the kurtosis increases^[1] (i.e. becomes more leptokurtic). Large values of

and

yield a distribution that is more platykurtic.

Moments

Let

be a random variable distributed with the skewed generalized t distribution.The

moment (i.e.

), for

, is:

Special Cases

Special and limiting cases of the skewed generalized t distribution include the skewed generalized error distribution, the generalized t distribution introduced by McDonald and Newey,^[6] the skewed t proposed by Hansen, the skewed Laplace distribution, the generalized error distribution (also known as the generalized normal distribution), a skewed normal distribution, the student t distribution, the skewed Cauchy distribution, the Laplace distribution, the uniform distribution, the normal distribution, and the Cauchy distribution. The graphic below, adapted from Hansen, McDonald, and Newey,^[2] shows which parameters should be set to obtain some of the different special values of the skewed generalized t distribution.

Skewed generalized error distribution

Generalized t distribution

Skewed t distribution

Skewed Laplace distribution

Generalized error distribution

The Generalized Error Distribution (also known as the generalized normal distribution) has the pdf:

Skewed normal distribution

Student's t-distribution

Skewed Cauchy distribution

Also note that the mean, variance, skewness, and kurtosis of the skewed Cauchy distribution are all undefined.

Laplace distribution

Uniform Distribution

Normal distribution

Cauchy Distribution

References

External links

Notes

1. ^¹²{{cite journal | last1 = Hansen | first1 = C. | last2 = McDonald | first2 = J. | last3 = Newey | first3 = W. | year = 2010 | title = Instrumental Variables Estimation with Flexible Distributions | url = | journal = Journal of Business and Economic Statistics | volume = 28 | issue = | pages = 13–25 | doi=10.1198/jbes.2009.06161}}
2. ^¹Hansen, C., J. McDonald, and P. Theodossiou (2007) "Some Flexible Parametric Models for Partially Adaptive Estimators of Econometric Models" Economics: The Open-Access, Open-Assessment E-Journal
3. ^¹{{cite journal | last1 = McDonald | first1 = J. | last2 = Michelfelder | first2 = R. | last3 = Theodossiou | first3 = P. | year = 2009 | title = Evaluation of Robust Regression Estimation Methods and Intercept Bias: A Capital Asset Pricing Model Application | url = | journal = Multinational Finance Journal | volume = 15 | issue = 3/4| pages = 293–321 | doi = 10.17578/13-3/4-6 }}
4. ^¹²McDonald J., R. Michelfelder, and P. Theodossiou (2010) "Robust Estimation with Flexible Parametric Distributions: Estimation of Utility Stock Betas" Quantitative Finance 375-387.
5. ^¹²{{cite journal | last1 = McDonald | first1 = J. | last2 = Newey | first2 = W. | year = 1998 | title = Partially Adaptive Estimation of Regression Models via the Generalized t Distribution | url = | journal = Econometric Theory | volume = 4 | issue = 3| pages = 428–457 | doi = 10.1017/S0266466600013384 }}
6. ^¹Savva C. and P. Theodossiou (2015) "Skewness and the Relation between Risk and Return" Management Science, forthcoming.
7. ^¹²³⁴{{cite journal | last1 = Theodossiou | first1 = P | year = 1998 | title = Financial Data and the Skewed Generalized T Distribution | url = | journal = Management Science | volume = 44 | issue = 12–part–1| pages = 1650–1661 | doi=10.1287/mnsc.44.12.1650}}