Skewed generalized t distribution

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]

Definition

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Probability density function

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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.   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.

In the original parameterization[1] of the skewed generalized t distribution,

 

and

 .

These values for   and   yield a distribution with mean of   if   and a variance of   if  . 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

 

and a variance of

 

The   parameter controls the skewness of the distribution. To see this, let   denote the mode of the distribution, and

 

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

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Let   be a random variable distributed with the skewed generalized t distribution. The   moment (i.e.  ), for  , is:  

The mean, for  , is:

 

The variance (i.e.  ), for  , is:

 

The skewness (i.e.  ), for  , is:

 
 

The kurtosis (i.e.  ), for  , is:

 
 
 

Special Cases

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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,[8] 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.

 
The skewed generalized t distribution tree

Skewed generalized error distribution

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The Skewed Generalized Error Distribution (SGED) has the pdf:

 
 

where

 

gives a mean of  . Also

 

gives a variance of  .

Generalized t-distribution

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The generalized t-distribution (GT) has the pdf:

 
 

where

 

gives a variance of  .

Skewed t-distribution

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The skewed t-distribution (ST) has the pdf:

 
 

where

 

gives a mean of  . Also

 

gives a variance of  .

Skewed Laplace distribution

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The skewed Laplace distribution (SLaplace) has the pdf:

 
 

where

 

gives a mean of  . Also

 

gives a variance of  .

Generalized error distribution

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The generalized error distribution (GED, also known as the generalized normal distribution) has the pdf:

 
 

where

 

gives a variance of  .

Skewed normal distribution

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The skewed normal distribution (SNormal) has the pdf:

 
 

where

 

gives a mean of  . Also

 

gives a variance of  .

The distribution should not be confused with the skew normal distribution or another asymmetric version. Indeed, the distribution here is a special case of a bi-Gaussian, whose left and right widths are proportional to   and  .

Student's t-distribution

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The Student's t-distribution (T) has the pdf:

 
 

  was substituted.

Skewed Cauchy distribution

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The skewed cauchy distribution (SCauchy) has the pdf:

 
 

  and   was substituted.

The mean, variance, skewness, and kurtosis of the skewed Cauchy distribution are all undefined.

Laplace distribution

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The Laplace distribution has the pdf:

 
 

  was substituted.

Uniform Distribution

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The uniform distribution has the pdf:

 
 

Thus the standard uniform parameterization is obtained if  ,  , and  .

Normal distribution

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The normal distribution has the pdf:

 
 

where

 

gives a variance of  .

Cauchy Distribution

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The Cauchy distribution has the pdf:

 
 

  was substituted.

References

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  • Hansen, B. (1994). "Autoregressive Conditional Density Estimation". International Economic Review. 35 (3): 705–730. doi:10.2307/2527081. JSTOR 2527081.
  • Hansen, C.; McDonald, J.; Newey, W. (2010). "Instrumental Variables Estimation with Flexible Distributions". Journal of Business and Economic Statistics. 28: 13–25. doi:10.1198/jbes.2009.06161. hdl:10419/79273. S2CID 11370711.
  • Hansen, C.; McDonald, J.; Theodossiou, P. (2007). "Some Flexible Parametric Models for Partially Adaptive Estimators of Econometric Models". Economics: The Open-Access, Open-Assessment e-Journal. 1 (2007–7): 1. doi:10.5018/economics-ejournal.ja.2007-7. hdl:20.500.14279/1024.
  • McDonald, J.; Michefelder, R.; Theodossiou, P. (2009). "Evaluation of Robust Regression Estimation Methods and Intercept Bias: A Capital Asset Pricing Model Application" (PDF). Multinational Finance Journal. 15 (3/4): 293–321. doi:10.17578/13-3/4-6. S2CID 15012865.
  • McDonald, J.; Michelfelder, R.; Theodossiou, P. (2010). "Robust Estimation with Flexible Parametric Distributions: Estimation of Utility Stock Betas". Quantitative Finance. 10 (4): 375–387. doi:10.1080/14697680902814241. S2CID 11130911.
  • McDonald, J.; Newey, W. (1988). "Partially Adaptive Estimation of Regression Models via the Generalized t Distribution". Econometric Theory. 4 (3): 428–457. doi:10.1017/s0266466600013384. S2CID 120305707.
  • Savva, C.; Theodossiou, P. (2015). "Skewness and the Relation between Risk and Return". Management Science.
  • Theodossiou, P. (1998). "Financial Data and the Skewed Generalized T Distribution". Management Science. 44 (12–part–1): 1650–1661. doi:10.1287/mnsc.44.12.1650.
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Notes

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  1. ^ a b c d Theodossiou, P (1998). "Financial Data and the Skewed Generalized T Distribution". Management Science. 44 (12–part–1): 1650–1661. doi:10.1287/mnsc.44.12.1650.
  2. ^ a b Hansen, C.; McDonald, J.; Newey, W. (2010). "Instrumental Variables Estimation with Flexible Distributions". Journal of Business and Economic Statistics. 28: 13–25. doi:10.1198/jbes.2009.06161. hdl:10419/79273. S2CID 11370711.
  3. ^ Hansen, 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
  4. ^ McDonald, J.; Michelfelder, R.; Theodossiou, P. (2009). "Evaluation of Robust Regression Estimation Methods and Intercept Bias: A Capital Asset Pricing Model Application" (PDF). Multinational Finance Journal. 15 (3/4): 293–321. doi:10.17578/13-3/4-6. S2CID 15012865.
  5. ^ a b McDonald J., R. Michelfelder, and P. Theodossiou (2010) "Robust Estimation with Flexible Parametric Distributions: Estimation of Utility Stock Betas" Quantitative Finance 375-387.
  6. ^ a b McDonald, J.; Newey, W. (1998). "Partially Adaptive Estimation of Regression Models via the Generalized t Distribution". Econometric Theory. 4 (3): 428–457. doi:10.1017/S0266466600013384. S2CID 120305707.
  7. ^ Savva C. and P. Theodossiou (2015) "Skewness and the Relation between Risk and Return" Management Science, forthcoming.
  8. ^ Hansen, B (1994). "Autoregressive Conditional Density Estimation". International Economic Review. 35 (3): 705–730. doi:10.2307/2527081. JSTOR 2527081.