The Skewed Generalized T Distribution Tree Package Vignette

Transcription

1 The Skewed Generalized T Distribution Tree Package Vignette Carter Davis Setember, 05 Summary The Skewed Generalized T Distribution is a univariate 5-arameter distribution introduced by Theodossiou 998) and known for its extreme flexibility. Secial and limiting cases of the SGT distribution include the skewed generalized error distribution, the generalized t distribution introduced by McDonald and Newey 988), the skewed t roosed by Hansen 994), the skewed Lalace distribution, the generalized error distribution also known as the generalized normal distribution), the skewed normal distribution, the student t distribution, the skewed Cauchy distribution, the Lalace distribution, the uniform distribution, the normal distribution, and the Cauchy distribution. This vignette describe the skewed generalized t distribution and its various secial cases. Booth School of Business, University of Chicago

2 Contents A. Introduction B. Skewed Generalized T Distribution C. Skewed Generalized Error Distribution D. Generalized T Distribution E. Skewed T Distribution F. Skewed Lalace Distribution G. Generalized Error Distribution H. Skewed Normal Distribution I. Student T Distribution J. Skewed Cauchy Distribution K. Lalace Distribution L. Uniform Distribution M.Normal Distribution N. Cauchy Distribution

3 A. Introduction The skewed generalized t distribution SGT) is a highly flexible distribution with many secial cases. The SGT distribution has five arameters: µ, σ, λ,, and q. The grahic below shows the secial cases of the SGT distribution and which arameters must be set to obtain these distributions. This grahic was adated from Hansen, McDonald, and Newey 00). Note that the SGT arameters have the following restrictions: {σ,λ,,q) : σ > 0, < λ <, > 0,q > 0} ) It is imortant to note that if any of the arameters are not within these bounds in any of the [dqr]sgt functions, then a warning message is issued and NaNs roduced. Note that and q are allowed to be Inf. The section that describes the SGT distribution gives the closed form definition of the moments and shows how the arameters influence the moments. We show below though how the arameters influence the moments of the distribution. There are two otions that are imortant to note in the [dqr]sgt functions: mean.cent and var.adj. The mean.cent otion is either TRUE or FALSE. If mean.cent is TRUE, then µ is the mean of the distribution. If mean.cent is FALSE, then µ is the mode of the dis-

4 Figure : Visualizing the Flexibility of the Skewed Generalized T Distribution Panel A: µ Equals the Mean Panel B: σ Controls the Variance Panel C: λ Controls the Skewness Panel D: and q Control the Kurtosis Panel E: and q Control the Kurtosis Exlanation: This figure shows the flexibility of the SGT distribution. The black curve in each grah has arameter values: µ = 0, σ =, λ = 0, =, and q = 00, with both mean.cent and var.adj are TRUE. This aroximates a standard normal df very closely. All other curves change just one arameter. 3

5 tribution. In the resence of skewness of course, the mean equals the mode and the mean.cent otion makes no difference. Imortantly, mean.cent can only be TRUE if q >. If q and mean.cent is TRUE, then a warning will be given and NaNs roduced. The var.adj is either TRUE, FALSE, or a ositive scalar numeric of length one). If var.adj is TRUE, then the σ arameter is scaled such that σ is the variance of the distribution. If var.adj is FALSE, then if q >, the variance, for q >, is simly roortional to σ, holding all other arameters fixed. If var.adj is a ositive scalar then, then σ is scaled by the value of var.adj. The SGT section below shows how this is done. Imortantly, var.adj can only be TRUE if q >. If var.adj is TRUE and q then a warning will be given and NaNs roduced. If var.adj is a nonositive scalar, then a warning is issued and var.adj is assumed FALSE. It is imortant to note that the h th moment of the SGT distribution is only defined if q > h. Thus the h th moments reorted in this vignette only hold true if q > h. The λ arameter controls the skewness of the distribution. To see this, let M denote the mode of the distribution, and note that M f SGT x;µ,σ,λ,,q)dx = λ ) Since < λ <, the robability left of the mode, and therefore right of the mode as well, can equal any value in 0,) deending on the value of λ. Thus the SGT distribution can be highly skewed as well as symmetric. If < λ < 0, then the distribution is negatively skewed. If 0 < λ <, then the distribution is ositively skewed. If λ = 0, then the distribution is symmetric. Finally, and q control the kurtosis of the distribution. As and q get smaller, the kurtosis increases i.e. becomes more letokurtic). Large values of and q yield a distribution that is more latykurtic. The remainder of this vignette outlines the roerties of the SGT distribution and its secial cases. At the to of each section, a line of R code is given to show how to obtain that secial case of the SGT distribution. For examle, the normal distribution 4

6 section gives the line of code: > dsgtx, mu, sigma, lambda = 0, =, q = Inf, mean.cent, var.adj) which shows that the normal distribution has the arameters µ and σ free, while the other aramers should be equal to their given values to obtain the normal distribution. 5

7 B. Skewed Generalized T Distribution > dsgtx, mu, sigma, lambda,, q, mean.cent, var.adj) The Skewed Generalized T Distribution has the df: f SGT x;µ,σ,λ,,q) = vσq / B,q ) x µ+m ) + +q qvσ) λsignx µ+m)+) where m = ) vσλq B,q ) B,q if mean.cent = TRUE and 0 otherwise. Note that mean.cent = TRUE and q is an error and NaNs will be roduced. Also ) v = q 3λ +) B 3,q ) B,q 4λ B,q ) B,q ) if var.adj = TRUE, v = if var.adj = FALSE, and v is the value of var.adj if var.adj is a ositive scalar. Note that var.adj = TRUE and q is an error and NaNs will be roduced. h r=0 The h th moment i.e. E[X EX)) h ]), for q > h, is: ) ) ) h +λ) r+ + ) r λ) r+) λ) h r vσ) h q h B r+,q r B,q r r h+ ) h r+ B,q ) h r The mean, for q >, is: ) vσλq B,q µ+ ) m B,q thus if mean.cent = TRUE, the mean, for q >, is simly µ. 6

8 The variance, for q >, is: ) vσ) q 3λ +) B 3,q ) B,q 4λ B,q ) B,q ) thus if var.adj = TRUE, the variance, for q >, is simly σ. The skewness, for q > 3, is: B q 3/ λvσ) 3 ) 3 8λ B B,q,q ) λ ) ) B,q,q ) 3 B,q ) + +λ ) ) 4 B,q B,q 3 ) ) The kurtosis, for q > 4, is: q 4/ vσ) 4 ) 4 48λ 4 B B,q B,q ) 4 +4λ +3λ ) ) B,q B,q ) 3,q ) 3λ +λ ) ) B,q B,q ) 4 B,q 3 ) + +0λ +5λ 4) B ) 3 5,q B,q 4 ) ) 7

9 C. Skewed Generalized Error Distribution > dsgtx, mu, sigma, lambda,, q = Inf, mean.cent, var.adj) The Skewed Generalized Error Distribution has the df: lim f SGTx;µ,σ,λ,,q) q = f SGED x;µ,σ,λ,) = e x µ+m vσ+λsignx µ+m))) vσγ/) where m = vσλγ π + ) if mean.cent = TRUE and 0 otherwise. Also π+3λ )Γ v = 3 ) 6 λ Γ ) πγ + ) Γ ) if var.adj = TRUE, v = if var.adj = FALSE, and v is the value of var.adj if var.adj is a ositive scalar. The h th moment i.e. E[X EX)) h ]) is: h r=0 ) h +λ) r+ + ) r λ) r+) ) ) λ) h r vσ) h Γ h rγ r+ r r h+ ) h+ r Γ ) The mean is: vσλγ µ+ π + ) m thus if mean.cent = TRUE, the mean is simly µ. 8

10 The variance is: vσ) π+3λ )Γ ) 6 λ Γ ) πγ 3 + ) Γ ) ) thus if var.adj = TRUE, the variance is simly σ. The skewness is: λσ 3 π 3/ Γ ) 6+ λ Γ + ) 3 Γ ) 34) ) π +3λ Γ + ) Γ +4π 3/ +λ ) Γ ) ) 4 ) 3 The kurtosis is: σ 4 π Γ ) 356) λ 4 Γ + ) 4 Γ 4+ π 3/ λ +λ ) Γ + ) Γ ) +3) 4+ πλ +3λ ) Γ + ) Γ ) 4 +π +0λ +5λ 4) Γ ) ) 5 ) 3 9

11 D. Generalized T Distribution > dsgtx, mu, sigma, lambda = 0,, q, mean.cent, var.adj) The Generalized T Distribution has the df: f SGT x;µ,σ,λ = 0,,q) = f GT x;µ,σ,,q) = ) ) vσq / B,q x µ + +q qvσ) where ) v = B,q ) q / 3 B,q if var.adj = TRUE, v = if var.adj = FALSE, and v is the value of var.adj if var.adj is a ositive scalar. Note that var.adj = TRUE and q is an error and NaNs will be roduced. The h th moment i.e. E[X EX)) h ]), for q > h, is: / ) + ) ) h) vσ) h q h B h+,q h ) B,q The mean, for q >, is µ. The variance, for q >, is: ) vσ) q / B 3,q ) B,q thus if var.adj = TRUE, the variance, for q >, is simly σ. The skewness, for q > 3, is 0. 0

12 The kurtosis, for q > 4, is: ) vσ) 4 q 4/ B 5,q 4 ) B,q

13 E. Skewed T Distribution > dsgtx, mu, sigma, lambda, =, q, mean.cent, var.adj) The Skewed T Distribution has the df: f SGT x;µ,σ,λ, =,q) = f ST x;µ,σ,λ,q) = vσπq) / Γq) Γ +q) x µ+m + qvσ) λsignx µ+m)+) ) +q where m = vσλq/ Γ ) q π / Γ ) q + if mean.cent = TRUE and 0 otherwise. Note that mean.cent = TRUE and q / is an error and NaNs will be roduced. Also v = q / 3λ +) ) 4λ q π Γ q Γq) )) / if var.adj = TRUE, v = if var.adj = FALSE, and v is the value of var.adj if var.adj is a ositive scalar. Note that var.adj = TRUE and q is an error and NaNs will be roduced. h r=0 The h th moment i.e. E[X EX)) h ]), for q > h/, is: ) ) h +λ) r+ + ) r λ) r+) λ) h r vσ) h q h B r+,q r r r h+ The mean, for q > /, is: µ+ vσλq/ Γ q π / Γ ) m q + ) ) B,q B,q) h r+ ) h r ) thus if mean.cent = TRUE, the mean, for q > /, is simly µ.

14 The variance, for q >, is: vσ) q 3λ +) ) 4λ q π Γ q Γq) )) thus if var.adj = TRUE, the variance, for q >, is simly σ. The skewness, for q > 3/, is: B q 3/ λvσ) 3 B,q) 3 8λ B,q ) λ ) ) B,q,q ) ) 3 B,q + +λ ) ) B B,q,q 3 ) ) The kurtosis, for q >, is: q vσ) 4 B,q) 4 B 48λ 4 B,q ) 4 +4λ +3λ ) ) B,q B,q ) ) 3,q 3λ +λ ) ) B B,q,q ) B,q 3 ) + +0λ +5λ 4) B ) 3 ) ) 5,q B,q 3

15 F. Skewed Lalace Distribution > dsgtx, mu, sigma, lambda, =, q = Inf, mean.cent, var.adj) The Skewed Lalace Distribution has the df: lim f SGTx;µ,σ,λ, =,q) q where = f SLalace x;µ,σ,λ) = e x µ+m vσ+λsignx µ+m)) vσ m = vσλ if mean.cent = TRUE and 0 otherwise. Also v = [ +λ )] if var.adj = TRUE, v = if var.adj = FALSE, and v is the value of var.adj if var.adj is a ositive scalar. The h th moment i.e. E[X EX)) h ]) is: h r=0 The mean is: ) h +λ) r+ + ) r λ) r+) ) λ) h r vσ) h r r h+ Γr +) µ+vσλ m thus if mean.cent = TRUE, the mean is simly µ. The variance is: vσ) +λ ) 4

16 thus if var.adj = TRUE, the variance is simly σ. The skewness is: 4vσ) 3 λ3+λ ) The kurtosis is: 4vσ) 4 +4λ +λ 4 ) 5

17 G. Generalized Error Distribution > dsgtx, mu, sigma, lambda = 0,, q = Inf, mean.cent, var.adj) The Generalized Error Distribution has the df: lim f SGTx;µ,σ,λ = 0,,q) q where = f GED x;µ,σ,) = e x µ vσ ) v = Γ Γ 3 ) ) vσγ/) / if var.adj = TRUE, v = if var.adj = FALSE, and v is the value of var.adj if var.adj is a ositive scalar. The h th moment i.e. E[X EX)) h ]) is: + ) h) vσ) h ) ) Γ h+ ) Γ The mean is µ. The variance is: vσ) Γ Γ 3 ) ) thus if var.adj = TRUE, the variance is simly σ. The skewness is 0. The kurtosis is: vσ) 4 Γ Γ 5 ) ) 6

18 H. Skewed Normal Distribution > dsgtx, mu, sigma, lambda, =, q = Inf, mean.cent, var.adj) The Skewed Normal Distribution has the df: lim f SGTx;µ,σ,λ, =,q) q where = f SNormal x;µ,σ,λ) = e x µ+m vσ+λsignx µ+m))) vσ π m = vσλ π if mean.cent = TRUE and 0 otherwise. Also [ π 8λ +3πλ ) v = π ] if var.adj = TRUE, v = if var.adj = FALSE, and v is the value of var.adj if var.adj is a ositive scalar. The h th moment i.e. E[X EX)) h ]) is: h r=0 The mean is: ) h +λ) r+ + ) r λ) r+) ) λ) h r vσ) h Γ ) ) r+ r r h+ π) h+ r µ+ vσλ π m thus if mean.cent = TRUE, the mean is simly µ. The variance is: vσ) π 8λ +3πλ ) π 7

19 thus if var.adj = TRUE, the variance is simly σ. The skewness is: vσ) 3 λ6λ π +5λ )) π 3/ The kurtosis is: vσ) 4 9λ 4 +6πλ 5+λ )+3π +0λ +5λ 4 )) 4π 8

20 I. Student T Distribution We resent two arameterizations of the Student T Distribution. The first arameterization has the arameters µ, σ, and q free. The second is consistent with the tyical student t distribution arameterization, where µ and σ are fixed at 0 and resectively, and q = d/, where d is the degrees of freedom and the only free arameter. First Parameterization: > dsgtx, mu, sigma, lambda = 0, =, q, mean.cent, var.adj) The Student T Distribution has the df: f SGT x;µ,σ,λ = 0, =,q) = f T x;µ,σ,q) = vσπq) / Γq) Γ +q) x µ+m qvσ) + ) +q where q v = q ) / if var.adj = TRUE, v = if var.adj = FALSE, and v is the value of var.adj if var.adj is a ositive scalar. Note that var.adj = TRUE and q is an error and NaNs will be roduced. The h th moment i.e. E[X EX)) h ]), for q > h/, is: ) + ) h) vσ) h q h Γ h+ ) )) Γ q h π Γq) The mean, for q > /, is µ. The variance, for q >, is: vσ) q q 9

21 thus if var.adj = TRUE, the variance, for q >, is simly σ. The skewness, for q > 3/, is 0. The kurtosis, for q >, is: 3vσ) 4 q 4q )q ) Second Parameterization: Let df equal d, the degrees of freedom of a tyical student t distribution arameterization. > dsgtx, mu = 0, sigma =, lambda = 0, =, q = df/, + mean.cent = FALSE, var.adj = sqrt)) The Student T Distribution has the df: f SGT x;µ = 0,σ =,λ = 0, =,q = d/) = f T x;d) = Γ ) d+ πd) / Γd/) x +) d+ d Note that we substituted v = from the first arameterization into this one. The h th moment i.e. E[X EX)) h ]), for d > h, is ) + ) h) d h/ Γ h+ ) Γ d h )) π Γd/) The mean, for d >, is 0. The variance, for d >, is: d d The skewness, for d > 3, is 0. 0

22 The kurtosis, for d > 4, is: 3d d 4)d )

23 J. Skewed Cauchy Distribution > dsgtx, mu, sigma, lambda, =, q = /, mean.cent = FALSE, + var.adj = FALSE) The Skewed Cauchy Distribution has the df: f SGT x;µ,σ,λ, =,q = /) = f SCauchy x;µ,σ,λ) = vσπ x µ + vσ) λsignx µ)+) ) The h th moment i.e. E[X EX)) h ]) is undefined. Thus the mean, variance, skewness, and kurtosis are all undefined. If mean.cent or var.adj are TRUE, NaNs are roduced. NOTE: If var.adj = sqrt), then v =, which gives a df more consistent with the standard arameterization of the Cauchy distribution: = f SCauchy x;µ,σ,λ) = σπ x µ + σ λsignx µ)+) )

24 K. Lalace Distribution > dsgtx, mu, sigma, lambda = 0, =, q = Inf, mean.cent, var.adj) The Lalace Distribution has the df: lim f SGTx;µ,σ,λ = 0, =,q) q where e x µ vσ = f Lalace x;µ,σ) = vσ v = if var.adj = TRUE, v = if var.adj = FALSE, and v is the value of var.adj if var.adj is a ositive scalar. The h th moment i.e. E[X EX)) h ]) is: + ) h ) vσ) h Γh+) The mean is µ. The variance is: vσ) thus if var.adj = TRUE, the variance is simly σ. The skewness 0. The kurtosis is: 4vσ) 4 3

25 L. Uniform Distribution > dsgtx, mu, sigma, lambda, = Inf, mean.cent, var.adj) The Uniform has the df: lim f SGTx;µ,σ,λ,,q) where = f unif x;µ,σ) = { vσ if x µ vσ 0 otherwise v = 3 if var.adj = TRUE, v = if var.adj = FALSE, and v is the value of var.adj if var.adj is a ositive scalar. The h th moment i.e. E[X EX)) h ]) is: vσ) h+ vσ) h+) vσh+) The mean is µ. The variance is: vσ) 3 thus if var.adj = TRUE, the variance is simly σ. The skewness is 0. The kurtosis is: vσ) 4 5 4

26 M. Normal Distribution > dsgtx, mu, sigma, lambda = 0, =, q = Inf, mean.cent, var.adj) The Normal Distribution has the df: lim f SGTx;µ,σ,λ = 0, =,q) q where = f Normal x;µ,σ) = e x µ vσ ) vσ π v = if var.adj = TRUE, v = if var.adj = FALSE, and v is the value of var.adj if var.adj is a ositive scalar. The h th moment i.e. E[X EX)) h ]) is: ) + ) h vσ) h Γ h+ )) π The mean is µ. The variance is: vσ) thus if var.adj = TRUE, the variance, for q >, is simly σ. The skewness is 0. The kurtosis is: 3vσ) 4 4 5

27 N. Cauchy Distribution > dsgtx, mu, sigma, lambda = 0, =, q = /, mean.cent = FALSE, + var.adj = FALSE) The Cauchy Distribution has the df: f SGT x;µ,σ,λ = 0, =,q = /) = f Cauchy x;µ,σ) = ) vσπ x µ + vσ) The h th moment i.e. E[X EX)) h ]), for q > h, is: undefined. Thus the mean, variance, skewness, and kurtosis are all undefined. If mean.cent or var.adj are TRUE, NaNs are roduced. NOTE: If var.adj = sqrt), then v =, which gives the standard arameterization of the Cauchy df: = f Cauchy x;µ,σ) = σπ x µ ) ) + σ 6

28 References [] Hansen, B. E., 994, Autoregressive Conditional Density Estimation, International Economic Review 35, [] Hansen, C., J. B. McDonald, and W. K. Newey, 00, Enstrumental Variables Estimation with Flexible Distributions, Journal of Business and Economic Statistics 8, 3-5. [3] McDonald, J. B. and W. K. Newey, 988, Partially Adative Estimation of Regression Models via the Generalized t Distribution, Econometric Theory 4, [4] Theodossiou, Panayiotis, 998, Financial Data and the Skewed Generalized T Distribution, Management Science 44,