Jan Purczyński, Kamila Bednarz-Okrzyńska Estimation of the shape parameter of GED distribution for a small sample size

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1 Jan Purczyńki, Kamila Bednarz-Okrzyńka Etimation of the hape parameter of GED ditribution for a mall ample ize Folia Oeconomica Stetinenia 4()/,

2 Folia Oeconomica Stetinenia DOI: 0.478/foli ESTIMATION OF THE SHAPE PARAMETER OF GED DISTRIBUTION FOR A SMALL SAMPLE SIZE Prof. Jan Purczyńki Szczecin Univerity Faculty of Management and Economic of Service Department of Quantitative Method Cukrowa 8, Szczecin, Poland jan.purczynki@wzieu.pl Kamila Bednarz-Okrzyńka, MA Szczecin Univerity Faculty of Management and Economic of Service Department of Quantitative Method Cukrowa 8, Szczecin, Poland kamila.bednarz@wzieu.pl Received 8 October 03, Accepted July 04 Abtract In thi paper a new method of etimating the hape parameter of generalized error ditribution (GED), called approximated moment method, wa propoed. The following etimator were conidered: the one obtained through the maximum likelihood method (MLM), approximated fat etimator (AFE), and approximated moment method (AMM). The quality of etimator wa evaluated on the bai of the value of the relative mean quare error. Computer imulation were conducted uing random number generator for the following hape parameter: = 0.5, =.0 (Laplace ditribution) =.0 (Gauian ditribution) and = 3.0. Keyword: etimating parameter of GED ditribution. JEL claification: C0, C3, C46.

3 36 Jan Purczyńki, Kamila Bednarz-Okrzyńka Introduction In the paper a ditribution which i the generalization of Gauian ditribution or Laplace ditribution will be conidered. The generalized error ditribution (GED) include the pecific cae of the Laplace ditribution and Gauian ditribution. The denity function of which i given by equation () i called the Generalized Error Ditribution (GED) or Generalized Gauian Ditribution (GGD). Due to a changing value of the hape parameter (equation ), the ditribution enable modeling of variou phyical and economic variable. GGD ha been applied in image recognition, ignal diturbance modeling and peech modeling. Furthermore it i widely applied in image compreion, where it i ued for modeling the ditribution of the dicrete coine tranform (DCT) coefficient. GED i uccefully applied in modeling the ditribution of rate of return for tock indexe and companie 3. The ditribution with the o called fat tail have been applied in modeling time-varying conditional variance, among other 4 and, where for the GARCH model etimation, GED wa ued a a conditional ditribution 5. Thi paper focue on the problem preent in an etimation of the hape parameter in the cae of a mall ample ize. We aumed the following ymbol for GED denity: where Γ(z) Euler gamma function. λ f ( x) µ Γ = exp( λ x ) () For =, GED turn into the Laplace ditribution (biexponential): λ f ( x) = exp( λ x µ ) () In the cae of =, a normal ditribution i obtained: where: λ =. σ λ f ( x) = exp( λ ( x µ ) ) (3) π For implicity reaon, it i aumed that on the bai of the ample, the etimation of parameter µ wa determined: N µ = x k (4) N k = and conequently, a erie of value x k ha been centralized by ubtracting µ.

4 Etimation of the Shape Parameter of GED Ditribution for a Small Sample Size 37 Hence the denity of the following form i conidered: λ f ( x) = exp( λ x ) Γ (5) GED characteritic and parameter etimation method were firt mentioned in Subbotin, yet the application of the ditribution to tatitical iue i owed to Box&Tiao 6.. Selected method of GED parameter etimation.. Maximum likelihood method (MLM) By applying MLM, the logarithm of likelihood function: from condition N ln( L( λ, )) = N ln( λ) + N ln λxk Γ k= we obtain ln( L( λ, ) λ ln( L( λ, ) = 0 ; = 0 (6) and N λ = N (7) x k k = N xk ln xk N k= gw ( ) = +Ψ + ln xk = 0 N N k= xk k= (8) d dz. where: Ψ ( z) = ln Γ( z) From equation (8) the hape parameter i derived and from equation (7) parameter λ.

5 38 Jan Purczyńki, Kamila Bednarz-Okrzyńka.. Approximated fat etimator In Krupińki and Purczyńki the method of GED parameter etimation baed on abolute moment wa applied 7. An abolute moment of order m i given by: m Em = x f ( x) dx (9) From equation (5) and (9) we obtain: m + Γ Em = m λ Γ (0) Moment etimator E m ha a form: N m Em = xk N k = Auming two different value of moment m and m in equation (0) and eliminating parameter λ, we obtain: () E G () = = ( Em ) m + Γ m m m m m m m + m Γ Γ () The etimation of the hape parameter i obtained in the form of an inverion function of the function G(). The following form of the inverion function wa propoed: ln( G ( ) a c ˆ = b (3) Table contain value of coefficient a, b and c for particular et of moment value. The coefficient were derived through computer imulation taking into account the mallet value of the error RRMSE (equation 9), i.e. mean quare approximation wa conducted, hence the name of the method approximated fat etimator (AFE).

6 Etimation of the Shape Parameter of GED Ditribution for a Small Sample Size 39 Table. Coefficient value of model (3) Moment value a b c Etimate ŝ m = 0.; m = ŝ 0 m = 0.5; m = ŝ m = ; m = ŝ m = ; m = ŝ 3 Source: author own calculation. An algorithm propoed in Krupińki and Purczyńki i the following 8. Baed on equation (3) and Table, the value of the hape parameter etimate ŝ i determined. Conequently, in accordance with dependency (4), the final etimate ŝf i calculated: f ˆ ˆ3 for ˆ >.85 ˆ for < ˆ.85 = ˆ for 0.5 < ˆ ˆ0 for ˆ 0.5 (4) Finally, baed on the etimate ŝf, the parameter etimate λˆ i determined: where m = m or m = m. λ m + m Γ f ˆ = Em Γ f ˆ (5).3. Approximated moment method In thi paper a modification of the method provided in Krupińki and Purczyńki i propoed 9. The modification i related mainly to the form of the inverion function. By auming in equation (0) m = m and eliminating the parameter λ, we obtain: m + Γ E Γ g = = m m + ( Em ) Γ (6)

7 40 Jan Purczyńki, Kamila Bednarz-Okrzyńka The invere function of the function g (equation 6) take the following form dependent on moment value m = m : m = 0.5; m = 0.50; 70 55g for g ˆ 0 = 5.7g for.079 < g < g for g.3 (7a) m = 0.5; m =.0; 8 6g for g.7 ˆ = 9 5.5g for g >.7 (7b) m =.0; m =.0; 7 9g for g ˆ = 3 5g for g > (7c) m = ; m = 4;.98 g for g 6 ˆ 3 =.3 6g for g > 6 (7d) The final etimate ŝ a i obtained from the following dependency: ˆ3 for ˆ >.9 ˆ for.4 < ˆ.9 a ˆ = ˆ for 0.53 < ˆ.4 ˆ0 for ˆ 0.53 (8) Uing equation (5) the etimate of parameter λˆ value i obtained. In order to differentiate from the previouly ued method AFE, the propoed method i called approximated moment method AMM.. Reult of computer imulation In order to ae the quality of particular etimator, numerical experiment were conducted uing a random number GED generator for elected value of the hape parameter: = 0.5, = (Laplace ditribution) = (Gauian ditribution) and = 3. The computer imulation conited in performing K = 000 iteration, and on their bai determining the etimation error RRMSE (Relative Root Mean-Squared Error):

8 Etimation of the Shape Parameter of GED Ditribution for a Small Sample Size 4 where: K ˆ k d RRMSE = (9) K k= d ŝ k etimation of parameter value for k-imulation, exact value of the hape parameter. d In the paper the error RRMSE wa determined (equation 9), ince it i directly related to the error MSE (Mean-Squared Error): K k d. K k = where: MSE = ( ˆ ) MSE RRMSE = (0) d The mean-quared error i important a uch, ince it encompae both the error of the etimator bia and it variance: where: V(ŝ) variance of the etimator, b(ŝ) bia of the etimator. ( ˆ) ( ˆ) MSE = V + b () When preenting the reult of computer imulation, the label rme wa ued in place of RRMSE. In order to olve equation (8) the biection method wa applied. The calculation were conducted for a changing ample ize (number of obervation) N = 3, 4,, 0. During the calculation it wa noted that the hape parameter error i influenced by the way of determining the mean value, which i ubtracted in the proce of centralizing. It wa epecially noticeable in the cae of the AFE method, for which the ubtracting of the mean value (equation 4) yielded the error rme everal time larger than in the cae when the median wa ubtracted. Therefore in ubequent figure the value of rme for the ample centralized by mean of the median were preented.

9 4 Jan Purczyńki, Kamila Bednarz-Okrzyńka Figure depict value of error for the generator with the hape parameter = 0.5. It hould be noticed that the ample with N = 3 element ha a very large relative error for ML and AFE the error equal everal hundred percent. For other value of N AFE method ha the error of 00%. 3.5 rmeml i rmef i rmea i N i Dotted line with x rmeml preent the value obtained by MLM. Dahed line with circle rmf correpond to AFE method (equation 3, 4). Solid line with rectangle rmea mark the value of error obtained through AMM method (equation 7, 8). Fig.. Value rme obtained for GED generator with the hape parameter = 0.5 Source: author own calculation. Figure (Laplace ditribution) and 3 (Gauian ditribution) how coniderable imilarity: ML method yield the larget error, and the mallet error AMM method. Yet the larger the number of N element, the more imilar the error of particular method. Figure,, 3 how the large error of ML for the number of element N = 3. Figure N mut be ubtantially large to reveal the advantage of ML method leading to efficient and unbiaed etimator. Thi iue wa dicued in Meigen, Meigen 0.

10 Etimation of the Shape Parameter of GED Ditribution for a Small Sample Size rmeml i rmef i 0.6 rmea i N i The ymbol ued in thi figure are the ame a in Figure. Fig.. Value rme obtained for GED generator with the hape parameter = (Laplace ditribution) Source: author own calculation rmeml i rmef i rmea i N i The ymbol ued in thi figure are the ame a in Figure. Fig. 3. Value rme obtained for GED generator with the hape parameter = (Gauian ditribution) Source: author` own calculation.

11 44 Jan Purczyńki, Kamila Bednarz-Okrzyńka rmeml i rmef i rmea i N i The ymbol ued in thi figure are the ame a in Figure. Fig. 4. Value rme obtained for GED generator with the hape parameter = 3 Source: author own calculation. In the cae of = 0.5 and = 3 the larget error can be oberved for AFE reult. The reaon for it lie in the form of the function decribed by equation (3) ince it may lead to a complex number, in which cae it i adviable to take into account the real part of the derived value. For the generator with = 0.5 the complex number appeared in one-fourth of reult.. qml i qf i 0.9 qa i N i The following curve refer to: MLM-qML (dotted line with x); AFE-qF (dahed line with circle); AMM-qA (olid line with rectangle). Fig. 5. Ratio of the value of error rme yielded by centralizing with the median to the value of rme for the ubtracted mean value Laplace ditribution generator Source: author own calculation.

12 Etimation of the Shape Parameter of GED Ditribution for a Small Sample Size 45 Figure 5 preent the value of the rme error obtained by centralizing with the median, divided by the rme value for the ubtracted mean value Laplace generator. The value maller than one prove that ubtracting the median yield a maller value of error than ubtracting the arithmetic mean. It i particularly viible for the AMM. Coefficient qml for = obtain the value 0.96 and for = 3 qml = 0.9. It mean that when uing ML method, centralizing hould be done by ubtracting the median. The ituation i different for coefficient qa, which for = lightly exceed and for = 3 i about.. Thi mean that when uing the AMM method, the method of centralizing depend on the value of. For <.5 it i adviable to ubtract the median and for >.5 the arithmetic mean hould be ued. Concluion In thi paper a new method of etimating the hape parameter of GED ditribution approximated moment method wa propoed. The quality of the propoed etimator wa compared with the quality of the etimator obtained through the maximum likelihood method (MLM) and approximated fat etimator (AFE). The quality of etimator wa evaluated on the bai of the value of the relative mean quare error determined uing GED random number generator with the following hape parameter: = 0.5, = (Laplace ditribution) = (Gauian ditribution) and = 3. The method with the mallet error i the one propoed in thi paper the approximated moment method AMM. A far a other two method are concerned, for = and = maller value of rme are provided by AFE method. Neverthele, for = 0.5 and = 3 MLM i the mot accurate. Attention ha been drawn to the fact that the final reult depend on whether centralizing i conducted uing the arithmetic mean or the median. In the cae of MLM, it i adviable to ubtract the median. And for AMM, the etimated value of hape parameter ŝ hould be additionally taken into account. It tem from the fact that for the Laplace ditribution, the median i the more efficient etimator than the arithmetic mean etimator. For the normal ditribution the ituation i revere and the arithmetic mean etimator i more efficient. The propoed method i particularly ueful for a mall ample ize N 00. For a large ample ize, the mallet error i yielded by the etimator obtained by MLM.

13 46 Jan Purczyńki, Kamila Bednarz-Okrzyńka Note Kokkinaki, Nandi (005), pp Krupińki, Purczyńki (007), pp Weron, Weron (998), pp Hieh (989), pp Nelon (99), p Subbotin (93); Box, Tiao (96). 7 Krupińki, Purczyńki (006). 8 Ibidem. 9 Ibidem. 0 Meigen, Meigen (006). Reference Box, G.E.P. & Tiao, G.C. (96). A further look at robutne via Baye theorem. Biometrika, 49 (3/4). Hieh, D.A. (989). Teting for nonlinear dependence in daily foreign exchange rate change. Journal of Buine, 6. Kokkinaki, K. & Nandi, A.K. (005). Exponent parameter etimation for generalized Gauian probability denity function with application to peech modeling. Signal Proceing, 85. Krupińki, R. & Purczyńki, J. (006). Approximated fat etimator for the hape parameter of generalized Gauian ditribution. Signal Proceing, 86 (4). Krupińki, R. & Purczyńki, J. (007). Modeling the ditribution of DCT coefficient for JPEG recontruction. Signal Proceing: Image Communication, (5). Meigen, S. & Meigen, H. (006). On the modeling of mall ample ditribution with generalized Gauian denity in a maximum likelihood framework. IEEE Tranaction on Image Proceing, 5 (6). Nelon, D.B. (99). Conditional heterokedaticity in aet return: A new approach. Econometrica, 59 (). Subbotin, M.T.H. (93). On the law of frequency of error. Mathematicheki Sbornik, 3. Weron, A. & Weron, R. (998). Inżynieria finanowa. Warzawa: WNT.