Estimation of Peaked Densities Over the Interval [0,1] Using Two-Sided Power Distribution: Application to Lottery Experiments

Transcription

1 MPRA Munich Peronal RePEc Archive Etimation of Peaed Denitie Over the Interval [0] Uing Two-Sided Power Ditribution: Application to Lottery Experiment Krzyztof Konte Artal Invetment 8. April 00 Online at MPRA Paper No. 378 poted 9. April 00 00:43 UTC

2 Etimation of Peaed Denitie Over the Interval [0] Uing Two-Sided Power Ditribution: Application to Lottery Experiment Krzyztof Konte Artal Invetment Waraw 3 Abtract Thi paper deal with etimating peaed denitie over the interval [0] uing twoided power ditribution (Kotz van Dorp 004). Such data were encountered in experiment determining certainty equivalent of lotterie (Konte 00). Thi paper ummarize the baic propertie of the two-ided power ditribution (TP) and it generalized form (GTP). The GTP maximum lielihood etimator a reult not derived by Kotz and van Dorp i preented. The TP and GTP are ued to etimate certainty equivalent denitie in two data et from lottery experiment. The obtained reult how that even a two-parametric TP ditribution provide more accurate etimate than the mooth three-parametric generalized beta ditribution GBT (Libby Novic 98) in one of the conidered data et. The three-parametric GTP ditribution outperform GBT for thee data. The reult are however the very oppoite for the econd data et in which the data are greatly cattered. The paper demontrate that the TP and GTP ditribution may be extremely ueful in etimating peaed denitie over the interval [0] and in tudying the relative utility function. JEL claification: C0 C3 C4 C6 C C5 C8 C9 D03 D8 D87 Keyword: Denity Ditribution; Maximum Lielihood Etimation; Lottery experiment; Relative Utility Function. In memoriam to Profeor Samuel Kotz who adly paed away on March 6 00 at the age of 80. I never had the honor of nowing or meeting Profeor Kotz but it wa he together with Rene van Dorp who invented the two-ided power ditribution. Quite coincidentally I began woring with thi ditribution on March Over the next couple of day I obtained the reult which form the ubject of thi and two other upcoming paper. I had planned to end thi paper to Prof. Kotz for hi comment and only learnt of hi paing while maing my final reviion. The author i grateful to Stefan Traub from the Department of Economic Univerity of Bremen Ulrich Schmidt from the Department of Economic Univerity of Kiel and Katarzyna Idziowa from the Center of Economic Pychology and Deciion Science Kozmini Univerity Waraw for maing the reult of their experiment available. 3 Contact: ul. Chrościciego 93/ Waraw Poland onte@artal.com.pl onte000@yahoo.com tel.:

3 . Introduction Peaed data are becoming more and more frequently oberved epecially in financial application. Such data may be decribed uing the Laplace ditribution which i an alternative to mooth ditribution function. Le examined are denitie defined over a bounded interval. Thi ha mainly been due to the lac of any ditribution uited to analyze uch cae. It wa only recently that Kotz and van Dorp (004) introduced a few type of two-ided power ditribution defined over the interval [0]. However a imple maximum lielihood etimator wa preent only for it baic form. Every other type required a recurive optimization procedure. Peaed data over the interval [0] were encountered in experiment determining certainty equivalent of lotterie (Konte 00). The denitie were etimated uing beta BT and generalized beta GBT (Libby Novic 98) ditribution. The obtained reult howed that uch mooth function may not be appropriate in all cae and that the ue of a peaed ditribution may give better reult. The aim of thi paper i therefore to preent the application of a two-ided power ditribution TP and it generalized form GTP in the conidered cae. The maximum lielihood etimator for GTP a reult not derived by Kotz and van Dorp i preented beforehand. The baic TP and GTP propertie are firt ummarized in Point and 3. The TP and GTP maximum lielihood etimator for unimodal denitie are preented in Point 4. The derivation of TP and GTP maximum lielihood etimator i demontrated in Point 5 where non-unimodal denitie are alo analyzed. Point 6 demontrate the ue of TP and GTP maximum lielihood etimator for two data et conidered previouly by Konte. Point 7 ummarize the paper and conclude that the TP and GTP ditribution may be extremely ueful in etimating peaed denitie over the interval [0] and in reearching the relative utility function.. Propertie of Two-Sided Power Ditribution.. The two-ided power ditribution TP preented here i the Standard Two-Sided Power (STSP) ditribution a conidered by Kotz and van Dorp (004). It ha two parameter: ( r λ γ) TP ; r γ λ = r γ λ γ γ if if 0 r λ λ< r. (.)

4 The TP implifie to a uniform ditribution for γ = a triangular ditribution for γ = a power ditribution for λ = and a reflected power ditribution for λ = 0. Sample TP hape are preented in Figure.. The function i unimodal for γ > uniform for γ = and U- haped for γ <. Pleae note that there are ome minor error in the figure preented by Kotz and van Dorp (004) a well in the textual decription (pp. 7-7). Figure.. Sample hape of two-ided power ditribution. On the left λ = 0.5 in the middle λ = 0.5 on the right λ = The value of γ parameter i equal to the denity at λ... The TP ha an extremely intereting feature in that it mode i given by the λ parameter: and the value at the mode by the γ parameter ( r λ γ) Mode TP ; = λ (.) ( r ) TP = λ; λ γ = γ. (.3) Thi mae thi ditribution unique in that it parameter may be eaily determined from it hape..3. The mean value of TP i equal to: ( ) λ γ + Mean TP ( r; λ γ) =. γ +.4. The Cumulative TP ditribution i defined a: (.4) ( r λ γ) CTP ; γ r λ if 0 r λ λ = γ r ( λ) if λ < r. λ (.5) which allow after inverting the quantile to be determined: ( r λ γ) Quantile q;tp ; q γ λ if 0 q λ λ = q γ ( λ) if λ< q. λ (.6) 3

5 In the pecial cae where q = 0.5 the median i given by: ( r λ γ) Median TP ;.5. The variance of TP i given by: γ γ γ λ if λ = ( λ)( λ) γ if λ <. ( ) ( ) ( γ + ) ( γ + ) γ γ λ λ Variance TP ( r; λ γ) =. (.7) (.8) 3. Propertie of the Generalized Two-Sided Power Ditribution 3.. The generalized two-ided power ditribution GTP preented here i the Generalized Standard Two-Sided Power (GSTSP) ditribution a conidered by Kotz and van Dorp and i a pecial cae of the Uneven Standard Two-Sided Power (USTSP) ditribution. It ha three parameter and i defined by: ( r λ γ δ) GTP ; r φ λ = r φ λ γ δ if if 0 r λ λ< r. (3.) where γδ φ =. λδ + λ γ ( ) (3.) The function i unimodal for γ > and δ > J-haped for γ > and δ < invere J- haped for γ < and δ > uniform from 0 to λ for γ = uniform from λ to for δ = U- haped with anti-mode at λ for γ < and δ < i power ditribution for λ = and i a reflected power ditribution for λ = 0. Figure 3.. Sample hape of generalized two-ided power GTP ditribution. On the left λ = 0.5 in the middle λ = 0.5 and on the right λ = All but the firt hape are referred to a non-unimodal throughout the paper. Some 4

6 ample hape are given in Figure 3.. Only unimodal curve are preented a they are of mot interet in thi paper and in practical application. An important cae (a will be dicued in 6.3) i when either γ or δ aume an infinite value. The GTP then reduce to a bounded TP ditribution defined over the interval [λ ] or [0 λ] repectively. 3.. The GTP mode i given by it λ parameter: and the value at the mode i given by: 3.3. The mean value of GTP i equal to: ( r λ γ δ) Mode GTP ; = λ (3.3) ( r ) GTP = λ; λ γ δ = φ. (3.4) γ λ Mean GTP ( r; λ γ δ) = + λδ +. ( γ + )( δ + ) λδ + ( λ) γ (3.5) 3.4. The Cumulative GTP i defined a: ( r λ γ δ) CGTP ; γ λφ r if 0 r λ γ λ = δ ( λ) φ r if λ< r. δ λ (3.6) which allow after inverting the ditribution quantile to be calculated: ( p λ γ δ) Quantile q;gtp ; q γ γ λφ λ if 0 q λφ γ = ( ) ( ) δ q δ λφ λ if < q. ( λ) φ γ (3.7) In the pecial cae where q = 0.5 the median i given by: ( r λ γ δ) Median GTP ; γ γ λφ λ if λφ γ = δ λφ δ ( λ) if <. ( λ) φ γ (3.8) 3.5. The GTP variance i given by: 5

7 ( ) ( ) ( γ) ( γ)( γ δ) ( γ)( γ)( γ γδ δ ) ( )( ) ( ) ( γ δ) γ( γ) ( δ) ( ) ( )( ) ( ) ( ) 3 4 ( + λ+ λ + λ + λ ) F A B C D E Variance GTP ( r; λ γ δ) = G where A= γ + γ + γ B= + + C= D= γ γ δ γ + γ γδ + δ 3 E= + + F = γδ G= + γ + γ + δ + δ γ λ δλ. (3.9) Equation (3.9) ha an unpleaant form but i given here a variance i an important meaure of ri in many financial application. Beide it i required in the denity-baed mean regreion approach (Konte 00). 4. Maximum Lielihood Etimator for TP and GTP Ditribution 4.. Although Kotz and van Dorp provided a maximum lielihood etimator for the TP ditribution they did not preent a imilar olution for GTP. Intead they propoed a recurive maximum lielihood procedure for the more general Uneven Standard Two-Sided (USTSP) ditribution. It appear however that uch a olution doe exit for GTP. The only retriction i that thi imple and traightforward etimator only concern unimodal denitie which in any cae eem to be the only intereting one in mot practical application. The olution for non-unimodal denitie i alo preent but require olving a nonlinear equation and more detailed conideration. At thi point the maximum lielihood etimator for TP and GTP in the cae of unimodal denitie are demontrated without giving any detail of how they were derived. Thee are provided in Point 5 together with the non-unimodal denity analyi. 4.. An important feature of both TP and GTP lielihood function i that they may have multiple maxima. That thee local maxima only appear at the ample point ait in finding the global maximum. Importantly the lielihood value at thee point can be calculated quite imply by uing a formula and do not require any optimization algorithm. Thi produce a very different approach than that commonly ued for mooth ditribution. The etimation procedure i baed on checing the value of the lielihood function at 6

8 the ample point and electing the greatet value. Knowing the point at which the lielihood function achieve it maximum allow the ought parameter to be derived once again by uing a formula without recoure to any optimization algorithm The maximum lielihood etimator in the cae of TP i preented in a lightly different manner than that adopted by Kotz and van Dorp. The etimator of the λ parameter i: λ= ˆ r (4.) where r denote the value of the th point from the ordered ample at which the log-lielihood function defined a: LogL = w ln w (4.) achieve it maximum 4. In (4.) denote the ample count and w = w + w (4.3) w w i i= = ln r r ( r) = ( r ) i + i= + ln (4.4) (4.5) where r i denote the value of the i th point from the ordered ample. Once the point i determined the maximum lielihood etimator of γ i given by: ˆ γ = (4.6) w where w i calculated at point In the cae of GTP the maximum lielihood etimator of the λ parameter i alo given by (4.) i.e. it i the value r of the th point from the ordered ample at which the loglielihood function defined a: ( ( ) ) LogL = w + ln ln r w + r w (4.7) achieve it maximum. In (4.7) w w and w + are defined a above. Once the point i 4 w NB: Kotz and van Dorp do not provide thi form and a different expreion viz. e i maximized. 7

9 determined the maximum lielihood etimator of γ and δ are given by: ˆ γ = ( r ) w w w + r (4.8) and ˆ δ = + r w w w + r (4.9) where the value of w GTP wa not provided by Kotz and van Dorp. w + and r are calculated at the point. A mentioned thi reult for 5. Derivation of Maximum Lielihood Etimator for TP and GTP 5.. Point 5 i devoted olely to the derivation of the maximum lielihood etimator for TP and GTP together with an analyi of non-unimodal denitie. Thi i a quite technical and detailed ubject and may therefore be ipped by reader more intereted in practical etimation reult. 5.. The derivation of the maximum lielihood etimator for TP will be preented firt. Thi i a different derivation than that preented by Kotz and van Dorp but it follow their line of reaoning. Let u conider the lielihood function in the interval between point and + from the ordered ample. According to (.) thi may be expreed a: L γ + P P = γ λ ( λ) γ (5.) where P i= i= + = ri and + P ( ) = ri. The interval between the lower bound of the ditribution (i.e. the value 0) and the firt ample point may be alo conidered. In thi cae = 0 P = and P = ( r) 0 0 i. Similarly for the interval between the lat ample point and the i= upper bound of the ditribution (i.e. the value ) = logarithm of (5.) reult in the log-lielihood function: = i i= P r ( )( ) ( ) ( ) ( ) and P + =. Taing the LogL = lnγ + γ ln P lnλ + γ ln P ln λ. (5.) The econd derivative of (5.) with repect to λ i: 8

10 dλ λ ( λ) d LogL = ( γ ) + (5.3) which i alway poitive for γ >. It follow that in the cae of unimodal denitie the loglielihood function i alway convex between the point and + and it reache it maximum value at one of thee point. Moving from one interval to another however change the log-lielihood function (5.) a it depend on. The corollary of thi i that the log-lielihood function i not differentiable at the ample point and local maxima may appear there. It i therefore ufficient to chec the value of the log-lielihood function at the ample point to find it global maximum. Thi can be done a follow. The maximum value of the log-lielihood function at point can be found by determining it firt derivative with repect to γ and comparing the reult to 0: d LogL dγ = + ln P ln r + ln P ( ) ln( r ) = 0 (5.4) γ where λ ha been ubtituted with r. Equation (5.4) may be repreented in a impler form uing (4.3) (4.5): d LogL dγ = w = 0. (5.5) γ Solving (5.5) lead to the maximum lielihood etimator of γ (4.6). Subtituting thi reult into the log-lielihood function (5.) and rearranging yield the form (4.) which doe not depend on the ditribution parameter and i therefore eay to maximize The procedure for GTP i imilar. According to (3.) and (3.) the lielihood function may be expreed a: L γ + γδ P P = λδ ( λ) γ + λ ( λ) δ. (5.6) Taing the logarithm of (5.6) yield the log-lielihood function: LogL { lnγ lnδ ln λδ ( λ) γ } = ( γ )( P λ) + ( δ ) P ( ) ( λ) ln ln ln ln. (5.7) It econd derivative with repect to λ i: ( ) ( )( ) ( ) ( ) + ( ) d LogL γ δ γ δ = + + dλ λ λ λδ λ γ (5.8) which i alway poitive for γ > and δ >. Applying the reaoning preented in point 5. for 9

11 TP it i ufficient to chec the value of the log-lielihood function at the ample point in order to find it global maximum. Thee can be found a follow. The firt derivative of the log-lielihood function (5.7) with repect to γ and δ are compared to 0: d LogL r δ = w = 0 dγ γ r δ + ( r ) γ d LogL r + = w = 0 dδ δ r δ + ( r ) γ (5.9) (5.0) where (4.4) and (4.5) were ued to implify the reult. Solving the et of equation (5.9) and (5.0) reult in the maximum lielihood etimator of γ (4.8) and δ (4.9). It i worth noting that the econd pair of olution alo exit with the root in the denominator being negative. Thi pair however lead to very high value of γ of δ and ditribution which do not fit the data. Subtituting the etimated value (4.8) and (4.9) into the log-lielihood function (5.7) reult in the form (4.7) which doe not depend on the ditribution parameter and i ued for maximization It i intereting (albeit from a theoretical rather than a practical point of view) to determine the maximum lielihood etimator for non-unimodal denitie. In thi cae the econd derivative of the log-lielihood function may aume a negative value. Thi would indicate that the log-lielihood function i concave and it maximum may be located between point and +. The TP etimator will be conidered firt In order to find a maximum the firt derivative of the log-lielihood function (5.) with repect to λ i compared to 0: d LogL = ( γ ) = 0 dλ λ λ (5.) which reult in: ˆ λ=. (5.) Equation (5.) determine the value of λ at which the log-lielihood function (5.) achieve it maximum. It can eaily be checed whether thi value i located between point and +. If not then the log-lielihood function achieve the maximum either at point or at point + a in the cae of unimodal denitie. If the value of λ determined by (5.) i located inide the conidered interval then the 0

12 log-lielihood value may be calculated a follow. Firt we define w w and w + more generally than (4.3) (4.4) and (4.5) a function of λ within the interval [ + ] rather than the value at point : ( λ) ( λ) ( λ) w = w + w (5.3) ri = ln λ i= ( ) w λ ( ) w λ ( r) = ( λ) i + i= + ln. (5.4) (5.5) Clearly (5.3) - (5.5) reult in the ame value a (4.3) - (4.5) for λ= r i.e. at the bound of the conidered interval [ + ]. Repeating the tep preented in point 5.. yield the loglielihood value (5.6) which ha a very imilar form to (4.): LogL = w( λ) ln. w( λ) (5.6) The only difference i that w i now determined for uing (4.3). λ= uing (5.3) rather than for λ= r The procedure preented above mut be repeated for all the interval [ + ] in order to find the global maximum of the log-lielihood function. Very imilarly to unimodal denitie the maximum lielihood etimator of γ i then given by: ˆ γ = (5.7) w ( λ) which i calculated uing (5.3) for the determined value of λ and The firt derivative of the log-lielihood function (5.7) with repect to λ i compared to 0: 5.5. The cae of the GTP i more complex. which reult in two olution: ( ) + ( ) + ( ) ( ) ( ) + d LogL λ γ λδ λ γ λδ = = 0 dλ λ λ γ λδ (5.8)

13 γ λ= γ + ( ) δ (5.9) γ λ=. (5.0) γ δ Here determining whether the obtained value of λ are located within the conidered interval [ + ] i not poible a it wa in the cae of TP becaue γ and δ are not nown. One poibility i therefore to maximize the log-lielihood function (5.7) and to chec the obtained value of λ. Such a procedure however i not o convenient becaue (5.7) ha 3 parameter The ta can be implified a follow. Calculating the firt derivative of the log-lielihood function (5.7) with repect to γ and δ reult in: d LogL λδ = w = dγ γ λδ + ( λ) γ ( λ) 0 d LogL λ + = w( λ) = 0 dδ δ λδ + ( λ) γ (5.) (5.) where (5.4) and (5.5) are ued to implify the reult (cf. (5.9) and (5.0)). Solving the et of equation (5.) and (5.) yield etimator of γ and δ a function of λ (cf.(4.8) and (4.9)) ˆ γ λ = ( ) w ( λ) + ( λ) w( λ) w( λ) λ (5.3) and ˆ δ( λ) =. + λ w( λ) w( λ) w( λ) + λ (5.4) (4.7)) Subtituting (5.3) and (5.4) to (5.7) yield the log-lielihood function (cf. ( ) LogL ( λ) = w( λ) + ln ln λ w( λ) ( λ) w( λ) (5.5) which i a function of only one parameter λ and i therefore much eaier to maximize than (5.7). A maximizing a function i equivalent to olving it firt derivative with repect to 0 the preented way of proceeding may be further invetigated Returning to the full form of the log-lielihood function by ubtituting (5.3) - (5.5) to (5.5) calculating it firt derivative with repect to λ then putting bac (5.3) -

14 (5.5) in order to implify the reult and comparing it to 0 yield: + w( λ) m+ w( λ ) ( + m) d LogL w( ) ( ) w( ) m λ λ λ λ = + = 0. dλ λ λ λw λ λ w λ ( ) + ( ) ( ) (5.6) Solving (5.6) with repect to λ and leaving all the w(λ) expreion on the right-hand ide give the following three nonlinear equation: λ= m w + w( λ) ( λ) + w( λ) + ( λ) ( λ) 4w w (5.7) λ= (5.8) ( λ) ( λ) + 4w w λ= + (5.9) which can be olved numerically with repect to λ. Each equation may have 0 or more olution The maximum of the log-lielihood function (5.7) in the conidered interval [ + ] may be any of λ value determined by olving (5.7) (5.8) and (5.9). The whole maximum lielihood procedure for non-unimodal denitie would therefore require the following tep: Solve (5.7) (5.8) and (5.9) numerically with repect to λ Chec whether any of the reulting λ value are located within the interval [ + ] Calculate γ and δ for the λ value() uing (5.3) and (5.4) Calculate the econd derivative of the log-lielihood function uing (5.8) and chec whether thi value i negative (a poitive value would indicate a minimum of the loglielihood function) If the above condition are atified then calculate the value of the log-lielihood function uing (5.5). If any one of them i not atified then the maximum i either at point or + a i the cae with unimodal denitie. Calculate the log-lielihood value at the bound of the interval uing (5.5) or (4.7). Choe the greatet value of the log-lielihood function. The decribed procedure mut be repeated for all interval [ + ] in order to find 3

15 the global maximum The procedure decribed in may be implified if olving (5.7) - (5.9) i replaced by maximizing (5.5). Such a procedure hould be more efficient a fewer equation and poible λ value have to be examined A preented in thi Point both the TP and GTP maximum lielihood etimator are fairly eay to calculate for unimodal denitie. The procedure for non-unimodal denitie i not o traightforward but can be eaily implemented by a oftware pacage. 6. Etimation of Lottery Reult 6.. The TP and GTP maximum lielihood etimator are here ued in an application conidered in a previou paper by Konte (00). Two data et are examined. Set - the experimental data preented by Traub and Schmidt (009) whoe reearch concerned the relationhip between WTP (Willingne to Pay) and WTA (Willingne to Accept). Set - the experimental data of Idziowa (009) whoe reearch concern the quetion of whether the form in which probability i preented ha any impact on the hape of the probability weighting function. 6.. Konte (00) propoed a two-tage regreion procedure to decribe the experimental reult. In the firt tage the relative certainty equivalent are determined uing the tranformation: ce A r= (6.) P A where r denote the relative certainty equivalent ce denote the certainty equivalent P = Max(x) = the maximum lottery outcome and A = Min(x) = the minimum lottery outcome. The relationhip (6.) enure that r aume value in the range [0 ]. The denitie of the relative certainty equivalent for given value of probability are then etimated. The remaining part of thi paper concentrate on thee etimation reult. In the econd tage the obtained denitie are ued to etimate the relationhip between the relative certainty equivalent and the probability of winning the main prize: ( ) r Q p = (6.) where p denote the probability of winning the prize and Q denote a relative utility function which ha the form of a cumulative denity function defined over the range [0]. Further detail are available in Konte (00). 4

16 6.3. Although the maximum lielihood etimator for TP and GTP preented in Point 4 are fairly imple and traightforward a few practical comment are in order. Firt the TP and GTP maximum lielihood etimator appear to be computationally extremely fat when compared with tandard etimator of mooth ditribution which require running optimization oftware. The difference may be a much a two order of magnitude when compared with the GBT etimation The calculation time can be even further reduced whenever there i a large ample whoe data aume a limited number of value. Thi i becaue it i not neceary to calculate the value of the log-lielihood function (4.) or (4.7) for all the ample point but only for ditinct point. In financial application however the ample value are uually all different and any attempt to reduce the number of point will only lengthen the procedure. Second the procedure preented here only hold for unimodal denitie. In any cae when γ or δ aume a value le than the obtained reult may not be the maximum lielihood etimator. Thee cae were imply excluded from further conideration in the practical tudy preented in thi Point. Thi i a ound approach when the reearcher ha reaon to believe that the ought ditribution i unimodal. A tated γ and δ value le than appeared for ample point located away from the mode and the reulting ditribution exhibited a U- hape pointing to the anti-mode of the denity. The log-lielihood function value were low in thee cae which made it natural to eliminate uch olution. Third olution where either γ or δ aumed a value of infinity were alo encountered. Thi i a corollary of w or w + auming a value of 0 (cf. (4.8) and (4.9)). Such olution were alo excluded from thi reearch. The motivation for excluding uch olution even if they provide the highet lielihood value may not be all that obviou. However an infinite value for either γ or δ reduce the GTP to a bounded TP defined over part of the interval [0] and the lielihood function value appear to be incommenurable with ditribution defined over the entire interval The TP maximum lielihood etimation reult for Set are preented in Figure 6.. Each box preent an etimation reult for a ingle probability. A can be een the TP which ha parameter perform better than the GBT which ha 3 parameter in 7 out of cae. Thi i indicated by the maximized value of the log-lielihood function. Even without checing thee value the TP curve appear to be better uited for uch peaed data than a mooth ditribution. 5

17 Figure 6.. Denitie of r for repective probabilitie in Set. The TP etimation reult are mared in red and the GBT etimation reult are mared with dahed red line. The tp and gbt name indicate the maximized value of the log-lielihood function. The parenthee contain the mode median and mean of the ditribution calculated uing the formula given in Point (for TP) and in Konte (00) for GBT. The preented etimation how that the data are poitively ewed for low probabilitie and negatively ewed for high probabilitie 6

18 6.5. Figure 6. preent the etimation reult for Set uing the GTP maximum lielihood etimator. Figure 6.. Denitie of r for repective probabilitie in Set. The GTP etimation reult are mared in red and the GBT etimation reult are mared with dahed red line. The gtp and gbt name indicate the maximized value of the log-lielihood function. The parenthee contain the mode median and mean of the ditribution calculated uing the formula given in Point 3 (for GTP) and in Konte (00) for GBT. 7

19 Here the GTP etimator outperform GBT in all cae. Thi how that GTP i much better uited to decribe peaed denitie than the commonly ued mooth ditribution. The trong correpondence of the mode with the probability value i epecially worth noting. Thi how that the mot liely relative certainty equivalent i roughly equal to the probability Figure 6.3 preent the etimation reult for Set uing the GTP maximum lielihood etimator. In thi cae however the GTP perform poorly compared with GBT with only one cae out of nine yielding a better reult than the GBT. Thi how that GTP i not a ditribution to be ued when the data are cattered. Figure 6.3. Denitie of r for repective probabilitie in Set. The GTP etimation reult are mared in red and the GBT etimation reult are mared in dahed red. The gtp and gbt value indicate the maximized log-lielihood function. The parenthee contain the mode median and mean of the ditribution. 8

20 6.7. In the econd phae of the regreion procedure decribed in 6.. the relationhip between relative certainty equivalent and probabilitie i etimated. Thi i done uing the denity function hown in Figure and 6.3. The reult are only briefly preented here a the regreion procedure digree from the main ubject of thi paper. Figure 6.4 preent the etimation of the relative utility function Q for Set uing the GTP denitie hown in Figure 6.. The mean quantile (including median) and mode (maximum lielihood) regreion etimation are hown on the one graph. A can be een the lottery valuation are only nonlinear with probability when median and mean are conidered. Such nonlinearity i not confirmed for mode. The parameter α and β are both equal to in thi cae which mean the relationhip i linear. It follow that the mot liely lottery valuation are cloe to their expected value and that the mot liely behavior of a group i fully rational. Figure 6.4. Etimation reult of the relative utility function Q for Set. MN Mean (Leat Square); Q Lower Quartile; MED Median; Q3 Upper Quartile; ML Maximum Lielihood (Mode). The orange area mar data between the lower and upper quartile. The functional form of Q i a cumulative beta ditribution with parameter α and β. 7. Concluion Thi paper preented etimation procedure for peaed denitie over the interval [0]. The two-ided power ditribution TP and the generalized ditribution GTP (Kotz and van Dorp 004) were conidered for thi purpoe. The paper preented the TP and GTP maximum lielihood etimator. The latter wa not derived by Kotz and van Dorp. The obtained etimation demontrated that GTP outperform generalized beta ditribution GBT (Libby Novic 98) in one of the conidered data et. However the reult for much cat- 9

21 tered data a preent in the econd et are oppoite. The obtained denitie are then ued to etimate the relative utility function (Konte 00). The previouly preented reult that lottery valuation are only nonlinear with probability when mean and median are conidered i confirmed. Such nonlinearity diappear for mode. The paper demontrate that TP and GTP ditribution may be extremely ueful in etimating peaed denitie over the interval [0] and in reearching the relative utility function. Reference:. Idziowa K. (009). Determinant of the probability weighting function preented under name Katarzyna Domurat at SPUSM Conference Rovereto Italy Augut 009 paper Id = 8 Konte K. (00). Denity Baed Regreion for Inhomogeneou Data; Application to Lottery Experiment MPRA Paper Available at SSRN: 3. Kotz S. van Dorp J. R. (004). Beyond Beta; Other Continuou Familie of Ditribution with Bounded Support and Application World Scientific Publihing Singapore. 4. Libby D. L. Novic M. R. (98). Multivariate generalized beta ditribution with application to utility aement Journal of Educational Statitic 7 pp Traub S. Schmidt U. (009). An Experimental Invetigation of the Diparity between WTA and WTP for Lotterie Theory & Deciion 66 (009) pp