Wind speed analysis with the upper truncated quasi Lindley distribution

Transcription

1 Joural of Statistical ad Ecoometric Methods, vol.4, o.4, 015, ISSN: (prit), (olie) Sciepress Ltd, 015 Wid speed aalysis with the upper trucated quasi Lidley distributio Emrah Altu 1,* ad Gamze Ozel Abstract We propose a ew distributio with extra shape parameter amed a trucated quasi Lidley distributio which is more flexible tha may well-kow distributios. Mathematical ad statistical properties of trucated quasi Lidley distributio are give oly for upper trucated versio of quasi Lidley distributio to reduce the mathematical complexity. A importat property of the upper trucated quasi Lidley distributio is that it ca have bathtub-shaped failure rate fuctio. We preset some of its mathematical properties icludig ordiary momets, quatile ad momet geeratig fuctios, order statistics. The method of maximum likelihood to estimate the model parameters is discussed ad the behavior of maximum likelihood estimator is studied. The importace of the ew distributio is illustrated by meas of the wid speed data ad the capability i modelig wid speed is evaluated. The results idicate that the trucated versio of quasi Lidley distributio ca provide better fits tha Expoetial, 1 Departmet of Statistics, Hacettepe Uiversity, 06800, Akara, Turkey. emrahaltu@hacettepe.edu.tr (E. Altu). Departmet of Statistics, Hacettepe Uiversity, 06800, Akara, Turkey. * Correspodig author. Article Ifo: Received : September 7, 015. Revised : October 1, 015. Published olie : December 1, 015.

2 18 Wid speed aalysis with the upper trucated quasi Lidley distributio Lidley, quasi Lidley ad Weibull distributios i estimatig wid speed distributio. Therefore, trucated quasi Lidley ca be a alterative for use i the assessmet of wid eergy potetial. Mathematics Subject Classificatio: 60E05; 60E10; 6G30; 6P30 Keywords: Trucated Quasi Lidley Distributio; Maximum Likelihood Estimatio; Quatile Fuctio, Order Statistics; Wid Speed 1 Itroductio The Lidley was itroduced by Lidley [1] as a ew distributio useful to aalyze failure time data especially i applicatios modelig stress-stregth reliability. The motivatio of the Lidley distributio arises from its ability to model failure time data with icreasig, decreasig, uimodal ad bathtub shaped hazard rates. The distributio represets a good alterative to the expoetial failure time distributio that suffer from ot exhibitig uimodal ad bathtub shaped failure rates []. Ghitay et al. [3] obtaied some properties of the Lidley distributio ad showed that the Lidley distributio provides a better modelig for some applicatios tha the expoetial distributio. Mazucheli ad Achcar [4] also foud that may of the mathematical properties are more flexible tha those of the expoetial distributio ad proposed the Lidley distributio as a possible alterative to the expoetial distributio. The probability distributio fuctio (pdf) ad cumulative distributio fuctio (cdf) of the Lidley distributio are give by, respectively, θ 1+θ -θx f L(x) = (1+ x)e, x > 0, θ> 0 θ+ 1+θx θ+ 1 -θx F L(x) = 1- e, x> 0, θ> 0 (1)

3 Emrah Altu ad Gamze Ozel 19 The Lidley distributio is a mixture of expoetial ( θ ) ad gamma (, θ ) distributios with their mixig proportios are (1 (1 θ ) + ad ( θ ( 1 θ) ) +. Let us poit out that some researchers proposed ew classes of distributios based o modificatios of the Lidley distributio, icludig also their properties. The geeralized Lidley distributio was proposed i Zakerzadeh ad Dolati [5]. The Poisso-Lidley distributio was itroduced by Sakara [6]. The zero-trucated Poisso-Lidley distributio ad geeralized Poisso-Lidley distributio were cosidered i Ghiaty et al. [7]. Recetly, Shaker ad Mishra [8] have itroduced a two parameter quasi Lidley (QL) distributio which is the particular case of the Lidley distributio. They showed that the QL distributio provides better fits tha the Lidley distributio based o differet data sets. The QL distributio is defied by its pdf ad cdf as, respectively, θ( α+ x θ) α+ 1 θx f QL(x) = e, x > 0, θα>, 0 1+α+θx = > θα> α+ 1 θx F QL(x) 1 e, x 0,, 0 I Eq. (), the QL distributio reduces to the Lidley ad gamma (, θ ) distributios for α = θ ad α = 0, respectively. A trucated distributio is defied as a coditioal distributio that results from restrictig the domai of the statistical distributio. Hece, trucated distributios are used i cases where occurreces are limited to values which lie above or below a give threshold or withi a specified rage. If occurreces are limited to values which lie below a give threshold, the lower (left) trucated distributio is obtaied. Similarly, if occurreces are limited to values which lie above a give threshold, the upper (right) trucated distributio arises [9, 10]. Trucated versios of the well-kow statistical distributios are proposed by may researchers to model the trucated data i various fields. Zhag ad Xie [11] studied the characteristics of the trucated Weibull distributio ad illustrated the applicability of this distributio to modelig lifetime data. Ahmed et al. [1] ()

4 0 Wid speed aalysis with the upper trucated quasi Lidley distributio proposed the trucated versio of the Birbaum-Sauders (BS) distributio ad showed that trucated BS distributio is more appropriate tha the classical BS model for describig the fiacial loss data from a commercial bak. Recetly, Sigh et al. [13] have itroduced the trucated versio of the Lidley distributio ad discussed statistical properties of proposed distributio ad showed that trucated versio of the Lidley distributio provides a better modelig tha Weibull, Lidley ad expoetial distributios based o a real data. As far as we kow, trucated versio of the QL distributio has ot bee ivestigated. Therefore, the aim of this study is to obtai a trucated QL distributio which is more flexible tha the expoetial, Lidley, quasi Lidley, ad lower trucated quasi distributios. Accurately modelig wid speed is critical i estimatig the wid eergy potetial of a certai regio. Several statistical distributios have bee studied i order to model wid speed data smoothly. The Erlag, iverse ormal ad Gumbel distributios preseted as wid speed distributios i [7], while a geeralized extreme value distributio was used i [14]. Chag [15] itroduced mixture trucated ormal distributios, while Usta ad Katar [16] proposed certai flexible families of distributios as a alterative to the expoetial distributio i estimatig wid speed distributio. Philippopoulos et al. [17] compares various distributios ad shows that the gamma distributio could be a efficiet alterative to the expoetial distributio. Cosequetly, the metioed studies [18 1] emphasize that expoetial distributio does ot preset good performace i the modelig of wid speed data i compariso for all wid types ecoutered i ature, such as low or high, skewed or kurtotic, or skewed ad kurtotic wid speed. Thus, i order to miimize errors i wid speed estimatio, it is ecessary to select the most appropriate distributio for the descriptio of wid speed measured for a specific area. A upper trucated distributio is applicable to this situatio where the rage of radom variable is bouded from above by a ukow cut-off poit, called a trucatio poit. I other words, if the values of

5 Emrah Altu ad Gamze Ozel 1 radom variables are observed i the iterval [0, T], the the upper trucated distributio ca be used. The wid speed measuremets are geerally observed i the rage of [0, V], ad therefore, the upper trucated distributio ca be applied to model the wid speed data. Therefore, this paper also proposes, for the first time, the use of the upper trucated QL (UTQL) distributio, i modelig wid speed data. I additio, a compariso is made betwee UTQL distributio ad well kow distributios usig wid speed data. The paper is orgaized as follows: Sectio itroduces the probability desity, cumulative desity, survival ad hazard fuctios of UTQL distributio. The the plots of the proposed distributio for several values of parameters are preseted. I Sectio 3, some statistical properties of the UTQL distributio, such as ordiary momets, momet geeratig ad quatile fuctios, skewess ad kurtosis, order statistics are derived ad mea ad variace for the several values of the parameters are obtaied. Sectio 4 provides the maximum likelihood estimatio of the model parameters. The wid speed data is used to evaluate the performace of trucated versios of the UTQL distributio i Sectio 5. Sectio 6 cocludes the study with the obtaied results ad certai suggestios for further research. Upper Trucated Quasi Lidley Distributio.1. The probability desity ad cumulative desity fuctios The cdf of a double trucated distributio is give by F(x) F(v) G(x) =, v< x <ζ, < v<ζ< (3) F( ζ) F(v) where v ad ζ are the itervals of the trucated distributio. The pdf of the trucated distributio is defied as

6 Wid speed aalysis with the upper trucated quasi Lidley distributio f(x) g(x) =, v< x <ζ, < v<ζ< F( ζ) F(v) where f(x) ad F(x) are the pdf ad the cdf of the baselie distributio, respectively. It is easily see from Eq. (4) that the distributio reduces the baselie distributio for v = 0 ad ζ. Eq. (4) is called as the upper trucated distributio for ay baselie distributio for v = 0. If f QL(x) i Eq. () is cosidered as a baselie distributio ad usig the Eqs. (3) ad (4), we obtai the pdf ad the cdf of the UTQL distributio respectively as follows: θx θ( α+θx)e g UTQL(x) =, x> 0, θ, α> 0, 0< x<ζ θζ ζθ (1 +α)(1 e ) +ζθe θ (( +α)( ) +θ ) θ ( ζ e x) 1 1 e x x G UTQL(x) =, x> 0, θ, α> 0, 0< x<ζ ζθ (1 +α)(1 e ) +ζθ The desity fuctio i Eq. (5) is much more flexible tha the Lidley ad quasi Lidley desity fuctios. Thus it ca allow for greater flexibility of the tails. It ca exhibit differet behavior depedig o the parameter values. Note that the probability desity i Eq. (5) is reduced the quasi Lidley distributio for ζ = 0. To obtai the mode of the UTQL distributio, we give the first derivate of (5) as ( ) θx g UTQL(x) e x 1 ζθ θ α+θ = x α + 1 e 1 + ζθe ζθ ( )( ) (4) (5) (6) (7) Solvig the Eq. (7) for zero poit, gives 1 α x =. Whe a < 1, θ 1 α x = is θ the maximum poit of the distributio. The, the mode of the UTQL distributio ca be defied as 1 α, α< 1 Mode = θ (8) 0, o/w Figure 1 illustrates some of the possible shapes of the pdf of the UTQL distributio for selected values of the parameters.

7 Emrah Altu ad Gamze Ozel 3 Figure 1: Plots of the UTQL desity fuctios for give θ = 0.5, 1, 1.5 α = 0.5, 1, 1.5 ad ζ=10 As see from Figure 1, decreasig θ parameter yields to fat-tail structure ad θ parameter specifies the shape of the distributio. I fact, plots i Figure 1 reveals that the mode of the pdf icreases as θ. It is evidet that the UTQL distributio is much more flexible tha the Lidley distributio, i.e. the additioal shape parameter θ allow for a high degree of flexibility of the UTQL distributio. Hece, the ew model ca be very useful i may practical situatios for modelig positive real data sets such as wid speed data... Survival ad Hazard Rate Fuctios Cetral role is playig i the reliability theory by the ratio of the probability

8 4 Wid speed aalysis with the upper trucated quasi Lidley distributio desity ad survival fuctios. The survival fuctio of the UTQL distributio is obtaied from Eq. (6) as θ (( +α)( ) +θ ) θ ( ζ e x) 1 1 e x x S(x) = 1 G UTQL (x) = 1 ζθ (1 +α)(1 e ) +ζθ Figure represets plots of the survival fuctio for the UTQL distributio for several parameter values α, θ ad ζ. (9) Figure : Plots for the survival fuctios of UTQL distributio for give θ = 0.5, 1, 1.5 α = 0.5, 1, 1.5 ad ζ=10 The other characteristic of iterest of a radom variable is the hazard rate fuctio. It ca be loosely iterpreted as the coditioal probability of failure, give it has survived to time t. We obtai the hazard fuctio of the UTQL distributio as

9 Emrah Altu ad Gamze Ozel 5 h(x) θx θ( α+θx)e g (x) (1 +α)(1 e ) +ζθe θ (( )( ) ) +α +θ 1 θζ (1 +α)(1 e ) +ζθ θζ ζθ UTQL = = 1 G ( x) x UTQL (x) θ ζ e 1 1 e x Figure 3 illustrates the hazard rate fuctio of the UTQL distributio for differet values of the parameters α, θ ad ζ. Figure 3: Plots for the hazard fuctios of the UTQL distributio for give θ = 0.5, 1, 1.5 α = 0.5, 1, 1.5 ad ζ=10 Figure 3 shows that the hazard rate fuctio of the UTQL distributio is flexible for several values of parameters. It is see i Figure 3 that the hazard rate fuctio are icreasig.

10 6 Wid speed aalysis with the upper trucated quasi Lidley distributio 3 Mai Properties We derive computatioal sum represetatios ad explicit expressios for the ordiary ad cetral momets, skewess, kurtosis, geeratig ad quatile fuctios, order statistics of X. These expressios ca be evaluated aalytically or umerically usig packages such as Mathematica, Matlab ad Maple Momets Some key features of a distributio such as skewess ad kurtosis ca be studied through its momets. We derive closed-form expressios for the ordiary momets, geeratig fuctio, skewess ad kurtosis of X. We obtai the ordiary momets of the UTQL distributio as ζ r θ r θx E(X ) = x ( α+θx)e dx ( α+ 1) F( ζ) 0 ( ) ( ) r ( ) Γ r+ 1 Γ r+ 1, ζθ 1 = θ α+ 1 F( ζ) where Γ (.) ad Γ (.,.) are gamma ad upper icomplete gamma fuctios, respectively, ad defied as follows: s 1 x (s) x e dx, 0 (10) Γ = (11) s 1 x (s, t) x e dx. t Γ = From Eq. (10), the mea ad variace of the UTQL distributio are obtaied usig the first ad the secod ordiary momets as ζθ ζθ α e + ζθ α e + ζ θ + αζθ + E(X) =µ=, ζθ ζθ θ( α e + ζθ α e + 1) (1)

11 Emrah Altu ad Gamze Ozel 7 E(X ) = ζθ ζθ 3 3 α 6e + 6ζθ α e + 3ζ θ + ζ θ + αζ θ + αζθ + 6 ζθ ζθ θ ( α e +ζb α e + 1). The mea ad variace ( σ ) of the UTQL distributio are preseted i Table 1 for differet values of parameters α, θ ad ζ. Note that variace { } V(X) E(X ) E(X) =σ = is foud from Eq. (1). Table 1: Mea ad variace of the UTQL distributio for several values of the θ parameters α ξ = 5 ξ = 10 ξ = 15 µ σ µ σ µ σ Table 1 shows that the mea ad variace decrease whe θ icreases for a fixed α. Besides, for a fixed θ, while mea decreases, variace icreases whe the α icreases. Assumig α ad θ is kept fix, the mea ad variace icrease wheξ icreases.

12 8 Wid speed aalysis with the upper trucated quasi Lidley distributio Further, the cetral momets ( µ r ) ad cumulats ( κ r ), r = 1,,..., of the UTQL distributio ca be obtaied from r k r r r ( 1) 1 r k k k= 0 µ = µ µ ad r k r r r r ( 1) 1 r k k= 0 κ =µ µ µ (13) Here κ 1 =µ 1, κ =µ µ, κ =µ µµ+ µ, κ =µ µµ µ + µµ µ etc. The skewess 3/ 1 3/ γ =κ κ ad kurtosis cumulats. 4 / γ =κ κ are also computed from the secod, third ad fourth 3.. Momet Geeratig Fuctio The momet geeratig fuctio (mgf) is widely used as a alterative way to aalytical results compared with workig directly with pdf ad cdf. Here, we give a formula for the mgf x ξt ξθ αθ( e 1) ξθ ξθ ( θ t) (( α + 1)( e 1) + ξθe ) M (t) = tx M(t) = E(e ) of X as ξ ξθ ( 1 ( t) ( e ( t 1) ) ( t) ) t θ θ ξθ ξ + θ ξθ ( )( ) α + 1 e 1 + ξθe Usig the Eq. (14), the ordiary momets of the UTQL distributio ca be also calculated. ξθ (14) 3.3. Quatile Fuctio The quatile fuctio of a probability distributio is the iverse of the cdf ad defied as follows: Q(p) = if{x : p F(x)} = 1 F (x) (15) We obtai the quatile fuctio of the UTQL distributio from Eqs. (6) ad

13 Emrah Altu ad Gamze Ozel 9 (15) as Q UTQL ξθ (1 +α+ξθ) ( ( ) ) 1+ α +W (1 +α) e 1 p p e + (p) = θ where W(.) is a Lambert-W fuctio which is a multi-valued complex fuctio defied as the solutio of the equatio W(z) exp[w(z)] = z (17) Here, z is a complex umber [1]. Simulatig UTQL distributio radom variable is straightforward. Let p be a uiform variable o the uit iterval (0, 1). Thus, by meas of the iverse trasformatio method, the radom variable X give by ξθ (1 +α+ξθ) ( ( ) ) 1+ α +W (1 +α) e 1 p p e + X = θ (16) (18) 3.4. Skewess ad Kurtosis The effects of the parameters o the skewess ad kurtosis of X ca be based o quatile fuctio i Eq. (16). There are may heavy tailed distributios for which this measure is ifiite. So, it becomes uiformative precisely whe it eeds to be. The Bowley s skewess is based o quartiles: ad the Moors kurtosis is based o octiles: Q(3 / 4) Q(1/ ) + Q(1/ 4) S = Q(3 / 4) Q(1/ 4) Q(7 / 8) Q(5 / 8) Q(3 / 8) + Q(1/ 8) K =, Q(6 / 8) Q( / 8) where Q (.) represets the quatile fuctio of X. These measures are less sesitive to outliers ad they exist eve for distributios without momets. Skewess measures the degree of the log tail ad kurtosis is a measure of the

14 30 Wid speed aalysis with the upper trucated quasi Lidley distributio degree of tail heaviess. Whe the distributio is symmetric, S= 0 ad the whe the distributio is right (or left) skewed, S> 0 (or S < 0). As K icreases, the tail of the distributio becomes heavier. From Eq. (16), skewess ad kurtosis of the UTQL distributio are obtaied ad preseted i Table. Table : Skewess ad kurtosis of the UTQL distributio for several values of the θ parameters α ξ = 5 ξ = 10 ξ = 15 Skewess Kurtosis Skewess Kurtosis Skewess Kurtosis The values i Table idicate a arrow rage for the skewess of X, similary, the kurtosis does ot vary much.

15 Emrah Altu ad Gamze Ozel Order Statistics Order statistics make their appearace i may areas of statistical theory ad practice. Suppose X 1,X,...,X is a radom sample from the UTQL distributio. Let 1:, :,..., : X X X be a ordered statistics ad g : () t represets the pdf of the s X s : s+ i s 1 i s UTQL θ UTQL = i= 1 i UTQL UTQL G (t, ) g (t, θ) g s: (t) ( 1) B(s, s+ 1) G ( ξθ, ) G (t, θ) where Bs (, s+ 1) is beta fuctio, g UTQL(t) ad g UTQL(t) are the pdf ad cdf of the UTQL distributio, respectively. For s = 1, the pdf of the first order statistics is defied as t (( +α)( ) +θ ) ζθ θt θ θt ( t)e e e 1 1 e t θ α+θ θζ ζθ ζθ s (1 +α )(1 e ) +ζθ e (1 +α )(1 e ) +ζθ i g 1: (t) = ( 1) (19) i+ 1 i= 1 ζθ θγ θγ e e (( 1+α)( 1 e ) +θγ) B( i, i + 1) ζθ (1 +α )(1 e ) +ζθ Similarly, the pdf of X : ca be obtaied by takig s=. i 4 Parameter Estimatio Several approaches for parameter poit estimatio were proposed i the literature but the maximum likelihood estimatio (MLE) method is the most commoly employed. The MLEs ejoy desirable properties ad ca be used whe costructig cofidece itervals ad regios ad also i test statistics. I this sectio, the parameter estimatio of the UTQL distributio is obtaied usig the maximum likelihood estimatio (MLE) procedure. Let x 1, x,... x be a radom sample from the UTQL distributio. The likelihood fuctio of the UTQL

16 3 Wid speed aalysis with the upper trucated quasi Lidley distributio distributio is give as θ x i θ i1 = θζ θζ i (1 )(1 e ) e (0) + α θζ i= 1 L( α, θ, ζ x) = ( α+ x θ)e ad the log-likelihood fuctio is obtaied as θζ θζ ( )( ) i i (1) l L( α, θ, ζ x) = l( θ) l 1+α 1 e θζ e + l( α+ x θ) θ x i= 1 i= 1 The log-likelihood fuctio ca be maximized either directly by usig the SAS (PROCNLMIXED) or the Ox program (sub-routie MaxBFGS) or by solvig the oliear likelihood equatios obtaied by differetiatig Eq. (1). Here, the log-likelihood fuctio is maximized by solvig oliear equatios obtaied by differetiatig the log- likelihood fuctio. The first derivatives of the log-likelihood fuctio with respect to the parameters are l L( α, θ, ζ x) ξ e (1 +α) ξ e (1 +ξθ) x ξθ θ θ ( ) α + θ ξθ ξθ i = + + x ξθ i () (1 + α) 1 e ξθe i= 1 xi i= 1 θξ l L( αθζ,, x) 1 e 1 = (1 )(1 e θξ ) e θξ + (3) α + α ξθ i= 1α + x iθ The give equatios ca be solved usig iterative methods such as Newto ˆ ξ = max x,,..., 1 x x, ˆ ξ is the Raphso method. The MLE of ξ is take as { } largest value i sample. Let θ 0 ad α 0 be the iitial values of θ ad α, respectively. The, we have log L log L log L θ θα ˆ θ θ 0 θ = log L log L ˆ α α log L 0 θα α α where (4)

17 Emrah Altu ad Gamze Ozel 33 l L( α, θ, ζ x) ( ζ e +ζe ( α+ 1) ζ θe ) = + ζθ ζθ ζθ ζθ ζθ (( 1) ( e 1) e ) θ θ α + + ζθ ( ζ θe ζ e +ζ e ( α+ 1)) 3 ζθ ζθ ζθ i ζθ ζθ (( α+ 1) ( e 1) +ζθe ) i= 1( α+θxi ) x, (5) ζθ ll( αθζ,, x) (e 1) 1 =, (6) α ( α + θx) ζθ ζθ (( α + 1)( e 1) + ζθe ) i= 1 i = θ α θζ ( 1 )( 1 e ) + α ζθe xi +. ( α+ x θ) ( )( ) ζθ ζθ ζθ ζθ ζθ ζθ l L( αθζ,, x) e ζ 1+α 1 e ζθe (1 e ) ζ θe e ( ζ+α) ζθ i= 1 i (7) 5 Applicatio I this sectio, we provide applicatios to a real data set to demostrate the potetiality of the trucated versios of quasi Lidley distributio. A set of real test data represetig wid speed reported by [] is used to evaluate the performace of the trucated quasi Lidley distributio. It cosists of the observatios listed i Table 3. As see from Table 3, each data poit represets the average wid speed over some time period. The, the descriptive statistics for the wid speed data are give i Table 4. Table 4 shows that the wid speed data is positively skewed.. Accordig to kurtosis value, wid speed data distributio is flatter tha a ormal distributio with a wider peak. This meas that the probability for extreme values of wid speed is less tha for a ormal distributio, ad the values are wider spread aroud the mea. Here, we have fitted the data with the expoetial, Lidley, QL, upper, lower ad double trucated Quasi Lidley distributios. We obtai the pdf of the lower trucated Quasi Lidley (LTQL) ad double trucated Quasi Lidley

18 34 Wid speed aalysis with the upper trucated quasi Lidley distributio (DTQL) distributio as θ(v x) θe ( α+θx) g LTQL (x) = ; x > v; θ, α> 0 ( α+θ v + 1) DTQLD θx θe ( α + θx) ζθ vθ e ( α+ζθ+ 1) e ( α+ vθ+ 1) g = ; v, ζ, θ, α> 0 (8) (9) Table 3: Wid speed data (m/s) Table 4: Descriptive statistics for the wid speed data Miimum Media Mea Maximum Variace Skewess Kurtosis The log-likelihood fuctio of UTQL ad DTQL ca be obtaied usig the Eq. (8) ad (9) to estimate the parameters of UTQLD ad DTQL distributios.

19 Emrah Altu ad Gamze Ozel 35 We estimate the ukow parameters of the distributios by the maximum likelihood. Table 5 represets the fittig summary of these distributios icludig the estimates of parameters, log-likelihood, Akaike iformatio criterio (AIC), Corrected Akaike iformatio criterio (AICC), ad Bayesia iformatio criterio (BIC) which are calculated by give equatios, AIC = log(l) + k, k(k + 1) AICC = AIC + (30) ( k 1) BIC = log(l) + k log() where k is the umber of parameters ad is the sample size. Accordig to the AIC, AICC ad BIC statistics values, the DTQL distributio gives the best fittig to the wid speed data. Figure 4 represets the fittig performace of the DTQL distributio graphically, icludig the quatile-quatile (Q-Q) plot, probability-probability (P-P) plot, empirical ad theoretical desities ad empirical ad theoretical cumulative distributio fuctios. Table 5: Maximum likelihood estimates, AIC, AICC, BIC statistics values uder cosidered distributios based o wid speed data Distributio Parameter Estimatios LogL AIC AICC BIC Exp( λ ) ( ˆ λ ) = , , ,9 381,0941 W(, ) k λ ( ˆ ˆ) k, λ = (.93, 5.75) 146,496 96,991 97,17 96,7059 L( θ ) ( ˆ θ ) = ,55 351, ,16 350,9613 QL ( θα, ) ( ˆ θα, ˆ ) = ( 0.390, ) 166,46 336,94 337,1 336,6387 UTQL( θαζ,, ) ( ˆ θαζ, ˆ, ˆ) = (0.69, 0.013,10.4) 156, ,73 319,08 318,304 LTQL(,, v) θα ( ˆ ˆ v ˆ ) DTQL(,, v, ) θα ζ ( ˆ ˆ vˆ ˆ) θα,, = (0.390, 0.000, ) 166,46 338,94 339,8 338,496 θα,,, ζ = (0.387, 0.014,,10.4) 143,643 95,86 95,88 94,7153

20 36 Wid speed aalysis with the upper trucated quasi Lidley distributio Desity Empirical ad th Empirical quatiles Q-Q plot Data Theoretical quatiles CDF Empirical ad th Empirical probabilities P-P plot Data Theoretical probabili Figure 4: Fittig performace of the DTQL distributio based o wid speed data Figure 4 shows that the wid speed data fits very well to the DTQL distributio. 6 Coclusio I this study we propose a ew model, the so-called the distributio which exteds the Lidley distributio i the aalysis of data with real support. A obvious reaso for obtaiig a stadard distributio is because the geeralized form provides larger flexibility i modelig real data. We derive expasios for the momets, quatile fuctio, order statistics, survival fuctio, hazard fuctio, momets ad for the momet geeratig fuctio. The estimatio of parameters is approached by the method of maximum likelihood, also the iformatio matrix is derived. The wid speed data is modeled by the lower, upper ad double trucated quasi Lidley distributios, expoetial, Lidley, quasi Lidley ad Weibull distributios to evaluate the performace of the trucated versios of quasi Lidley distributio. Fittig performace of these distributios are compared

21 Emrah Altu ad Gamze Ozel 37 accordig to AIC, AICC ad BIC statistics values ad clearly the wid speed data is best modeled by DTQL distributio. The preset study might provide differet ad useful isights to scietists dealig with wid eergy by firstly itroducig a trucatio parameter ito QL distributio. Refereces [1] Lidley, D.V., Fiducial distributios ad Bayes theorem, J. Royal Stat. Soc. Series B, (1958), 0, [] Bakouch, H.S., Bader, M.A., Al-Shaomrai, A.A. Marchi, V.A.A. ad Louzada, F., A exteded Lidley distributio, Joural of The Korea Statistical Society, 41, (01), [3] Ghitay, M.A., Atieh, B. ad Nadarajah S., Lidley distributio ad its Applicatios, Math. Cmput. Simul., 81(11), (008), [4] Mazucheli, J. ad Achcar, J.A., The Lidley distributio applied to competig risks lifetime data, Computatio Methods Programs, 104, (011), [5] Zakerzadeh, H. ad Dolati, A., Geeralized Lidley distributio, Joural of Mathematical Extesio, 3(), (009), [6] Sakara, M., The discrete Poisso Lidley distributio, Biometrics, 6, (1970), [7] Ghitay, M.E., Al-Mutairi, D.K. ad Nadarajah, S., Zero-trucated Poisso Lidley distributio ad its applicatio, Mathematics ad Computers i Simulatio, 79, (008), [8] Shaker, R. ad Mishra, A., A quasi Lidley distributio, Africa Joural of Mathematics ad Computer Sciece Research, 6(4), (013), [9] Dusit C. ad Cohe A.C., Estimatio i the sigly trucated Weibull distributio with a ukow trucatio poit, Commu. Statist-Theory Meth., 13(7), (1984),

22 38 Wid speed aalysis with the upper trucated quasi Lidley distributio [10] Zhag T. ad Xie M., O the upper trucated Weibull distributio ad its reliability implicatios, Reliability Egieerig ad System Safety, 96, (011), [11] Ahmed, S.E., Castro-Kuriss, C., Leiva, V. ad Sahueza, A., A trucated versio of the Birbaum-Sauders distributio with a applicatio i fiacial risk, Pakista Joural of Statistics, 6(1), (010), [1] Sig, S. K., Sig, U. ad Sharma, V. K., The Trucated Lidley Distributio: Iferece ad Applicatio, Joural of Statistics Applicatios & Probability, 3(), (014), [13] Zhou J., Erdem E., Li G. ad Shi J., Comprehesive evaluatio of wid speed distributio models: A case study for North Dakota sites, Eergy Covers Maage, 51(7), (010), [14] Bauer E., Characteristic frequecy distributios of remotely sesed i situ ad modeled wid speeds, It. J. Climatol, 16, (1996), [15] Chag T.P., Estimatio of wid eergy potetial usig differet probability desity fuctios, Appl. Eergy, 88(5), (011), [16] Usta I. ad Katar Y.M., Aalysis of some flexible families of distributios for estimatio of wid speed distributios, Appl. Eergy, 89(1), (01), [17] Philippopoulos K., Deligiorgi D. ad Karvouis G., Wid speed distributio modelig i the Greater Area of Chaia, Greece, It. J. Gree Eergy, 9(), (01), [18] Akdag S.A., Bagiorgas H.S. ad Mihalakakou G., Use of two-compoet Weibull mixtures i the aalysis of wid speed i the Easter Mediterraea, Appl. Eergy, 87(8), (010), [19] Carta J.A. ad Ramirez P., Use of fiite mixture distributio models i the aalysis of wid eergy i the Caaria Archipelago, Eergy Covers Maage, 48, (007),

23 Emrah Altu ad Gamze Ozel 39 [0] Soukissia T., Use of multi-parameter distributios for offshore wid speed modelig: The Johso SB distributio, Appl. Eergy, 111, (013), [1] Morga V.T., Statistical distributios of wid parameters at Sydey, Australia, Reew Eergy, 6(1), (1995), [] Seguro, J.V. ad Lambert, T.W., Moder estimatio of the parameters of the Weibull wid speed distributio for wid eergy aalysis, Joural of Egieerig ad Idustrial Aerodyamics, 85, (000),