Kurtosis Statistics with Reference to Power Function Distribution

Transcription

1 ISSN Journal o Statistis Volume, 06. pp. -0 Abstrat Kurtosis Statistis with Reerene to Power Funtion Distribution Azaz Ahmad and Ahmed Saeed Akhter Pearson statistis o skewness and kurtosis gave alse impression to assess the peakedness and tailedness or skewed (moderately, J-shape or reverse J-shape) Distributions. A number o alternate measures were suggested in literature by Hosking (99), Blest (00), Elamir and Seheult (00), and Fiori and Zenga (005) that provided better interpretation than the Karl Pearson statistis. Power Funtion Distribution has the harateristis o symmetri, J-shape or reverse J- shape with varying magnitude o its shape parameter. In this paper, we derived the Blest s statistis o skewness and kurtosis, L-skewness and L-kurtosis and Trimmed L-skewness and Trimmed L-kurtosis or Power Funtion Distribution. Comparison is made with Karl Pearson statistis. Keywords Power untion distribution, Blest s measure, L-moments, Trimmed L-moments. Introdution The oldest and the most ommon measure o skewness and kurtosis is the standard ourth moment by Pearson (905). These measures oten onentrate only on symmetri Distribution. It does not provide true inormation about peakedness and tailedness or skewed Distribution. Government Postgraduate College, Okara, Pakistan azazpu@yahoo.om College o Statistial and Atuarial Sienes, University o the Punjab, Quaid e Azam Campus, Lahore, Pakistan asakther@yahoo.om

2 Azaz Ahmad and Ahmed Saeed Akhter Poor perormane o standardized entral moment o skewness and kurtosis may lead to seek alternate statistis and deinition to study the Distribution shape harateristis. Inluded in these Blest s measures (Blest, 00), L-skewness and L-kurtosis (Hosking, 99) and TL-skewness and TL-kurtosis (Elamir and Seheult, 00). Rahila and Memon (0) also onduted suh kind o omparison or Weibull Distribution but they just ompared the Pearson with Blest s measure. Both Blest s and Karl Pearson measures are based on higher moments o a Distribution. So these measures an t evaluate or those Distributions like Cauhy and Inver Rayleigh Distribution whose higher moments do not exists. So how an we study the desription o suh Distributions? Answer is obtained by evaluating another alternate measures, the L-moments and Trimmed L-moments. So in our study, we omputed also L-skewness and L-kurtosis, Trimmed L-skewness and Trimmed L- kurtosis or Power Funtion Distribution with the Blest s measures and Pearson. Statistial Distributions have long been employed in the assessment o semiondutor devie and produt reliability. The use o the Exponential Distribution whih is requently preerred over mathematially more omplex Distributions, suh as the Weibull and the Lognormal among others, suggest that most engineers avor the appliation o simpler models to obtain ailure rates and reliability igures quikly. It is, thereore, proposed that the Power Funtion Distribution be onsidered as a simple alternative whih, in some irumstanes, may exhibit a better it or ailure data and provide more appropriate inormation about reliability and Hazard Rates (Menioni,996). The Distribution Funtion o Power Funtion Distribution as x F( x) 0 x b shape parameter sale parameter b 0 (.). Skewness and Kurtosis Statistis or Power Funtion Distribution. Karl Pearson Statistis o Skewness and Kurtosis: Karl Pearson measure o skewness is the standard third moment oeiient as

3 Kurtosis Statistis with Reerene to Power untion distribution X E standard ourth moment oeiient or kurtosis as X E We evaluated or Power Funtion Distribution as b r r x E( x ) x. dx b r r b Ex ( ) r ( ) 0 b b ( ) b ( ) ( ) ( ) b b b b b b b ( )( ) ( ) () ( ) () ( ) ( )( ) b b ( ) ( ) ( )( ) ( ) ( )( ) 5 (..) (..) (..) (..) (..5)

4 6 Azaz Ahmad and Ahmed Saeed Akhter 6 ( ) ( ) b b b b 6 ( )( ) ( ) ( ) ( ) b ( ) ( )( ) ( ) ( )( ) ( ) ( )( )() 6 ( ) ( )( ) ( )( )( ) b ( ) ( )( )( ) Now using eq. (..) to eq. (..6), we obtain Karl Pearson measures o skewness and kurtosis respetively as ( )( ) ( )( ) ( )( ) (..6) (..7) (..8) ( )( ) It is interesting to note that the third and ourth moment about mean are the untion o sale and shape parameters but the measures o skewness and kurtosis are untion o shape parameter only. Power Funtion Distribution is symmetri or = and negatively skewed or > and positively skewed or <. The Figure shows the skewness and kurtosis or Power Funtion Distribution with hanging the values o shape parameter. The Figure indiates the oeiient o kurtosis has a value at two dierent points = 0. and =.87. For = 0. the Distribution is positively skewed (.099 ) or =.87 the Distribution is negatively skewed ( ) or = the Distribution is symmetri. As we noted that skewness dereased or inreasing value o shape parameter, kurtosis also dereased rapidly or < but or > it inreased gradually. We are ousing the Distribution in three ases = > <

5 Kurtosis Statistis with Reerene to Power untion distribution. Blest s Statistis or Kurtosis: Blest suggested a new measures o kurtosis by whih the eet o any skewness is deleted, allowing omparison o Distribution on the basis o kurtosis alone (Blest, 00). Blest proposed a new measure o entral tendeny alled Meson (rom Greek mesos, meaning middle ) is denoted by and the standardized value o meson is meson is zero i.e E X 0 Ex ( ) r r 7, third moment about and the r th moment about meson are deined as So irst our moment about meson are Ex ( ) E( x ) E(x ) [ ] (..) Ex ( ) E( x ) E ( x ) ( ) E ( x ) ( x )( ) E( x ) E( x ) 0 ( ) Ex ( ) (..) E x ( ) [ ] E (x ) E (x ) (x ) ( ) (x )( ) E(x ) ( ) E((x ) E(x ) (..)

6 8 Azaz Ahmad and Ahmed Saeed Akhter Ex ( ) E( x ) E ( x ) ( ) E ( x ) ( x ) ( ) 6( x ) ( ) ( x )( ) ( ) E ( x ) E ( x ) 6 E( x ) E( x ) ( ) 6 0 ( ) 6 6 ( 6 ) ( ) (..) 6 Blest s proposed the ollowing Moment Ratio (..5) (..6) So rom the eq. (..), we evaluate third moment as E X 0 so, 0 (..7) so, the real roots o this equation is the measure o skewness in term o standardized value o meson.

7 Kurtosis Statistis with Reerene to Power untion distribution 9 put y in (..7) y y y 0 y y y y y 0 y y y Let, y 0 y ( y ) y 0 ( y ) y 0 Put y w w it redued in quadrati equation w 0 b b a w a a, b, ()( ) w () w w w (..8)

8 0 Azaz Ahmad and Ahmed Saeed Akhter w w w , w put bak into y w y w so / y y / / The trial solution aomplishing this mirale turns out to be the symmetrial expression as / / (..9) Blest s oeiient o skewness is derived or Power Funtion Distribution by putting the eq. (..) into eq. (..9) as / / ( )( ) ( )( ) / / ( ) ( ) (..0) / / ( )( ) ( )( ) / / ( ) ( ) is the untion o only shape parameter. And oeiient o kurtosis is adjusted or skewness, as the standardized ourth moment about the Meson is obtained by taking the eq. (..).

9 Kurtosis Statistis with Reerene to Power untion distribution ( ) (( ) ( )) {( ) ( ) () () } {( ) () } (( ) ) (..) From Figure, it is lear that both measures have same interpretation or symmetri Distribution. We an see that how Karl Pearson oeiient o skewness gives alse impression about skewness or Case II and III. Figure shows that or symmetri Distribution both measures o kurtosis are same but or skewed Distribution Blest measures always less than the Pearson kurtosis. The gap between two is smaller in Case II and it inreased in Case III.. Relation among Meson, Mean and Median: The gap between median and meson is another way measuring the degree o kurtosis. b Mean (..) b Median (..) / / b / ( )( ) Meson (..) / / ( )( ) ( )( ) / / / ( ) ( ) b b / ( )( ) / / ( )( ) ( )( ) / / ( ) ( ) (..)

10 Azaz Ahmad and Ahmed Saeed Akhter So the Figure shows the relation between three measure o entral tendeny or b= It is lear rom the Figure that Meson = Mean = Median or Case I Meson >Mean> Median or Case II Meson <Mean< Median or Case III So the Distributions or Case III are ar latter than Distributions in Case II.. L-kurtosis: L-moments are expetations o ertain linear ombinations o Order Statistis. L-moment exist or a real valued random variable X, i X has a inite mean. A Distribution whose mean exists is haraterized by its L-moments (Hosking, 99). The irst our L-moment or random variable X are deined by E( X ) x( F) df E ( X, X, ) x ( F )( F ) df (,,, ) ( )(6 6 ) E X X X x F F F df (,,,, ) ( )(0 0 ) E X X X X x F F F F df where, X is the k th Order statisti or sample size n, the limits on the integral are 0 to kn,. L-skewness and L-kurtosis is deined as and, respetively. K th moment o Order Statistis or sample size n rom Power Funtion Distribution is deined as b k nk n n! x x x ( k ) x ( k )!( n k)! b b b 0 n ( k ) dx Gamma k Gamma n k bn! ( k )!( n k)! Gamma n (..)

11 Kurtosis Statistis with Reerene to Power untion distribution () () () () b b ( ) ( )( ) b ( )( ) (..) () () () () () () b b 6b ( ) ( )( ) ( )( )( ) b ( )( ) ( ) ( ) ( )( ) b( ) (..) ( )( )( ) () () () () () () () () 6 b ( b ) ( b ) ( ) ( )( ) ( )( )( ) b ( )( )( )( ) b ( )( )( ) 9 ( )( ) 8 ( ) 6 ( ) ( )( )( ) b( )( ) (..) ( )( )( )( ) L-skewness is deined as (..5) L-kurtosis is deined as ()( ) (..6) ()()

12 Azaz Ahmad and Ahmed Saeed Akhter.5 Comparison Among Skewness and Kurtosis Statistis: It is interesting to note that all three measures o skewness have same interpretation or symmetri Distribution. For Case II, Karl Pearson showed the poor perormane in measuring the skewness but Blest s and L-skewness gave similar result or this ase. And or Case III L-skewness is less than the Blest s measure and Karl Pearson. Figure 6 indiates that L-kurtosis gives less weight to extreme tail distribution as ompared to Blest s and Pearson measure o kurtosis..6 Trimmed L-Moments: TL-moments deined by Elamir and Seheult (00) are generalization o L-moments that do not require the mean o underlying Distribution to exist. They are deined by r ( s, t) j r r ( ) E Xrs j: rst r j0 j (.6.) Here, s and t are positive integers. The ase s = t = 0 yields the original L- moments deined by Hosking( 990). The term trimmed is appropriate beause (, ) the deinition o st r does not involve the expetation o the s smallest or the t largest Order Statistis o sample o size r+s+t. The (r+s-j) th Order Statistis rom Power Funtion Distribution deined as b rs j t j ( r s t)! x x x E( X rs j: rst ) x dx ( r s j )!(t j)! b b b 0 x put y in eq() then x by b when x 0then y 0 when x bthen y ( r s t)! ( r s j )!(t j)! / rs j / j by y y dy 0 t b( r s t)! Gamma( r s / j) EX ( rs j: rst ) ( r s j )!(t j)! Gamma( r s t / ) ( st, ) ( st, ) r Trimmed L-moments ratio are deined as r are dimensionless ( st, ) measures o the shape o a Distribution. The lose orm o the irst our TLmoments o Power Funtion Distribution with various hoie o trimming i.e.

13 Kurtosis Statistis with Reerene to Power untion distribution (0,), (,0) and (,) are obtained and then evaluated the TL-skewness and TLkurtosis as r =, (0,) (0,) r r (.6.) (0,) (,0) r (,) r (.6.) (,0) r (,0) (.6.) (,) r (,).7 Comparison between L and TL Moment Ratios: Interesting to note that all measures have same interpretation or symmetri Distribution. For positively skewed Distribution, the size o trimming aet the amount o skewness as well as kurtosis. There is less skewness and peakednes or (,0) as ompared to other hoies o trimming. For negatively skewed Distribution, all measures are relatively equal exept or hoie (,0).. General Conlusion Moments are used to provide parameter estimation, itting o Distribution and empirial desription o data. In this paper, we are ousing the objetive o measuring numerial desription o Distribution. For this, we evaluate Karl Pearson Moment Ratio or Power Funtion Distribution. Karl Pearson Moment Ratio has aurate interpretation or only symmetri Distribution. It does not provide true amount o skewness and peakedness or heavy tailed Distributions, measuring the true amount we evaluate the alternate measures i.e Blest s measures, L-moments and TL-moments. Comparing Blest s measure with Karl Pearson, ounded that Blest s oinide with Karl Pearson when Distribution is symmetri but as the amount o skewness (positive or negative) inreases Blest s piture o existing peakedness o a Distribution beomes learer as it removes the eet o asymmetry. Pearson s measure in this regard is ound to be over pronouning the peakedness. The gap between median and meson is another way measuring the degree o kurtosis or a negatively (positively) skewed Distribution. Positively skewed Distributions o Power Funtion Distribution overs higher areas between meson and median and so, they are more peaked than its negatively skewed Distributions. 5

14 6 Azaz Ahmad and Ahmed Saeed Akhter Both Blest s and Karl Pearson measures are based on higher moments o a distribution. So these measures an t be evaluated or those Distributions like Cauhy and Inver Rayleigh Distribution whose higher moments do not exists. So how an we study the desription o a suh Distribution? Answer is obtained by evaluating another alternate measures the L-moments. Due to advantages o L- moments over the onvention moments many Distribution are analyzed by these moments. Linear ombination o Order Statistis o Power Funtion Distribution are used to ompute the L-moments. Furthermore, these moments are less sensitive in the ase o Outlier (Vogel and Fennensy 99). Comparison with Karl Pearson and Blest s, L-Moment Ratio always give less weight to heavy tailed Distributions. L-moments annot be deined or the Distributions whose mean do not exist. So we seek another alternative measures or suh Distribution i.e TL-moments. TL-moments as a generalization o the L- moments and with more advantages over L-moments and onventional moments. TL-moments assign zero weight to extreme observations, they are more Robust than L-moments when used to estimate rom a sample ontaining Outliers. Like L-moments, TL-moments also ompletely determine the distribution. Dierent hoies or the amount o trimming give dierent amount o skewness and peakedness or Power Funtion Distribution. We evaluate the TL-skewness and TL-kurtosis or trimming hoies (,0), (0,) and (,). And ompare with the L- skewness and kurtosis. Our all measures are oinide or symmetri Distributions. For positively skewed Distribution the TL-skewness and kurtosis or hoie (,0) give less weight as ompared to other hoies. The gap among the TL-skewness, kurtosis statistis and L-skewness and L-kurtosis beomes smaller and smaller as value o shape parameter inreases exept or hoie (,0). Aknowledgement We thank the reerees or their valuable and onstrutive omments and suggestions. Also thank to Hassan Humayun Khan or his support.

15 Kurtosis Statistis with Reerene to Power untion distribution 7, 5 Pearson oeiient o Skewness and Kurtosis Figure : Skewness and kurtosis or Power Funtion Distribution, 5 Comparison between Blest's and Pearson Kurtosis Figure : Kurtosis o Karl Pearson and Blest,.0.5 K.PEARSON SK BLEST'S SK Figure : Skewness o Karl Pearson and Blest

16 8 Azaz Ahmad and Ahmed Saeed Akhter Menson, Mean, Median MEAN MEDIAN MESON Figure : Relation between Meson, Mean and Median or b=,, K.PEARSON SK. L-SK BLEST'S SK.0.5 Figure 5: Skewness statistis.,, K.P.KURTOSIS BLEST'S KURTOSIS L-KURTOSIS Figure 6: Relation among kurtosis statistis

17 Kurtosis Statistis with Reerene to Power untion distribution,, 0,,,0, ,,,,0 L-Skewness and Trimmed L-Skewness Figure 7: L-skewness and TL-skewness,, 0,,,0, ,,,,0 L-Kurtosis and Trimmed L-kurtosis Figure 8: L-kurtosis and TL-kurtosis Reerenes. Ahsan, R. and Memon, A. Z. (0). A note on Blest s measure o kurtosis (with reerene to Weibull Distribution). Pakistan Journal o Statistis and Operation Researh, 8(), Blest, D. C. (00). A new measure o kurtosis adjusted or skewness. Australian and New Zealand Journal o Statistis, 5(), Elamir E. A. and Seheult A. H. (00) Trimmed L-moments. Computational Statistis and Data Analysis,, 99.

18 0 Azaz Ahmad and Ahmed Saeed Akhter. Fiori, A. M. and Zenga, M. (005). The meaning o kurtosis, the inluene untion and an early intuition by L. Faleshini. Statistia Anno, 65(), Hosking, J. R. M (99). Moments or L moments? An example omparing two measures o Distributional shape. Journal o the Amerian Statistial Assoiation, 6(), Menioni, M. and Barry, D. M. (996). The Power Funtion Distribution: A useul and simple Distribution to assess eletrial omponent reliability. Miroeletronis Reliability, 6(9), Moniem, A. I. B. and Selim, Y. M. (009). TL-moments and L-moments estimation or the Generalized Pareto Distribution. Applied Mathematial Sienes, (), Pearson, K. (905). The ault law and its generalizations by Fehner and Pearson. Biometrika, (/), Vogel, R. M. and Fennessey, N. M. (99). L-moment diagrams should replae produt moment diagram. Water Resoures Researh, 9,