On Differently Defined Skewness

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1 COPUTATIONAL ETHODS IN SCIENCE AND TECHNOLOGY 4(), 9-46 (008) O Differetly Defied Skewess Pomeraia Academy, Arciszewskiego, Slupsk, Polad sulewski@zis.pap.edu.pl (Received: 8 November 007; accepted: 0 Jauary 008, published olie: April 008) Abstract: Four defiitios of skewess are discussed: classic skewess, two Pearso s skewesses ad owley s skewess. The ability of these skewesses to express asymmetry is compared as well as the accuracy of their estimatio from ormal distributio is assessed. Key words: skewess, Johso s distributio, method of Parze, estimator desity fuctio. I. INTRODUCTION I statistical literature four differet defiitios of the skewess exist. eside the classic defiitio preseted i sectio III, also two Pearso s skewesses defied i sectios IV ad V, as well as owley s skewess described i sectio VI occur. To compare these skewesses certai probability distributio will be useful, i which through the chage of parameter values i a wide rage their asymmetry chage is possible. Distributios derived from the Gaussia distributio fit superbly for umerical experimets of this type. Johso s distributio of type S R ad S U, i which chagig the parameter values makes it possible to get the trasitio from the egative skewess to the positive oe, deserves special attetio i this aspect. I the preset work Johso s distributio of type S U was used for umerical experimets, whose domai i cotrast to Johso s distributio of type S R is a set of real umbers. I sectio VII ability of these skewesses to express asymmetry was compared as well as the accuracy of their estimatio from ormal distributio was assessed. II. DISTRIUTIONS DERIVED FRO THE GAUSSIAN DISTRIUTION A cumulative distributio fuctio of distributios derived from the ormal distributio is give by [4] ( ) ϑ( θ ) F x =Φ x,, () where Φ (). is a ormal cumulative distributio fuctio, ϑ ( x) is a icreasig fuctio of argumet x, whereas θ is a vector of parameters of the discussed distributio. Examples of distributios derived from the ormal distributio are: Logormal distributio, Chhikary s distributio, irbaum Sauders s distributio, Johso distributio of type SL, S, S U. The family of Johso s distributio describes the formula [5] x ε z = γ + η ψ λ trasformig radom variable x ito radom variable z depedet o ormal distributio N ( 0, ). A detailed discussio of Johso s distributios of type SL, S, S U together with umerous umerical examples is possible to fid i a book by Drapella [4]. As it is difficult to ivestigate properties of distributios with four parameters, i further cosideratios we will accept ε = 0, λ = ad γ = a/ b, η = /b. With these assumptios () takes the form ( x) () ψ a z =. () b The cumulative distributio fuctio for Johso s distributio of type S U radom variable x is give by

2 40 ( ) l x + x + a F( x) =Φ b whereas the desity fuctio is give by f ( x) = π b x + (4). (5) l ( x x ) a + + exp. b The desity fuctio of Johso s distributio of type S U for combiatios of parameter values preseted i Table was exemplified i Figs. ad. It follows from Figure that i Johso s distributio of type S U it is possible to chage the value of parameters to get the trasitio from egative skewess to a positive oe. III. THE CLASSIC SKEWNESS Let us ote that the classic skewess is calculated as [, 6] μ γ =, (6) K / μ where μ k for cotiuous distributio are cetral momets of the k-th order i form of [,, 6] μ k k = ( x ) f ( x) dx. (7) Cetral momets of Johso s distributio of type S U are impossible to defie aalytically, therefore mathematical eviromet athcad coutig the value of idefiite itegrals was used. The model computer implemetatio of classic skewess, writte i athcad, was itroduced below. a: b = Fig.. Desity fuctio of Johso s distributio of type S U for combiatios I-V of parameter values preseted i Table f( x): = ( ) x+ x + a l exp b π + b x : = x f( x) dx : = x ( ) f x dx : = x ( ) f x dx Fig.. Desity fuctio of Johso s distributio of type S U for combiatios VI-X of parameter values preseted i Table Table. Combiatios of parameter values Combiatio a b Combiatio a b I - VI 0. II - VII 0.6 III 0 VIII 0.9 IV IX. V X.5 ( ) μ : = ( ) μ : = + μ γ : = γ = 5.6. ( μ ) The relatio betwee classic skewess for Johso distributio of type S U ad values of parameters a ad b was preseted i Fgs. ad 4. It is otable that limγ = 0, limγ K =. (8) K b 0 b

3 O Differetly Defied Skewess 4 Fig.. Relatio betwee classic skewess for Johso distributio of type S U ad parameter a Fig. 6. Relatio betwee Pearso s skewess for Johso distributio of type S U ad parameter b To calculate the values of this coefficiet computatioal eviromet athcad was used. The relatio betwee Pearso s skewess for Johso distributio of type S U ad values of parameters a ad b was preseted i Figs. 5 ad 6. Oe should otice that limγ = 0. (0) b P V. EDIAN SKEWNESS Fig. 4. Relatio betwee classic skewess for Johso distributio of type S U ad parameter b IV. PEARSON S SKEWNESS Pearso s skewess (the mode skewess) is calculated as [7] P xmod γ =, (9) μ where is a mea value, x mode of distributio. mod The media skewess is give by γ x 0.5 =, () μ where is a mea value, x 0. 5 a media of distributio. This coefficiet is well-kow i literature as Pearso s secod skewess coefficiet [7]. Values of this coefficiet i athcad were couted. The relatio betwee media skewess for Johso distributio of type S U ad values of parameters a ad b are preseted i Fgs. 7 ad 8. Let us otice that Fig. 5. Relatio betwee Pearso s skewess for Johso distributio of type S U ad parameter a Fig. 7. Relatio betwee media skewess for Johso distributio of type S U ad parameter a

parameter b lim γ = 0., lim γ = 0., () a b 0 a limγ = 0, limγ = 0.

OWLEY S SKEWNESS owley s skewess is defied as [7] ( x0,75 x0,5 ) (

75 0,5 Fig. 0. Relatio betwee owley s skewess for Johso distributio

THE COPARISON OF SKEWNESS The ability of discussed skewesses to

Skewess for Johso distributio of type S U ad value parameters

about egative, zero ad positive asymmetry was received.

Values of quatiles of Johso distributio were calculated i Solver,

The relatio betwee owley s skewess for Johso distributio of type S U

4 4 Fig. 8. Relatio betwee media skewess for Johso distributio of type S U ad parameter b lim γ = 0., lim γ = 0., () a b 0 a limγ = 0, limγ = 0. () b VI. OWLEY S SKEWNESS owley s skewess is defied as [7] ( x0,75 x0,5 ) ( x0,5 x0,5 ) ( x x ) γ =, (4) 0,75 0,5 Fig. 0. Relatio betwee owley s skewess for Johso distributio of type S U ad parameter b VII. THE COPARISON OF SKEWNESS The ability of discussed skewesses to express asymmetry is show i Figs. ad. Skewess for Johso distributio of type S U ad value parameters cotaied i Table were compared o them, thaks to which distributio about egative, zero ad positive asymmetry was received. where x k are quatiles of the k-th order of distributio ( 0< k < ). Values of quatiles of Johso distributio were calculated i Solver, which is located i icrosoft Excel. The relatio betwee owley s skewess for Johso distributio of type S U ad values of parameters a ad b are preseted i Fgs. 9 ad 0. It is worthwhile markig that lim γ =, lim γ =, (5) a b 0 a limγ = 0, limγ =. (6) b Fig.. The ability of skewess to express asymmetry for combiatios I-V Fig. 9. Relatio betwee owley s skewess for Johso distributio of type S U ad parameter a Fig.. The ability of skewess to express asymmetry for combiatios VI-X

5 O Differetly Defied Skewess 4 Assessmets of estimatio accuracy of these skewesses were executed too, whe a sample x, x,..., x was draw from Gaussia distributio. Values xi ( i =,..., ) were geerated by meas of a fuctio NormLos, which was created i Visual asic for Applicatios (VA). Fuctio NormLos(m As Sigle, s As Sigle) Dim i As Iteger Dim sum As Sigle sum = 0 For i = To Let sum = sum + Rd Let NormLos = s (sum - 6) + m Ed Fuctio Fig. 4. The relatio betwee variace ad sample size The estimate of the sample momet of k-th order is give by k ˆ k = ( xi ) ; (7) i= however, the ubiased estimator of the cetral momet of -d ad -th order are calculated as [] ˆ μ ˆ μ ( x ˆ ) i = i= ( )( ) ( x ˆ ) i = i=, (8). (9) Ukow values of quatiles were replaced by appropriate order statistics [] [ ] umber = it +. (0) The sample mode is, accordig to the defiitio, a positio of maximum of the empirical desity fuctio. Figures ad 4 preset the relatio betwee variace calculated o the basis of 0 40 estimatios each from skewess ad a sample size. For 7 the smallest variace has the mode skewess. The classic skewess for 48 has the biggest variace, because as it is widely kow the accuracy of estimatio worses sigificatly alogside with the icrease of the order of cetral momets. Amog the aalysed skewesses, media skewess should be take ito accout, which for 8 has the smallest variace. To cofirm the above-quoted facts as well as i order to smooth-out empirical desity fuctios, the author employed the Parze ethod also kow as the kerel method [8, 9]. The empirical desity fuctio is composed of kerels. I this paper each kerel is of Gaussia form ( ) x x i = exp = π h h K z z z therefore the empirical desity fuctio is give by fˆ( x ) exp h i= (), () x x () i =. () π h The parameter h is a fuctio of sample size σ h( ) S = = S, () where S i= i i= i = x(), S x() = (4) Fig.. The relatio betwee variace ad sample size The computer implemetatio of estimatio of four skewesses, writte i VA, was itroduced below. Commets were placed after apostrophes.

6 44 Sub Estimate() 'declaratio of tables Dim edf(50, ) As Double Dim x() As Double Dim skewess() As Double 'declaratio of variables Dim m As Sigle, s As Sigle, mode As Double Dim xd As Double, xg As Double, krok As Double Dim q As Double, q As Double, q As Double Dim xc As Double, b As Double, idex As Log Dim i As Log, As Log, k As Log Dim sr As Double, m As Double, m As Double Dim c As Double, hor As yte, cc As Double Dim s As Double, s As Double, ds As Double Dim h As Double, j As Log, xx As Double Dim t As Double, max As Double Radomize Timer Worksheets("estimate").Select 'selectig worksheet "estimate" 'itroductio of cells to variables Let m = Cells(, ).Value 'mea value Let s = Cells(, ).Value 'stadard deviatio Let = Cells(, ).Value 'sample size ReDim x() ReDim skewess(040, 4) For k = To 040 For i = To Let x(i) = NormLos(m, s) 'recall to the fuctio 'sortig powrot: Let hor = 0 For i = To - If (x(i) <= x(i + )) The GoTo dalej Let b = x(i) Let x(i) = x(i + ) Let x(i + ) = b Let hor = dalej: If hor = The GoTo powrot 'the empirical desity fuctio - method of Parze Let s = 0 Let s = 0 For i = To Let ds = x(i) Let s = s + ds Let s = s + ds ds Let s = s / Let s = s / Let s = s - s s Let h = Sqr(s / ) Let c = / (Sqr( Applicatio.Pi())) Let xd = x() - h Let xg = x() + h Let krok = (xg - xd) / 50 For j = 0 To 50 Let xx = xd + j krok Let edf(j, ) = xx Let edf(j, ) = 0 For i = To Let t = (xx - x(i)) / h Let edf(j, ) = edf(j, ) + Exp(-0.5 t t) Let edf(j, ) = c edf(j, ) / / h Next j 'the sample mode max = edf(, ) For j = To 50 If edf(j, ) > max The max = edf(j, ): idex = j Next j mode = edf(idex, ) 'the sample quatiles

7 O Differetly Defied Skewess 45 q = x(it( 0.5) + ) q = x(it( 0.5) + ) q = x(it( 0.75) + ) 'the sample momets sr = 0 For i = To sr = sr + x(i) sr = sr / For i = To xc = x(i) - sr m = m + xc ^ m = m + xc ^ m = m / m = m / c = Sqr( ( - )) / ( - ) skewess(k, ) = (c m) / m ^ (.5) skewess(k, ) = (sr - mode) / Sqr(m / ( - )) skewess(k, ) = ((q - q) - (q - q)) / (q - q) skewess(k, 4) = (sr - q) / Sqr(m / ( - )) Next k 'itroductio of results to cells For i = To 040 Let Cells(i, ) = skewess(i, ) Let Cells(i, ) = skewess(i, ) Let Cells(i, ) = skewess(i, ) Let Cells(i, 4) = skewess(i, 4) Ed Sub 'classic skewess 'Pearso's skewess 'owley's skewess 'media skewess Fig. 5. The estimator desity fuctio of skewess obtaied with the Parze ethod for = 5 Fig. 7. The estimator desity fuctio of skewess obtaied with the Parze ethod for = 5 Figures 5-7 preset the estimator desity fuctio of skewess obtaied with the Parze ethod for { 5; 9; 5}. The samplig mode iflueces particularly for small wavy shape of desity fuctio of estimate of skewess. Refereces Fig. 6. The estimator desity fuctio of skewess obtaied with the Parze ethod for = 9 [] S. radt, Data aalysis, Warsaw 998 (i Polish). [] H. Cramer, athematical method i statistics, Warsaw 958 (i Polish). [] H. A. David, Order statistics, Wiley, New York, 970. [4] A. Drapella, Statistical iferece o the base skewess ad kurtosis, Pomeraia Pedagogical Academy, Slupsk 004 (i Polish).

46 [5] N. L. Johso, System of frequecy curves geerated by methods of traslatio, iometrika 6 (949). [6]. G. Kedal, A. Stuart, The advaced theory of statistics, Distributio theory,, Lodo 958. [7] J. F.

8 46 [5] N. L. Johso, System of frequecy curves geerated by methods of traslatio, iometrika 6 (949). [6]. G. Kedal, A. Stuart, The advaced theory of statistics, Distributio theory,, Lodo 958. [7] J. F. Keey, E. S. Keepig, athematics of statistics, Pt., rd ed. Priceto, NJ: Va Nostrad, 0-0 (96). [8] E. Parze, O estimatio of a probability desity ad mode, A. ath. Statist,, (96). [9]. W. Silverma, Desity estimatio, Chapma ad Hall, Lodo New York, 986. PIOTR SULEWSKI graduated i athematics i 996. Sice the he has bee workig at the Istitute of athematics at Pomeraia Academy i Słupsk. He received the PhD i reliability theory i 00 from the Systems Research Istitute of Polish Academy of Scieces i Warsaw. His research iterests cocer reliability mathematics ad computatioal methods i statistics. COPUTATIONAL ETHODS IN SCIENCE AND TECHNOLOGY 4(), 9-46 (008)

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