GEOMETRIC MEAN FOR NEGATIVE AND ZERO VALUES

Transcription

1 IJRRAS 11 (3) Jue 01 GEOMETRIC MEA FOR EGATIVE AD ZERO VALUES Elsayed A. E. Habib Depatmet of Mathematics ad Statistics, Faculty of Commece, Beha Uivesity, Egypt & Maagemet & Maketig Depatmet, College of Busiess, Uivesity of Bahai, P.O. Box 3038, Kigdom of Bahai ABSTRACT A geometic mea teds to dampe the effect of vey high values whee it is a log-tasfomatio of data. I this pape, the geometic mea fo data that icludes egative ad zeo values ae deived. It tus up that the data could have oe geometic mea, two geometic meas o thee geometic meas. Cosequetly, the geometic mea fo discete distibutios is obtaied. The cocept of geometic ubiased estimato is itoduced ad the iteval estimatio fo the geometic mea is studied i tems of coveage pobability. It is show that the geometic mea is moe efficiet tha the media i the estimatio of the scale paamete of the log-logistic distibutio. Keywods: Coveage pobability; geometic mea; logomal distibutio; obustess. ITRODUCTIO Geometic mea is used i may fields, most otably fiacial epotig. This is because whe evaluatig ivestmet etus ad fluctuatig iteest ates, it is the geometic mea that gives the aveage fiacial ate of etu; see, Blume (1974), Cheg ad Kaso (1985), Poteba (1988) ad Coope (1996). May wastewate dischages, as well as egulatos who moito swimmig beaches ad shellfish aeas, must test fo fecal colifom bacteia cocetatios. Ofte, the data must be summaized as a geometic mea of all the test esults obtaied duig a epotig peiod. Also, the public health egulatios idetify a pecise geometic mea cocetatio at which shellfish beds o swimmig beaches must be closed; see, fo example, Dape ad Yag (1997), Elto (1974), Jea (1980), Mooe (1996), Michoud (1981), Limpet et al. (001) ad Mati (007). I this pape, the geometic mea fo zeo ad egative values is deived. The data could have oe geometic mea, two geometic meas (bi-geometical) o thee geometic meas (ti-geometical). The oveall geometic mea might be obtaied as a weighted aveage of all the geometic meas. Theefoe, the geometic mea fo discete distibutios is obtaied. The cocept of geometic ubiased estimato is itoduced ad the poit ad iteval estimatio of the geometic mea ae studied based o logomal distibutio i tems of coveage pobability. The populatio geometic mea ad its popeties ae defied i Sectio. The geometic mea fo egative ad zeo values ae deived i Sectio 3. Estimatio of the geometic mea is defied i Sectio 4. The samplig distibutio of geometic mea is obtaied i Sectio 5. Simulatio study is peseted i Sectio 6. Appoximatio methods ae peseted i Sectio 7. A applicatio to estimatio of the scale paamete of log-logistic distibutio is studied i Sectio 8. Sectio 9 is devoted fo coclusios. 1 GEOMETRIC MEA Let X 1, X, be a sequece of idepedet adom vaiables fom a distibutio with, pobability fuctio p(x), desity fuctio f(x), quatile fuctio x F = F 1 x = Q u whee 0 < u < 1, cumulative distibutio fuctio F x = F X = F, the populatio mea μ = μ X ad the populatio media ν = ν X. 1.1 Populatio geometic mea The geometic mea fo populatio is usually defied fo positive adom vaiable as G = G X = = 1/ (1) by takig the logaithm log G = 1 This is the mea of the logaithm of the adom vaiable X, i.e, log () 419

2 IJRRAS 11 (3) Jue 01 Habib Geometic Mea fo egative ad Zeo Values Theefoe, log G = E log X = E (F ) (3) G = e E log X = e E (F) (4) See; fo example, Cheg ad Kaso (1985). 1. Popeties of geometic mea The geometic mea has the followig popeties: if 1. x = a, a costat, the G a = e E log a = e log a = a.. Y = bx, b > 0 costat, the G Y = e E log bx = e E log b+log X = bg X 3. Y = b, b > 0, the G X Y = e E log b X = e E log b log X = b. G X 4. X 1,, X ad Y 1,, Y k ae joitly distibuted adom vaiables each with G Xi ad G Yj ad Z = E log G Z = e k Y i k k = j =1 = e E log j =1 log Y j G k G Yi. k j =1 Y i 5. X 1,, X ae joitly distibuted adom vaiables with G Xi, ad Y = the G Y = e E log e E log = G Xi, the = c i, 6. X 1,, X ae joitly distibuted adom vaiables each with E( ), c i ae costats, ad Y = e a+b c E log ea+b i the G Y = e E X c i i. 7. X 1,, X ae idepedet adom vaiables with E( ), ad Y = e a+b, the E log ea+b G Y = e X i = e E a+b = e a+b E(). GEOMETRIC MEA FOR EGATIVE AD ZERO VALUES The geometic mea fo egative values depeds o the followig ule. Fo odd values of, evey egative umbe x has a eal egative th oot, the odd odd X = X (5).1 Case 1: if all X < 0 ad is odd The geometic mea i tems of th oot is G = ( ) = This is mius the th oot of the poduct of absolute values of X, the, (6) G = (7) Hece, The geometic mea fo egative values is log G = 1 log X i = E log X = E (F) (8) G = e E log X o G = G X (9) 40

3 IJRRAS 11 (3) Jue 01 Habib Geometic Mea fo egative ad Zeo Values This is mius the geometic mea fo the absolute values of X.. Case : egative ad positive values (bi-geometical). I this case it could use the followig ab = a b Cosequetly, ude the coditios that ad 1 ae odd the geometic mea is (10) G = 1 = + 1 = i= 1 +1 i= Thee ae two geometic meas (bi-geometical). The geometic mea fo egative values is (11) ad The secod fo positive values is G = 1, log G = E log X. (1) G = e E log X o G = G X (13) + G + =, log(g + ) = E log X + (14) i= 1 +1 ad G + = e E log X + = G X + (15) If oe value is eeded it might obtai a oveall geometic mea as a weighted aveage G = W 1 G + W G + = G, with p( < X < 0) G +, with p(0 < X < ) whee W 1 = 1 = p < X < 0 ad W = = p(0 < X < ). (16).3 Case 3: zeo icluded i the data (ti-geometical) With the same logic thee ae thee geometic meas (ti-geometical). G fo egative values with umbes 1, G + fo positive values with umbes, ad G 0 = 0 fo zeo values with umbes 3. It may wite the oveall geometic mea as G, with p( < X < 0) G = W 1 G + W G + + W 3 G 0 = G +, with p(0 < X < ) (17) G 0 = 0, with p(x = 0) whee W 3 = 3 = p(x = 0) ad = ae the total umbes of egative, positive ad zeo values..4 Examples The desity, cumulative ad quatile fuctios fo Paeto distibutio ae f x = αβ α x α 1, x > β, ad x F = β(1 F) 1/α (18) with scale β ad shape α; see, Elami (010) ad Fobes et al. (011). The mea ad the media ae The geometic mea is ad log G = μ = αβ α, ad ν = β α 1 1 log β(1 F) 1/α df 0 (19) = log β + 1 α = log βe 1 α, (0) 41

4 IJRRAS 11 (3) Jue 01 Habib Geometic Mea fo egative ad Zeo Values G = βe 1 α (1) The atios of geometic mea to the mea ad the media ae CG = G μ μ = (α 1)e 1 α, ad CG α ν = G ν = e 1 α () α Figue 1 atio of geometic mea to mea ad media fom Paeto distibutio Figue 1 shows that the geometic mea is quite less tha the mea fo small α ad appoaches quickly with iceasig α. O the othe had, the geometic mea is moe tha the media fo small α ad appoaches slowly fo lage α. Fo uifom distibutio with egative ad positive values, the desity is f x = 1 b a, a < x < b (3) the ad Two geometic meas ae b log G + = 1 b a log G = a b a dx = b log b b b a dx = a log a a b a (4) (5) G = e a log a a b a = a e a b a b log b b, ad G+ = e b a = b e b b a (6) The weights could be foud as W 1 = p a < X < 0 = a b a, ad W = p 0 < X < b = b b a The oveall geometic mea i tems of weighted aveage may be foud as (7) (8) 4

5 IJRRAS 11 (3) Jue 01 Habib Geometic Mea fo egative ad Zeo Values G = W b b b a a a b a b b b b a a a e e W1 = e e b a Table 1 gives the values of geometic mea ad mea fom uifom distibutio fo diffeet choices of a ad b. a b a Table 1 geometic mea ad mea fom uifom distibutio (a, b) (3,10) (0,1) (-,0) (-1,0) (-1,1) (-3,) (-11,) (-3,10) G G W W G μ Fo log-logistic distibutio the desity, cumulative ad quatile fuctios with scale β ad shape α ae α 1 α x 1 β β F α f x = 1 + x β, x > 0, ad x F = β 1 F α See, Johso et al. (1994). The mea ad media ae (9) The geometic mea is log G = μ = βπ/α si π/α The atios of geometic mea to mea ad media ae G μ = β si π/α = βπ/α π/α si π/α The Poisso distibutio has a pobability mass fuctio, ad ν = β (30) log β F 1 α df = log β, ad G = β 0 (31), ad G ν = β β = 1 (3) λx λ p x = e x!, x = 0,1,, (33) See, Fobes et al. (011). The geometic mea is G = G 0 = 0, G + = x=1 λx λ e x!, with pob. = e λ with pob. = 1 e λ Table gives the geometic mea ad mea fom Poisso distibutio fo diffeet choices of λ ad the umbe of tems used i the sum is 100. The biomial distibutio has a pobability mass fuctio (34) The geometic mea is p x = x px 1 p x, x = 0,1,,, (35) G = G 0 = 0, G + = x=1 with pob. = 1 p x px 1 p x, with pob. = 1 1 p Table gives the values of geometic mea ad mea fom biomial distibutio fo = 10 ad diffeet choices of p. (36) 43

6 IJRRAS 11 (3) Jue 01 Habib Geometic Mea fo egative ad Zeo Values Table geometic mea ad mea fom Biomial ad Poisso distibutios Poisso λ G W 3 = p G W = 1 p G μ Biomial, = 10 p G W G W G μ ESTIMATIO OF GEOMETRIC MEA Coside a adom sample X 1, X,, X of size fom the populatio. 3.1 Case 1: positive values If all X > 0, the opaametic estimato of the geometic mea is log g = = 1 i, ad g = e ad is the sample mea of logaithm of x, hece, E = E(log g) = log G. (37) 3. Case : egative ad positive values If thee ae egative ad positive values ad ude the coditios ad 1 ae odds, the estimato of geometic mea fo egative values is ad log( g ) = 1 = 1 1 i = 1 I x<0 i g = e 1 o g = e 1 (39) I x<0 is the idicato fuctio fo x values less tha 0. Fo positive values ad log g + = + = 1 i = 1 I x>0 i, g + = e + (41) whee x, 1, x +, ae the egative values ad thei umbes ad the positive values ad thei umbes, espectively. The estimated oveall geometic mea might be obtaied as weighted aveage, (38) (40) g = 1g + g + (4) 3.3 Case 3: egative, positive ad zeo values Whe thee ae egative, positive ad zeo values i the data ad ude the coditios that ad 1 ae odd, the estimated weighted aveage geometic mea is g = 1g + g (g 0 = 0) = 1g + g + (43) 44

7 IJRRAS 11 (3) Jue 01 Habib Geometic Mea fo egative ad Zeo Values ote that if thee ae egative values ad ad 1 ae eve umbes it might delete oe at adom. If thee ae zeo ad positive values oly i the data, the weighted aveage geometic mea is g = g g 0 Defiitio 1 Let θ is a estimato of θ. If G θ = θ, θ is geometically ubiased estimato to θ. = g + (44) Example let θ = g = e 1 ad θ = G, the 1 E log e G g = e E log g = e The g is geometically a ubiased estimato fo G. = e 1 E = e 1 log G = G (45) 4 SAMPLIG DISTRIBUTIOS I this sectio the samplig distibutio of g is studied fo egative, zeo ad positive values. Theoem 1 (ois 1940) Let all X > 0 be a eal-valued adom vaiable ad E log X ad E(log X) exist, the vaiace ad expected values of g ae σ g σ G, ad μ g G + σ G (46) This ca be estimated fom data as whee σ ad s s g s g, ad μ g g + s g (47) ae the populatio ad sample vaiaces fo, espectively. Coollay 1 If all X < 0, the vaiace ad the expected values of g ae σ g = σ Poof By usig the delta method o g = e the esult follows. G, ad μ g G + σ G (48) Theoem (Paze 008) If f is the quatile like (o-deceasig ad cotiuous fom the left) the f(y) has quatile Q u; f Y = f Q u; Y. If f is deceasig ad cotiuous, the Q u; f(y) = f Q 1 u; Y. Theoem 3 Ude the assumptios of 1. E log X ad E(log X ) exist ad. y = has appoximately omal distibutio fo lage with μ y = μ = log G ad vaiace σ y = σ, the Fo all X > 0, the geometic mea g = e y has appoximately the quatile fuctio Q u; g = e Q u; μ y, σ y (49) Fo all X < 0, the geometic mea g = e y has appoximately the quatile fuctio Q u; g = e Q 1 u; μ y, σ y (50) Q is the quatile fuctio fo omal distibutio. 45

8 IJRRAS 11 (3) Jue 01 Habib Geometic Mea fo egative ad Zeo Values Poof Whee the fuctio e y is a iceasig fuctio ad e y is a deceasig fuctio ad by usig theoem the esult follows; see, Gilchist (000) ad Asquith (011). Coollay If all X > 0, the lowe ad uppe 1 α % cofidece itevals fo G ae e Q α; μ y, σ y, e Q 1 α; μ y, σ y (51) If all X < 0, the lowe ad uppe 1 α % cofidece itevals fo G ae e Q 1 α; μ y,σ y, e Q α; μ y,σ y (5) Poof The esult follows diectly fom the quatile fuctio that obtaied i theoem 3. Coollay 3 Ude the assumptios of theoem 3 1. if all X > 0, the distibutioal momets of g ae μ +σ E g = e = Ge σ, G(g) = e μ = G, Mode g = e μ σ ad σ g = E(g) e σ 1 ad μ = log G = E() ad σ = σ.. if all X < 0, the distibutioal momets of g ae μ +σ E g = e, G( g) = G, Mode g = e μ σ ad σ g = E( g) e σ 1 ad μ = E( ) ad σ = σ. Poof Sice g ad g have logomal distibutios with mea μ ad σ, the esults follow usig the momets of logomal distibutio; see, Fobes et al. (011) fo momets of logomal. It is iteestig to compae the estimatio usig ois s appoximatio μ g ad s g ad distibutioal appoximatio E g ad σ g. The esults usig Paeto distibutio with diffeet choices of α, β ad ae give i Table 3: 1. The mai advatage of distibutioal appoximatio ove ois s appoximatio is that the distibutioal appoximatio has much less biased util i small sample sizes ad vey skewed distibutios (α = 1.5 ad.5 ).. The distibutioal ad ois s appoximatios have almost the same vaiace. 3. g = G g is geometically ubiased to G. Table 3 compaiso betwee ois ad distibutioal appoximatios mea ad vaiace fo g usig Paeto distibutio ad the umbe of eplicatios is β = 10 G μ g s g E g G g σ g α = α = α =

9 IJRRAS 11 (3) Jue 01 Habib Geometic Mea fo egative ad Zeo Values Theoem 4 Ude the assumptios of theoem 3 ad X 0, the geometic mea g = e + has appoximately Poof 1. a logomal distibutio with μ = μ + + log ad σ =. the lowe ad uppe 1 α % cofidece itevals fo G ae e Q α; μ,σ, e Q 1 α; μ,σ σ +, If s logomal (μ, σ ) the ax has logomal (μ + log a, σ ); see Joso et al. (1994). Sice e + has logomal with μ, ad a = / the the esult follows. +, σ + Theoem 5 Fo all values of X, the expected ad vaiace values of g = 1g + g + ae ad σ g 1 E g 1μ g + μ g+ σ g + (53) σ g+ 1 Cov g, g + (54) whee σ g 1 σ μ g ad σ g+ 1 σ + μ g+, Cov g, g + = E e E e 1 E e + μ g + 1 μ g+ + E e 1 σ 1 μ g, σ + μ g+, E e +, ad E e μ c + μ c 1 σ + 1 σ + c = e Poof Usig the delta method fo vaiace ad expected values the esults follow; see Johso et al. (1994). 47

10 IJRRAS 11 (3) Jue 01 Habib Geometic Mea fo egative ad Zeo Values Figue distibutio of g with logomal distibutio supeimposed usig 5000 simulated data fom Paeto distibutio with β = 1 ad α = 1.75 ad diffeet. The logomal distibutio gives a good appoximatio to the distibutio of g. Figue shows the distibutio of g usig simulated data fom Paeto distibutio with β = 1, α = 1.75 ad diffeet choices of ad Figue 3 shows the distibutios of g ad g usig simulated data fom omal distibutio with ( 100, 10) ad (0,1) ad = 5 ad 50, espectively. Figue 3 distibutio of g with logomal distibutio supeimposed usig 5000 simulated data fom omal distibutio with mea=m, ad stadad deviatio=s ad = 5 ad 50. Figue 3 distibutio of g with logomal distibutio supeimposed usig 5000 simulated data fom omal distibutio with mea=m, ad stadad deviatio=s ad = 5 ad SIMULATIO PROCEDURES I ode to assess the bias ad oot mea squae eo (RMSE) of g ad the coveage pobability of the cofidece iteval of G, a simulatio study is built. Seveal sceaios ae cosideed ad i each sceaio the simulated bias, 48

11 IJRRAS 11 (3) Jue 01 Habib Geometic Mea fo egative ad Zeo Values RMSE ae calculated. Futhe, the coveage pobability of the cofidece iteval is evaluated fo estimatig the actual coveage of the cofidece iteval by the popotio of simulated cofidece iteval cotaiig the tue value of G. The desig of the simulatio study is sample sizes: 5, 50, 75, 100; umbe of eplicatios: 10000; omial coveage pobability of cofidece iteval: 0.90, 0.95 ad The bias ad RMSE fo Poiso ad Paeto distibutios fo diffeet choices of paametes ad sample sizes ae epoted i Table 4. Fo Poisso distibutio, it is show that if λ is ea zeo the bias is egligible ad whe as it becomes fa away fom zeo the bias stat to icease fo small sample sizes ad egligible fo lage sample sizes. Fo Paeto distibutio, the bias is slightly oticeable fo small α (vey skewed distibutio), ad egligible fo lage values of α ad. Table 4 bias ad RMSE fo g fom Poisso ad Paeto distibutios Poisso λ = 0.1, G = λ = 1,G = λ = 3, G =.468 Bias RMSE Bias RMSE Bias RMSE Paeto β = 1, α = 1.5 β = 1, α = 3 β = 1, α = The simulatio esult fo coveage pobability usig log-logistic distibutio fo diffeet choices of is give i Table 5. As suggested by the esults obtaied fom the log-logistic distibutio fo diffeet values of α, the small the wost is the coveage pobability. While the sample size iceases, the coveage pobability is impoved. The, a elatively small sample size of 50 is sufficiet i ode to assue a good coveage pobability of the cofidece iteval. Table 5 coveage pobability of the cofidece iteval fo G fom log-logistic distibutio ad the umbe of eplicatio is Coveage pobability β = 1, α = 0.5, G = 1 β = 1, α = 1, G = β = 1, α = 5, G = 1 β = 1, α = 10, G = Moeove, Table 6 shows compaiso betwee the geometic mea, sample mea ad media fo egative values fom omal distibutio with μ = 100 ad σ = 5 i tems of mea squae eo. The table shows the geometic mea is moe efficiet that the media ad vey compaable to the sample mea whee efficiecy is at = 5 ad 0.95 at =

12 IJRRAS 11 (3) Jue 01 Habib Geometic Mea fo egative ad Zeo Values Table 6 Mea squae eo (MSE) ad efficiecy (eff.) with espect to mea usig simulated data fom omal distibutio ad the umbe of eplicatios is omal ( 100, 5) Mea media geometic mea MSE MSE eff. MSE eff APPROXIMATIO METHODS Whe it is difficult to obtai geometic mea exactly, the appoximatio may be useful i some cases; see, Cheg ad Lee (1991), Zhag ad Xu (011) ad Jea ad Helms (1983). Let X be a eal-valued adom vaiable. If E X ad E(X ) exist, the fist ad the secod ode appoximatio of geometic mea ae Theefoe, Example log G X log E X, ad log G X log E X e E(x) (55) σ X G X E X, ad G X E X e E(X) (56) μ +σ Fo logomal distibutio E(X) = e, σ X = μ e σ 1 ad the exact geometic mea is G X = e μ. The fist ad the secod ode appoximatios ae ad log G X μ + σ eσ 1 The atios of exact to appoximatio ae R 1 = log G X log E X = μ + σ, ad G X e μ + (57) G(exact) G(appox.) = eμ μ +σ e σ X σ, ad G X e μ +σ eσ 1 (58) = e σ ad R = e σ + eσ 1 Table 7 shows the fist ad the secod ode appoximatios fo g fom logomal distibutio. The fist ode σ σ appoximatio is good as log as < 0.10 ad the secod ode appoximatio is vey good as log as E(X) Table 7 atios of exact to appoximated geometic mea fom logomal dsitibutio σ σ/μ R R (59) E(X) < Fo omal distibutio E(X) = μ ad σ X = σ. The fist ad the secod ode appoximatios ae log G X log E X = log μ, ad G X μ (60) ad log G X log μe σ μ, ad G X μe σ μ (61) the, σ μ Geμ (6) Table 8 shows the simulatio esults of mea, media, the fist ad secod ode appoximatios fo g fom omal distibutio. 430

13 IJRRAS 11 (3) Jue 01 Habib Geometic Mea fo egative ad Zeo Values Table 8 sample mea, media, fist (g 1 ), secod ode (g ) appoximatios of g ad coefficiet of vaiatio fo simulated data fom omal distibutio ad = 5. σ = 1 σ = 10 μ x med σ/ μ g g APPLICATIO 7.1 Estimatio of the scale paamete of log-logistic distibutio Fom Fobes et al. (011) the log-logistic distibutio with scale β ad shape α is defied as f x = α β x β α x α The geometic mea ad media fo logistic distibutio ae β, x > 0, ad x F = β F 1 F 1 α (63) G = β, ad ν = β (64) Fo kow values of the shape paamete α the scale paamete β ca be estimated as Table 9 shows the bias, mea squae eo (MSE), efficiecy β = g, ad β = med(x) (65) MSE g MSE med ad geometic bias (g G) fo β with kow values of α usig simulated data fom log-logistic distibutio ad diffeet choices of ad the umbe of eplicatios is Table 9 bias, MSE, efficiecy ad geometic bias fo β = 10 fom log-logistic distibutio ad diffeet choice of α ad ad umbe of eplicatio is α = 0.5 med Bias MSE g Bias MSE MSE g Efficiecy MSE med Geometic Bias α = 1 Med Bias MSE g Bias MSE MSE g Efficiecy MSE med Geometic Bias α = 8 Med Bias MSE g Bias MSE MSE g Efficiecy MSE med Geometic Bias

14 IJRRAS 11 (3) Jue 01 Habib Geometic Mea fo egative ad Zeo Values Table 9 shows that 1. I geeal the geometic mea (g) is less biased tha media (med).. g has less mea squae eo (MSE) ad, theefoe, is moe efficiet tha med. 3. g is geometically ubiased estimato fo the paamete β. 8 COCLUSIO Thee ae may aeas i ecoomics, chemical, fiace ad physical scieces i which the data could iclude zeo ad egative values. I those cases, the computatio of geometic mea pesets a much geate challege. By usig the ule: fo odd umbes evey egative umbe x has a eal egative th oot, it is deived to the geometic mea as a mius of geometic mea fo absolute values. Theefoe, the data could have oe geometic mea, two geometic meas ad thee geometic meas. The oveall geometic mea is obtaied as a weighted aveage of all geometic meas. Of couse diffeet ules could be used. The sample geometic mea is poved to be geometically ubiased estimato to populatio geometic mea. Moeove, it is show that the geometic mea is outpefomed the media i estimatio the scale paamete fom log-logistic distibutio data i tems of the bias ad the mea squae eo whee geometic mea teds to dampe the effect of vey high values by takig the logaithm of the data. Its iteval estimatio is obtaied usig logomal distibutio ad it is show that the geometic mea had a good pefomace fo lage ad small sample sizes i tems of coveage pobability. ACKOWLEDGMET The autho is geatly gateful to associate edito ad eviewes fo caeful eadig, valuable commets ad highly costuctive suggestios that have led to claity, bette pesetatio ad impovemets i the mauscipt. REFERECES [1]. ois,. (1940) The stadad eos of the geometic ad hamoic meas ad thei applicatio to idex umbes. The Aals of Mathematical Statistics, 11, []. Blume, M.E. (1974) Ubiased estimatos of log-u expected ates of etu. Joual of the Ameica Statistical Associatio, 69, [3]. Elto, E.J., ad Gube, M.I. (1974) O the maximizatio of the geometic mea with a logomal etu distibutio. Maagemet Sciece, 1, [4]. Jea, W.H. (1980) The geometic mea ad stochastic domaice. Joual of Fiace, 35, [5]. Michaud, R.O. (1981) Risk policy ad log-tem ivestmet. Joual of Fiacial ad Quatitative Aalysis, XVI, [6]. Jea, W.H., ad Helms, B.P. (1983) Geometic mea appoximatios. Joual of Fiacial ad Quatitative Aalysis, 18, [7]. Cheg, D.C. ad Kaso, M.J. (1985) Cocepts, theoy, ad techiques o the use of the geometic mea i log-tem ivestmet. Decisio Scieces, 16, [8]. Poteba, J.M. ad Summes, L.H. (1988) Mea evesio i stock pices. Joual of Fiacial Eoomics,, [9]. Cheg, D.C. ad Lee, C.F. (1991) Geometic mea appoximatios ad isk policy i log-tem ivestmet. Advaces i Quatitative Aalysis of Fiace ad Accoutig, 6, [10]. Johso.L., Kotz S., ad Balakisha. (1994) Cotiuous uivaiate distibutios. Vol. 1 ad, Wiley Jho & Sos. [11]. Coope, I. (1996) Aithmetic vesus geometic mea estimatos: settig discout ates fo capital budgetig. Euopea Fiacial Maagemet,, [1]. Mooe, R.E (1996) Rakig icome distibutios usig the geometic mea ad a elated geeal measue. Southe Ecoomic Joual, 63, [13]. Dape,.R. ad Yag, Y. (1997) Geealizatio of the geometic mea fuctioal. Computatioal Statistics & Data Aalysis, 3, [14]. Gilchist W.G. (000) Statistical modelig with quatile fuctios. Chapma & Hall. [15]. Limpet, E., Stahel, W.A. ad Abbt, M. (001) log-omal distibutios acoss the scieces: keys ad clues. BioSciece, 51, [16]. Mati M.D. (007) The geometic mea vesus the aithmetic mea. Ecoomic Damage, [17]. Paze,E. (008) Uited statistics, cofidece quatiles, Bayesia statistics. Joual of Statistical Plaig ad Ifeece, 138, [18]. Elami, E.A.H. (010) Optimal choices fo timmig i timmed L-momet method. Applied Mathematical Scieces, 4, [19]. Asquith, W.H. (011) Uivaiate distibutioal aalysis with L-momet statistics usig R. Ph.D. Thesis, Depatmet of Civil Egieeig, Texas Tech Uivesity. [0]. Fobes G., Evas M., Hastig. ad Peacock B.(011) Statistical distibutios. Fouth Editio, Wiley Jho & Sos. [1]. Zhag, Q. ad Xu, Big (011) A ivaiace geometic mea with espect to geealized quasi-aithmetic-mea. Joual of Mathematical ad Applicatios, 379,