Chapter 3. Elementary statistical concepts. Summary. 3.1 Introductory comments

Transcription

1 Chapte 3 Elemetay statstcal cocepts Demets Kotsoyas Depatmet of Wate Resoces ad Evometal Egeeg Faclty of Cvl Egeeg, Natoal Techcal Uvesty of Athes, Geece mmay Ths chapte ams to seve as a emde ad syopss of elemetay statstcal cocepts, athe tha a systematc ad complete pesetato of the cocepts. tatstcs s the appled bach of pobablty theoy whch deals wth eal wold poblems, tyg to daw coclsos based o obsevatos. Two majo tasks statstcs ae estmato ad hypothess testg. tatstcal estmato ca be dstgshed paamete estmato ad pedcto ad ca be pefomed ethe o a pot bass (esltg a sgle vale, the expectato, o o a teval bass (esltg a teval whch the qatty soght les, assocated wth a ceta pobablty o cofdece. Uses of statstcal estmato egeeg applcatos clde the estmato of paametes of pobablty dstbtos, fo whch seveal methods exst, ad the estmato of qatles of dstbtos. tatstcal hypothess testg s also a mpotat tool egeeg stdes, ot oly typcal decso makg pocesses, bt also moe aalytcal tasks, sch as detectg elatoshps amog dffeet geophyscal, ad patclaly hydologcal, pocesses. All ths cocepts ae befly dscssed both a theoetcal level, to clafy the cocepts ad avod msses, ad a moe pactcal level to demostate the applcato of the cocepts. 3. Itodctoy commets tatstcs s the appled bach of pobablty theoy whch deals wth eal wold poblems, tyg to daw coclsos based o obsevatos. The coclsos ae oly feeces based o dcto, ot dedctve mathematcal poofs; howeve, f the assocated pobabltes ted to, they become almost cetates. The coclsos ae attbted to a poplato, whle they ae daw based o a sample. Althogh the cotet of tem poplato s ot stctly defed the statstcal lteate, loosely speakg we cosde that the tem descbes ay collecto of objects whose measable attbtes ae of teest. It ca be a abstacto of a eal wold poplato o of the epetto of a eal expemet. The poplato ca be fte (e.g. the poplato of the aal flows of the hydologc yea fo all hydologc bass of eath wth sze geate tha 00 km o fte ad abstactvely defed (e.g. the poplato of all possble aal flows of a hydologc bas. The tem sample descbes a collecto of obsevatos fom the patcla poplato (see defto secto 3... A mpotat cocept of statstcs s the estmato. It s dstgshed paamete estmato ad pedcto. I ode to clafy these cocepts, we cosde a poplato that s

2 3. Elemetay statstcal cocepts epeseted by a adom vaable wth dstbto fcto F (x; θ whee θ s a paamete. A paamete estmato poblem s a poblem whch the paamete s kow ad we seek a estmate of t. A pedcto poblem s a poblem whch the paamete s kow ad we seek a estmate of the vaable o a fcto of. As we wll see below, these two poblems ae dealt wth sg smla methods of statstcs, ad ths they ae both called estmato poblems. The eslts of the estmato pocedes ae called estmates. A estmate ca be ethe a pot estmate,.e. a mecal vale, o a teval estmate,.e. oe teval that cotas the vale soght wth a gve degee of cetaty. Covesely, fo a gve teval, statstcs ca calclate the coespodg degee of cetaty o, o the cotay, the degee of cetaty, that the qatty soght les wth the teval. Aothe mpotat aea of statstcs s hypothess testg that costttes the bass of the decso theoy. The pocess of hypothess testg eqes the fomlato of two statemets: the basc H 0, that s efeed to as the ll-hypothess, ad the alteatve hypothess H. We stat the pocess of testg by cosdeg that the ll hypothess s te ad we se the obsevatos to decde f ths hypothess shold be ejected. Ths s doe sg of statstcal methods. Althogh the hypothess testg s based o the same theoetcal backgod as the estmato, the dffeece les the examato of two alteatve models, whle the estmato we se oly oe model. The backgod fo all these cocepts s descbed ths chapte whle the ext chaptes seveal addtoal mecal examples ae gve. Of cose, statstcs clde may othe aeas, sch as the Bayesa aalyss, bt these ae ot coveed ths text. 3. Cocepts ad deftos 3.. ample We cosde a adom vaable wth pobablty desty fcto f(x. The vaable s defed based o a sample space Ω ad s coceptalzed wth some poplato. A sample of of sze (o legth of s a seqece of depedet detcally dstbted (IID adom vaables,,, (each havg desty f(x defed o the sample space Ω Ω Ω (Papols, 990, p. 38. Each oe of the vaables coespods to the possble eslts of a measemet o a obsevato of the vaable. Afte the obsevatos ae pefomed, to each vaable thee coespods a mecal vale. Coseqetly, we wll have a mecal seqece x, x,, x, called the obseved sample. The cocept of a sample s, theefoe, elated to two types seqeces: a abstactve seqece of adom vaables ad the coespodg seqece of the mecal vales. It s commo egeeg applcato to se the tem sample dstgshably fo both seqeces, omttg the tem obseved fom the secod seqece. Howeve, the two otos ae fdametally dffeet ad we shold be attetve to dstgsh each tme whch of the two cases the tem sample efes to.

3 3. Cocepts ad deftos 3 I statstcs t s assmed that the costcto of a sample of sze o the samplg s doe by pefomg epettos of a expemet. The epettos shold be depedet to each othe ad be pefomed de vtally the same codtos. Howeve, dealg wth atal pheomea ad egeeg t s ot possble to epeat the same expemet, ad ths samplg s a pocess of mltple measemets of the a atal pocess at dffeet tmes. As a coseqece, t s ot possble to ese that depedece ad same codtos wll hold. Noetheless, fo ceta statos we ca assme that the pevos codtos ae appoxmately vald (a assmpto eqvalet to smltaeosly assmg depedece, statoaty ad egodcty, cf. chaptes ad 4 ad ths we ca se classcal statstcal methods of statstcs to aalyse them. Howeve, thee ae cases whee these codtos (the depedece patcla ae fa fom holdg ad the se of classcal statstcs may become dageos as the estmatos ad feeces may be totally wog (see chapte tatstc A statstc s defed to be a fcto of a sample s adom vaables,.e. Θ g(,, ( vecto otato, Θ g(, whee : [,, ] T s kow as the sample vecto; ote that the spescpt T deotes the taspose of a vecto o matx. Fom the obsevatos we ca calclate the mecal vale of the statstc,.e. θ g(x,, x. Clealy, the statstc Θ s ot detcal wth ts mecal vale θ. I patcla, the statstc, as a fcto of adom vaables, s a adom vaable tself, havg a ceta dstbto fcto. Wheeas the mecal vale of the statstc s smply calclated fom the mathematcal expesso g(x,, x sg the sample obsevatos, ts dstbto fcto s dedcted based o theoetcal cosdeatos as we wll see late sectos. Typcal examples of commoly sed statstcs ae gve below Estmatos ad estmates A statstcs s sed to estmate a poplato paamete. Fo ay poplato paamete η, thee exsts oe o moe statstc of the fom Θ g(,, stable fo the estmato of ths paamete. I ths case we say that Θ g(,, s a estmato of the paamete η ad that the mecal vale θ g(x,, x s a estmate of η. Thee s ot a qe cteo to decde f a statstc ca be sed fo the estmato of a poplato paamete. Ofte the mathematcal expesso g(,, s fomlated as f η was a poplato paamete of a fte sample space detcal wth the avalable sample. Fo example, f we wsh to fd a estmato of the mea vale η m of a vaable, based o the sample (,, wth obsevatos (x,, x, we ca thk of the case whee s a dscete vaable takg vales (x,, x, each wth the same pobablty P( x /. I ths case, by defto of the mea (eq. (. - (.3 we fd that η (x + + x /. If the latte eqato we eplace the mecal vales wth the coespodg vaables, we obta the statstc Θ ( + + /. As we wll see, ths s the estmato of the mea

4 4 3. Elemetay statstcal cocepts vale of ay adom vaable, t s amed sample mea ad t s typcally deoted as. Howeve, ths empcal appoach does ot gve always a good estmato. Wheeas a estmato s ot defed by a stct mathematcal pocede the geeal case, seveal estmato categoes have goos deftos. Ths:. A statstc Θ s a based estmato of the paamete η f E[Θ] η. Othewse, t s a based estmato ad the dffeece E[Θ] η s called bas.. A estmato s Θ s a cosstet estmato of the paamete η f the estmato eo Θ η teds to zeo wth pobablty as. Othewse, the estmato s cosstet. 3. A statstc Θ s the best estmato of the paamete η f the mea sqae eo E[(Θ η ] s mmm. 4. A statstc Θ s the most effcet estmato of the paamete η f t s based ad wth mmm vaace (whee de to basedees the vaace eqals the estmato eo. It s easy to show that the estmato of the pevos example s a based ad cosstet estmato of the poplato mea m (see secto Moeove, fo ceta dstbtos fctos, t s best ad most effcet. I pactce, effots ae take to se based ad cosstet estmatos, whle the calclato of the best ad most effectve estmato s moe of theoetcal teest. Fo a ceta paamete t s possble to fd moe tha oe based o cosstet estmato. Ofte, the detemato of based estmatos s dffclt o mpossble, ad ths we may cotet wth the se of based estmatos Iteval estmato ad cofdece tevals A teval estmate of a paamete η s a teval of the fom (θ, θ, whee θ g (x,, x ad θ g (x,, x ae fctos of the sample obsevatos. The teval (Θ, Θ defed by the coespodg statstcs Θ g (,, ad Θ g (,, s called the teval estmato of the paamete η. We say that the teval (Θ, Θ s a γ-cofdece teval of the paamete η f P{Θ < η < Θ } γ (3. whee γ s a gve costat (0 < γ < called the cofdece coeffcet, ad the lmts Θ ad Θ ae called cofdece lmts. Usally we choose vales of γ ea (e.g. 0.9, 0.95, 0.99, so as the eqalty (3. to become ea ceta. I pactce the tem cofdece lmts s ofte (loosely sed to descbe the mecal vales of the statstcs θ ad θ, wheeas the same happes fo the tem cofdece teval. I ode to povde a geeal mae fo the calclato of a cofdece teval, we wll assme that the statstc Θ g(,, s a based pot estmato of the paamete η ad that ts dstbto fcto s F Θ (θ. Based o ths dstbto fcto t s possble to

5 3.3 Typcal pot estmatos 5 calclate two postve mbes ξ ad ξ, so that the estmato eo Θ η les the teval ( ξ, ξ wth pobablty γ,.e. P{η ξ < Θ < η + ξ } γ (3. ad at the same tme the teval ( ξ, ξ to be the as small as possble. Eqato (3. ca be wtte as P{Θ ξ < η < Θ + ξ } γ (3.3 Coseqetly, the cofdece lmts we ae lookg fo ae Θ Θ ξ ad Θ Θ + ξ. Althogh eqatos (3. ad (3.3 ae eqvalet, the statstcal tepetato s dffeet. The fome s a pedcto,.e. t gves the cofdece teval of the adom vaable Θ. The latte s a paamete estmato,.e. t gves the cofdece lmts of the kow paamete η, whch s ot a adom vaable. 3.3 Typcal pot estmatos I ths secto we peset the most typcal pot estmatos efeg to the poplato momets of a adom vaable espectvely of ts dstbto fcto F(x. Patclaly, we gve the estmatos of the mea, the vaace ad the thd cetal momet of a vaable. We wll ot exted to hghe ode momets, fstly becase t s dffclt to fom based estmatos ad secodly becase fo typcal sample szes the vaace of estmatos s vey hgh, ths makg the estmates extemely ceta. Ths s also the easo why egeeg applcatos momets hghe tha thd ode ae ot sed. Eve the estmato of the thd momet s accate fo a small sze sample. Howeve, the thd momet s a mpotat chaactestc of the vaable as t descbes the skewess of ts dstbto. Moeove, hydologcal vaables ae as a le postvely skewed ad ths a estmate of the skewess s ecessay. Apat fom the afoemetoed momet estmatos we wll peset the L-momet estmatos as well as the covaace ad coelato coeffcet estmatos of two vaables that ae sefl fo the smltaeos statstcal aalyss of two (o moe vaables Momet estmatos The estmatos of aw momets (momets abot the og of oe o two vaables,.e. the ( (s estmatos of ad (whee ad s ae chose teges, fomed accodg to the m m Y empcal method descbed secto 3..3, ae gve by the followg elatoshps: If the dstbto of Q s symmetc the the teval ( ξ, ξ has mmm legth fo ξ ξ. Fo osymmetc dstbtos, t s dffclt to calclate the mmm teval, ths we smplfy the poblem by splttg the ( 3. to the eqatos P{Θ < η ξ } P{Θ > η + ξ } ( γ /. The tems cofdece lmts, cofdece teval, cofdece coeffcet etc. ae also sed fo ths pedcto fom of the eqato.

6 6 3. Elemetay statstcal cocepts ~ M ( ~ ( s, M Y s Y (3.4 It ca be poved (Kedall ad tewat, 968, p. 9 that ~ ( ( ~ ( s ( s [ M ] m, E[ M ] m E (3.5 Coseqetly, the momet estmatos ae based. The vaaces of these estmatos ae Va ~ ( ( [ M ] m ( m Y ~ ( s (,s ( s [ ], Va[ M Y ] [ my ( my ] (3.6 ( It ca be obseved that f the poplato momets ae fte, the the vaaces ted to zeo as ; theefoe the estmatos ae cosstet. ( (s Typcal cetal momet estmatos,.e. estmatos of ad of oe ad two vaables, espectvely, ae those defed by the eqatos ( ( ( Y Y Mˆ ( ( s, M ˆ Y (3.7 These have bee fomed based o the empcal method descbed secto 3..3.These estmatos ae based (fo + s > ample mea The most commo statstc s the sample mea. As we have see secto 3..3, the sample mea s a estmato of the te (o poplato mea m E[] ad s defed by Y µ µ Y s (3.8 whch s a specal case of (3.4 fo. Its mecal vale x x (3.9 s called the obseved sample mea o smply the aveage. The symbols ad x shold ot be coceptally cofsed wth each othe o wth the te mea of the adom vaable,.e. m E[], whch s defed based o the eqatos (.0 o (. ad (.3. Nevetheless, these thee qattes ae closely elated. Implemetato of eqatos (3.5 ad (3.6 gves E [ ] E[ ] Va[ ] [ ] Va, (3.0 egadless of the dstbto fcto of. Ths, the estmato s based ad cosstet. Howeve, Va[ ] depeds o the depedece stcte of the vaables ; the fomla gve ( 3.0 holds oly f ae depedet. O the othe had, the fomla fo E[ ] holds always.

7 3.3 Typcal pot estmatos Vaace ad stadad devato A based estmato of the te (poplato vaace σ s: ( (3. It ca be poved (Kedall ad tewat, 968, p. 77 that whee Va (4 µ Va [ ] σ (4 4 (4 4 µ σ [ ] ( µ σ E µ + (4 3σ 3 4 (3. s the foth cetal poplato momet. The two last tems the expesso of [ ] ca be omtted fo lage vales of. Fom the expesso of [ ] E (3. we obseve that mltplcato of by /( eslts a based estmato of,.e. ( (3.3 s kow as sample vaace. Fo lage sample szes, the two estmatos ad ae pactcally the same. If the poplato s omally dstbted, t ca be show that Va 4 [ ] ( σ, Va[ ] 4 σ (3.4 ( The stadad devato estmatos commo se ae the sqae oots of the vaace estmatos, amely the ad ad ae ot based. Ths (Yevjevch, 97, p. 93. Kedall ad tewat, 968, p. 33, E (4 4 µ σ (3.5 4σ [ ] σ + O, Va[ ] + O whee the tems O(/ ae O(/ ae qattes popotoal to / ad /, espectvely, ad ca be omtted f the sample sze s lage eogh ( 0. fo If the poplato s omally dstbted, the followg appoxmate eqatos ca be sed E [ ] σ, Va[ ] σ (3.6 σ

8 8 3. Elemetay statstcal cocepts Fo 0 the eos of these eqatos ae smalle tha.5% ad.7% espectvely, whle fo 00 ae pactcally zeo. The coespodg eqatos fo ae E [ ] σ, Va[ ] σ (3.7 ( Fally, oe of the two followg estmatos of the coeffcet of vaato ca be sed: Cˆ Cˆ v, v (3.8 If the vaable s postve, the t ca be show that these estmatos ae boded fom above ( ˆ whle the same does ot hold fo the coespodg poplato Cv paametes. Obvosly, ths todces bas Thd cetal momet ad skewess coeffcet A based estmato of the te (poplato thd cetal momet (3 µ s gve by ( fo whch t ca be show (Kedall ad tewat, p that 3 Mˆ (3 (3.9 [ ˆ ( 3 ] ( ( (3 E M (3.0 µ It mmedately follows that a based (ad cosstet estmato of µ (3 s 3 ( ( ( ˆ (3 M (3. Fo lage sample sze, the two estmatos ae pactcally the same. The estmato of the skewess coeffcet s doe sg the followg estmato C s ˆ ˆ M Cs (3 3 Moe accate appoxmatos ae gve by E 4 [ ] 5 σ, Va[ ] 3 4 σ 3 ( 4 the eos of whch fo 0 ae less tha 0.005% ad 0.%, espectvely. The pecse eqatos ae E Γ [ ] ( σ Va [ ] Γ ( σ, Γ ( Γ ( The expesso of the estmato s vaace s qte complex ad s omtted (cf. Kedall ad tewat, 968, p. 33. If follows a omal dstbto the [ Cv ] Va $ C / v

9 3.3 Typcal pot estmatos 9 whch s ot based. The bas does ot ogate oly fom the fact that the two momet estmatos (meato ad deomato ae ot based themselves, bt also (maly fom the fact that C^ s s boded both fom above ad fom below, whlst the poplato C s s ot boded. Ths s de to the fte sample sze, whch detemes the ppe ad lowe lmt. Ths, t has bee show (Kby, 974; Walls et al., 974 that C^ s ( /. eveal appoxmate bas coecto coeffcets have bee poposed the lteate to be mltpled by C^ s estmated fom (3. to obta a less based estmate. Noe of them leads to a goos based estmato of the coeffcet of skewess. The fo most commo ae: ( (,, ( ( , ˆ (3.3 + C + + s The fst s obtaed f (3. the based momet estmatos ae eplaced by the based oes. The secod eslts f (3. we eplace the based thd momet estmato wth the based oe (Yevjevch, 978, p. 0. The thd oe has bee poposed by Haze ad the last oe has bee poposed by Bobée ad Robtalle (975, based o eslts by Walls et al. ( L-momets estmates ( ( Ubased estmates of the pobablty weghted momets ae gve by the followg b elatoshp (Ladweh et al., 979: ( b j x ( β ( (...( + + x( ( (...( (3.4 whee s the sample sze, ad x ( the odeed obsevatos so that x ( x ( x (. The estmates of the fst fo pobablty weghted momets ae: (0 b x( x ( b x( ( ( ( b x( ( ( (3.5 Notce that x ( s the lagest obsevato; the eqatos ae somewhat smple f the obsevatos ae odeed fom smallest to lagest bt t has bee the le egeeg hydology to pt the obsevatos descedg ode. The estmatos of the same qattes ae obtaed by eplacg x ( wth the vaable (, the so called ode statstc.

10 0 3. Elemetay statstcal cocepts ( ( ( 3 (3 b x( ( ( ( 3 Accodgly, the estmates of the fst fo L momets ae calclated by the eqatos elatg L momets ad pobablty weghted momets (see eqato (.3,.e., ( l ( l (0 b ( x ( b (0 b (3 l ( ( ( (3.6 b b b (4 l (3 ( ( b b b (0 b Covaace ad coelato A based estmato of the covaace σ Y of two vaables ad Y s: It ca be show (e.g. Papols, 990, p. 95 that Y ( ( Y Y (3.7 E (3.8 [ Y ] σ Y Theefoe, a based (ad cosstet estmato of σ Y s ( ( Y Y Y (3.9 kow as sample covaace. The estmato of the coelato coeffcet ρ Y s gve by the ext elatoshp, kow as the sample coelato coeffcet: R Y Y Y Y Y ( ( Y Y ( ( Y Y (3.30 The pecse dstbto fcto of ths estmato ad ts momets ae dffclt to deteme aalytcally; howeve, ths estmato s egaded appoxmately based. I may books, the deomato of (3.9 has the tem, whch s ot coect.

11 3.4 Typcal cofdece tevals 3.4 Typcal cofdece tevals 3.4. Mea kow poplato vaace Let be a adom vaable wth mea µ ad stadad devato σ. Accodg to the cetal lmt theoem ad eqato (3.0, the sample mea (the aveage of adom vaables N µ, σ, f s lage eogh. Moeove, t wll have wll have omal dstbto ( / pecsely ths omal dstbto espectvely of the sze, f the adom vaable s omal. The poblem we wsh to stdy hee s the detemato of the cofdece tevals of the mea µ fo cofdece coeffcet γ. We deote z (+γ/ the ((+γ/-qatle of the stadad omal dstbto N(0, (that s the vale z that coespods to o-exceedece pobablty (+γ/. Appaetly, de to symmety, z ( γ/ z (+γ/ (see Fg. 3.. Ths, z( + γ / σ z( + γ / σ P µ < < µ + γ (3.3 o eqvaletly P z z < + ( γ / ( + γ σ < µ + / σ γ (3.3 Eqato (3.3 gves the cofdece tevals soght. Fo the mecal evalato we smply eplace the estmato (3.3 wth ts mecal vale x. Fo coveece, Table 3. dsplays the most commoly sed cofdece coeffcets ad the coespodg omal qatles z (+γ/. We obseve that as the cofdece coeffcet teds to, whch meas that the elablty of the estmate ceases, the cofdece teval becomes lage so that the estmate becomes moe vage. O the cotay, f we choose a smalle cofdece coeffcet, a moe compact estmate wll eslt. I ths case, the cofdece teval wll be aowe bt the cetaty wll be hghe. f x ( Aea ( γ / Aea γ Aea ( γ / µ z + ( γ / σ µ z + Fg. 3. Explaatoy sketch fo the cofdece tevals of the mea. µ ( + γ / σ x

12 3. Elemetay statstcal cocepts Table 3. Typcal vales of the omal qatles z (+γ/ sefl fo the calclato of cofdece tevals. γ (+γ/ z (+γ/ We obseve fom (3.3 that the oly way to cease the accacy wthot ceasg the legth of the cofdece teval s to cease the sample sze by takg addtoal measemets. The pevos aalyss was based o the assmpto of kow poplato vaace, whch pactce t s ot ealstc, becase typcally all o fomato comes fom a sample. Howeve, the eslts ae of pactcal teest, sce (3.3 povdes a good appoxmato f the sample sze s lage eogh (> 30 ad f we eplace the poplato vaace wth ts sample estmate Mea kow poplato vaace The aalyss that we peset hee ca be sed fo kow poplato vaace ad fo ay sample sze. Howeve, ths aalyss has a estctve codto, that the adom vaable s omal, N(µ, σ. I ths case the followg coclsos ca be daw:. The sample mea has a omal dstbto N( µ σ,. Ths coclso s a / coseqece of a basc popety of the omal dstbto, specfcally the omal dstbto s closed de addto o, else, a stable dstbto.. The fcto of the sample vaace ( σ / follows the χ ( dstbto. Ths s coclded by the theoem of secto.0.4, accodg to whch the sm of the sqaes of a mbe of stadad omal vaables follows the χ dstbto. 3. The adom vaables ad ae depedet. Ths eslts fom a statstcal theoem (see e.g. Papols, 990, p.. 4. The ato ( µ /( / follows the tdet t( dstbto. Ths eslts by a theoem of.0.5. We deote t (+γ/ the [(+γ/]-qatle of the tdet t( dstbto (that s the pot t that coespods to exceedece pobablty (+γ/, fo degees of feedom. Becase of the symmety, t ( γ/ t (+γ/. Ths, o eqvaletly P t µ < < / + ( + γ / t( γ / γ (3.33

13 3.4 Typcal cofdece tevals 3 P t t < + ( + γ / ( + γ < µ / γ (3.34 Eqato (3.34 povdes the cofdece teval soght. Fo ts mecal evalato we smply eplace the teval estmatos of (3.34 the estmatos ad wth the coespodg sample estmates x ad s. Eve thogh (3.3 ad (3.34 ae cosdeably dffeet egadg the theoetcal gods ad the assmptos they ely po, fom a comptatoal pespectve they ae qte smla. Fthemoe, fo lage (>30 they pactcally cocde takg to accot that t ( + γ / z( + γ / (moe pecsely t( + γ / z( + γ / ( /( 3, fo degees of feedom. The two pevos aalyses do ot clde the case of a small sample sze, kow vaace ad o-omal dstbto. Ths case s ot coveed statstcs a geeal ad goos mae. Howeve, as a appoxmato, ofte the same methodology s also sed these cases, povded that the poplato dstbto s bell shaped ad ot too skewed. I geeal, the cases whee pecse cofdece tevals ca be detemed based o a cosstet theoetcal pocede, ae the excepto athe tha the le. I most of the followg poblems we wll se jst appoxmatos of the cofdece tevals A mecal example of teval estmato of the mea Fom a sample of aal flows to a esevo wth legth 5 (yeas, the sample mea s 0.05 hm 3 ad the sample stadad devato.80 hm 3. We wsh to deteme ( the 95% cofdece teval of the aal flow ad ( the sample sze fo 95% cofdece coeffcet that eables 0% pecso the estmato of the aal flow. ( We assme that the aal flows ae IID wth omal dstbto (secto.0. ad we se the eqato (3.34. Usg the table of the tdet dstbto (Appedx A3 o ay comptatoal method (see secto.0.5 we fd that fo 4 degees of feedom t (+γ/ t Coseqetly, the 95% cofdece teval s o / 5 < µ < / 5 ( hm < µ <.60 ( hm 3 Fo compaso, we wll calclate the cofdece teval sg eqato (3.3, eve thogh ths s ot coect. Fom Table 3. we fd z (+γ/ z Ths, the 95% cofdece teval s It wold ot be mathematcally coect to wte (3.34 eplacg the estmatos wth the estmates,.e. P{ / 5 < µ < / 5 } 0.95 We ote that µ s a (kow paamete (.e. a mbe ad ot a adom vaable, so t does ot have a dstbto fcto. Moeove, t s ot coect to say e.g. that wth 95% pobablty the mea vale les the teval (8.6,.94. The coect expesso wold be wth 95% cofdece.

14 4 3. Elemetay statstcal cocepts / 5 < µ < / 5 ( hm 3 o 8.63 < µ <.47 ( hm 3 The cofdece teval s ths case s a lttle smalle. ( Assmg that 30 we ca se (3.3. The followg eqato mst hold.96.8 / 0% 0.05 so 30. We obseve that the codto we have assmed ( 30 s vald. (If t wee ot vald we shold poceed wth a tal-ad-eo pocede, sg eqato ( Vaace ad stadad devato As the secto 0, we wll assme that the adom vaable has a omal dstbto N(µ, σ. As metoed befoe, ths case the fcto of the sample vaace / σ ( follows the χ ( dstbto. We deote χ + ad the [(+γ/]- ad [( γ/]-qatles, espectvely, of the ( γ / χ( γ / χ ( dstbto (the two ae ot eqal becase the χ dstbto s ot symmetc. Ths, we have P χ ( ( < χ ( γ / γ γ / < σ + (3.35 o eqvaletly P ( χ ( + γ ( ( γ < σ < / χ / γ (3.36 Eqato (3.36 gves the cofdece teval soght. It s easly obtaed that cofdece teval of the stadad devato s gve by P χ < σ < χ / ( + γ / ( γ γ ( A mecal example of teval estmato of stadad devato We wsh to deteme the 95% cofdece teval of the stadad devato of aal flow the poblem of secto The sample stadad devato s.8 hm 3. Wth the assmpto of omal dstbto fo the flow, we tlze eqato (3.37. Usg the χ dstbto table (Appedx o ay comptatoal method (see secto.0.4 we fd that fo 4 degees of feedom χ (+γ/ χ ad χ ( γ/ χ Ths, the 95% cofdece teval s σ < ( hm <

15 3.4 Typcal cofdece tevals 5 o.05 < σ < 4.4 ( hm Nomal dstbto qatle tadad eo I egeeg desg ad maagemet ( egeeg hydology patcla, the most feqet cofdece teval poblem that we face, coces the estmato of desg vales fo qattes that ae modelled as adom vaables. Fo stace, hydologcal desg we may wsh to estmate the esevo flow that coespods to a o-exceedece pobablty %, that s the % qatle of the flow. Let be a adom vaable wth dstbto F (x epesetg a atal qatty, e.g. a hydologcal vaable. Hee we assme that F (x s a omal dstbto N(µ, σ, whch ca be easly hadled, wheeas Chapte 6 we wll peset smla methods fo a epetoe of dstbtos beg commoly sed egeeg applcatos. Fo a gve o-exceedece pobablty F (x, the coespodg vale of the vaable ( symbolcally x, the -qatle wll be x µ + z σ (3.38 whee z the -qatle of the stadad omal dstbto N(0,. Howeve, ths eqato the poplato paametes µ ad σ ae kow pactce. Usg the pot estmates, we obta a estmate x ˆ x + z s, that ca be cosdeed as a vale of the adom vaable ˆ + z (3.39 Ths latte eqato ca be sed to deteme the cofdece teval of x. The pecse detemato s pactcally mpossble, de to the complexty of the dstbto fcto of ˆ. Hee we wll cofe o aalyss seekg a appoxmate cofdece teval, based o the assmpto that ˆ has omal dstbto. The mea of ˆ (3.5 to gve s gve fom eqato (.59, whch ca be combed wth (3.0 ad [ ] E[ ] + z E[ ] µ + zσ x E ˆ (3.40 assmg that s lage eogh ad omttg the tem O(/ E[ ]. Lkewse, the vaace of ˆ s gve by eqato (.6, whch ca be wtte as Va [ ˆ ] Va[ ] + z Va[ ] + z Cov[, ] (3.4 Gve that has omal dstbto, the thd tem of (3.4 s zeo (as metoed befoe, the vaables ad ae depedet. Combg (3.0 ad (3.6, we wte (3.4 as The aalyss hee has a appoxmate chaacte, ths we do ot dscmate betwee the estmatos ad, becase fo lage eogh the two estmatos ae vtally detcal.

16 6 3. Elemetay statstcal cocepts [ ˆ σ ] + σ σ z z + ε : Va (3.4 The qatty ε s kow lteate as stadad qatle eo o smply as stadad eo. Assmg that ˆ has a omal dstbto N(x, ε we ca wte ˆ x P z(+ γ / < < z(+ γ / γ ε (3.43 whee γ s the cofdece coeffcet. Eqvaletly, P { z ε < x < ˆ + z ε } γ ˆ (3.44 (+ γ / (+ γ / Replacg the pevos eqato the tem ε fom (3.4, ad the the stadad devato σ wth ts estmato, we obta the followg fal elatoshp P ˆ z z z γ + < x < ˆ ( + / + z( γ / + γ ( The latte eqato s a appoxmato, whose accacy s ceased a ceases. Moeove, t s vald oly the case of omal dstbto. Howeve, (3.44 s also sed fo othe dstbtos of the vaable, bt wth a dffeet expesso of the stadad eo ε ad a dffeet calclato method. The teested eade fo a geeal expesso of the stadad eo may coslt Kte (988, p The estmates of the cofdece lmts ae xˆ xˆ z s z s z xˆ xˆ (+ γ / +, + z( γ / ( Clealy these estmates ae fctos of o, eqvaletly, of the exceedece pobablty,. The depctos of those fctos a pobablty plot placed o ethe sde of the x cve ae kow as cofdece cves of x A mecal example of teval estmato of dstbto qatles Fthe to the mecal example of secto 3.4.3, we wsh to deteme the 95% cofdece teval of the aal flow that has exceedece pobablty (a % ad (b 99%. We ote that, becase of the small sample sze, we wll ot expect a hgh degee of accacy o estmates (ecall that the theoetcal aalyss assmed lage sample sze. We wll calclate fst the pot estmates (all ts ae hm 3. Fo the aal flow wth exceedece pobablty F 0.0 we have F 0.99 ad z.36. Ths, the pot estmate of xˆ Lkewse, fo the aal flow wth exceedece pobablty F 0.99 we have F 0.0 ad z.36, ths xˆ

17 3.4 Typcal cofdece tevals 7 We ca ow poceed the calclato of the cofdece lmts. Fo γ 95% ad z (+γ/.96, the lmts fo the flow wth exceedece pobablty % ae: xˆ xˆ Lkewse, the lmts fo exceedece pobablty 99% ae: xˆ xˆ Coelato coeffcet To calclate the cofdece lmts of the coelato coeffcet ρ of a poplato descbed by two vaables ad Y, we se the axlay vaable Z, defed by the so-called Fshe tasfomato: Z + R e Z l R tah Z (3.47 Z R e + whee R the sample coelato coeffcet. We obseve that fo < R < the age of Z s < Z <, whle fo R 0, Z 0. It ca be show that f ad Y ae omally dstbted, the Z has appoxmately omal dstbto N(µ Z, σ Z whee [ ] + ρ l, Z E Z σ Va[ Z ] (3.48 ρ 3 µ Z As a coseqece, f ζ (+γ/ s the (+γ/-qatle of the stadad omal dstbto, we obta ζ (+ γ / ζ (+ γ / P µ Z < Z < µ Z γ (3.49 o eqvaletly ζ P Z (+ γ / 3 < µ Z ζ (+ γ / < Z + γ 3 (3.50 Replacg µ Z fom (3.48 to (3.50 ad solvg fo ρ, ad also takg to accot the mootocty of the tasfomato (3.47, we obta P{R < ρ < R } (3.5 whee

18 8 3. Elemetay statstcal cocepts e R e Z ζ (+ γ / Z ± Z 3 Z + R e e Z Z Z Z l + + R R (3.5 To mecally evalate the cofdece lmts we mplemet eqatos (3.5 eplacg the estmatos wth the coespodg estmates (e.g. R, etc Paamete estmato of dstbto fctos Assmg a adom vaable wth kow dstbto fcto ad wth kow paametes θ, θ,, θ we ca deote the pobablty desty fcto of as a fcto f (x; θ, θ,, θ.. Fo kow x, ths, vewed as a fcto of the kow paametes, s called the lkelhood fcto. Hee, we wll exame the poblem of the estmato of these paametes based o a sample,,,. pecfcally, we wll peset the two most classcal methods statstcs, amely the momets method ad the maxmm lkelhood method. I addto, we wll peset a ewe method that has become popla hydology, the method of L momets. eveal othe geeal methods have bee developed statstcs fo paamete estmato, e.g. the maxmm etopy method that has bee also sed hydology (the teested eade s efeeced to gh ad Rajagopal, 986. Moeove, egeeg hydology may cases, othe types of methods lke gaphcal, mecal, empcal ad sem-empcal have bee sed. Examples of sch methods wll be gve fo ceta dstbtos chapte The method of momets The method of momets s based o eqatg the theoetcal momets of vaable wth the coespodg sample momet estmates. Ths, f s the mbe of the kow paametes of the dstbto, we ca wte eqatos of the fom whee (k m ad ae gve by m ( k mˆ, k,, K ( k, (3.53 ae the theoetcal aw momets, whch ae fctos of the kow paametes wheeas ( mˆ k m ( k k x f ( x, θ,, K θ dx ae the estmates, calclated fom the obseved sample accodg to (3.54 k k mˆ x (3.55 (

19 3.5 Paamete estmato of dstbto fctos 9 Ths, the solto of the esltg system of the eqatos gves the kow paametes θ, θ,, θ. I geeal, the system of eqatos may ot be lea ad may ot have a aalytcal solto. I ths case the system ca be solved oly mecally. Eqvaletly, we ca se the cetal momets (fo k > stead of the aw momets. I ths case, the system of eqatos s µ ( k ( k x, µ µ ˆ, k, K (3.56, whee ( µ m cetal momets gve by the s the poplato mea, ( ˆ (k x m the sample mea, the theoetcal µ ad ( µ ˆ k ( k µ, ( k x µ f ( x θ θ ;, K dx the coespodg sample estmates calclated by the elatoshp ( (3.57 k k µ ˆ x x (3.58 ( We ecall that the aw momets (3.55 ae based estmates, whle the cetal momets (3.58 ae based estmates. Nevetheless, based cetal momet estmates ae ofte sed stead of the based. Regadless of sg based o based estmates fo momets, geeal the method of momets does ot eslt based estmates of the paametes θ, θ,, θ (except specal cases Demostato of the method of momets fo the omal dstbto As a example of the mplemetato of the method of momets, we wll calclate the paametes of the omal dstbto. The pobablty desty fcto s: f ( x µ, σ ( x µ σ ; e (3.59 σ ad has two paamete, µ ad σ. Ths, we eed two eqatos. Based o (3.56, these eqatos ae π µ x (3.60, σ s whee the latte eqato we have deoted the theoetcal ad sample vaace (that s, the secod cetal momet of, by the moe commo symbols ad, espectvely. We kow (see secto.0. that the theoetcal momets ae Coseqetly, the fal estmates ae µ µ, σ σ σ s (3.6 µ x x, σ s ( x x (3.6

20 0 3. Elemetay statstcal cocepts Ths estmato of σ based, whle that of µ s based. If the last eqato we sed the based estmate of the vaace, the we wold have the deomato (- stead of. Eve ths case, the estmate of the σ wold ot be based, fo easos explaed secto As we have see ths example, the applcato of the method of momets s vey smple ad ths exteds to othe dstbto fctos The method of L momets The logc of the method of L momets s the same as the method of momets. If the dstbto has kow paametes, we ca wte eqatos of the fom λ k l k, k,,, (3.63 whee λ k ae the theoetcal L-momets, whch ae fctos of the kow paametes, ad l k the sample estmates. olvg ths system of eqatos we obta the L-momet estmates of the kow paametes of the dstbto. Becase L momets ae lea combatos of the pobablty weghted momets, (3.63 ca be wtte eqvaletly as β k b k, k 0,,, (3.64 whee β k s the pobablty weghted momet of ode k ad b k s ts estmate (see secto Estmates based o L-momets ae geeally moe elable tha those based o classcal momets. Moeove, the L-momet estmatos have some statstcally desable popetes e.g. they ae obst wth espect to otles, becase cotay to stadad momets, they do ot volve sqag, cbg, etc., of the sample obsevatos. I hydology, the L momets have bee wdely sed as descptve statstcs ad paamete estmato of seveal dstbtos. Examples of applcatos of the method ca be fod, amog othes, Kjeldse et al. (00, Koll ad Vogel (00, Lm ad Lye (003 ad Zadma et al. ( The maxmm lkelhood method Let be a adom vaable wth pobablty fcto f (x, θ, θ,, θ whee θ, θ,, θ ae paametes, ad,,, a sample of the vaable. Let f K, ( x K, x; θ, Kθ be, the jot dstbto fcto of the sample vecto : [,,, ] T. O ete obseved sample ca be thoght of as sgle obsevato of the vecto vaable. The dea behd the maxmm lkelhood method s that the pobablty desty f,, ( K at ths sgle pot wll be as hgh as possble (t s atal to expect a obsevato to le a aea wth hgh pobablty desty. We ca ths fd θ, θ,, θ, so that the fcto have a vale as hgh as possble at the pot (x, x,, x. I a adom sample, the vaables,,, ae depedet ad the jot pobablty desty fcto s, f,, K (

21 3.5 Paamete estmato of dstbto fctos f,,, K ( x, K, x ; θ, Kθ f ( x, θ θ K (3.65 ad, vewed as a fcto of the paametes θ, θ,, θ (fo vales of adom vaables eqal to the obsevatos x,, x s the lkelhood fcto of these paametes. Assmg that s dffeetable wth espect to ts paametes, the codto that maxmzes t s f,, K ( f ( x, K, x ; θ, K, θ, K, K, θ k 0, k, (3.66 Usg these eqatos, the kow paametes wll eslt. Howeve, the maplato of these eqatos may be complcated ad, stead of maxmzg the lkelhood, we may attempt to maxmze ts logathm L,, x θ ;, ( x, x ; θ, Kθ : l f ( x, K, x ; θ, Kθ l f (, K Kθ K (3.67 The fcto L( s called the log-lkelhood fcto of the paametes θ, θ,, θ. I ths case, the codto of maxmm s L ( x, K, x ; θ, Kθ ( x ; θ, Kθ ( x ; θ, Kθ f 0, k, θ f θ, K k k (3.68 olvg these eqatos we obta the vales of the kow paametes Demostato of the maxmm lkelhood method fo the omal dstbto We wll calclate the paametes of the omal dstbto sg the maxmm lkelhood method. The pobablty desty fcto of the omal dstbto s f ( x µ, σ Based o (3.65 we fom the lkelhood fcto ( x µ σ ; e (3.69 σ π ( x µ ( x,k, x; µ, σ (3.70 σ f e ( σ π ad takg ts logathm we fom the log-lkelhood fcto: L,K (3.7 x µ σ ( x, x ; µ, σ l( π l σ ( Takg the devatves wth espect of the kow paametes µ ad σ ad eqatg them to 0 we have L L 3 µ σ σ σ σ ( x m 0, + ( x 0 µ (3.7

22 3. Elemetay statstcal cocepts ad solvg the system we obta the fal paamete estmates: µ x x, σ ( x x s (3.73 The eslts ae pecsely detcal wth those of the secto 3.5., despte the fact that the two methods ae fdametally dffeet. The applcato of the maxmm lkelhood method s moe complex tha that of the method of momets. The detcal eslts fod hee ae ot the le fo all dstbto fctos. O the cotay, most cases the two methods yeld dffeet eslts. 3.6 Hypothess testg A statstcal hypothess s a hypothess elated to the vale of oe o moe paametes of a statstcal model, whch s descbed by a dstbto fcto. The hypothess testg s a pocess of establshg the valdty of a hypothess. The pocess has two possble otcomes: ethe the hypothess s ejected o accepted (moe pecsely: ot ejected. I ths secto we peset vey befly the elated temology ad pocede, whle ext chaptes we wll peset some applcatos. The eade teested fo a moe detaled pesetato of the theoy shold coslt statstcs books (e.g. Papols, 990, p , Fed et al., 988, p , whle fo a pesetato fo hydologcal applcatos s efeeced to Hsch et al. (993, p Temology Nll hypothess s the hypothess to be tested (symbolcally H 0. Usally, t s a hypothess of the fom θ θ 0, whee θ s paamete elated to a dstbto fcto of a gve vaable ad θ 0 s a mecal vale. Alteatve hypothess s a secod hypothess that shold ot be te at the same tme wth the ll hypothess (symbolcally H. It ca be smple, sch as θ θ, o (moe commoly composte, sch as θ θ 0, θ > θ 0 o θ < θ 0. Test statstc s a appopately chose sample statstc, that s sed fo the test (symbolcally Q. Ctcal ego s a teval of eal vales. Whe the test statstc vale les the ctcal ego the the ll hypothess s ejected (symbolcally R c ; see Fg. 3.. Oe-sded test s a test whee the alteatve hypothess s of the fom θ > θ 0 o θ < θ 0. I ths case the ctcal ego s a half le of the fom (q > q C o (q < q C, espectvely. Two-sde test s a test whee the alteatve hypothess f of the fom θ θ 0. I ths case the ctcal ego cossts of two half les (q < q L ad (q > q U.

23 3.6 Hypothess testg 3 f Q ( q, θ (a H :θ θ 0 H 0 : θ θ 0 Aea α/ Aea β Aea α/ Ctcal ego R c q L q / q q α Rego of o ejecto U α / Ctcal ego R c q f Q ( q, θ (b H :θ > θ 0 H 0 : θ θ 0 Aea β Aea α Rego of o ejecto q C q α Ctcal ego R c q Fg. 3. Explaatoy sketch of the cocepts elated to statstcal testg: (a a two-sded test, (b a oe-sded test. Paametc s a test whose hypotheses clde specfcato of the poplato dstbto fcto. No paametc s a test vald fo evey poplato dstbto fcto. Decso le s the le to eject o ot the ll hypothess, expessed as: eject H 0 f q R c Type Ι eo s the ejecto (based o the decso le of a te ll hypothess. Type ΙΙ eo s the acceptace (based o the decso le of a false ll hypothess gfcace level of a test s the pobablty of type I eo, amely the pobablty to eject a te ll hypothess. ymbolcally { Q R } α P c H 0 (3.74 Powe of a test s the pobablty of ejectg a false ll hypothess. ymbolcally, { Q R } p β P H (3.75 c

24 4 3. Elemetay statstcal cocepts whee β s the pobablty of type II eo, that s { Q R } β P c H ( Testg pocede The testg pocede cossts of the followg steps:. Fomlato of the ll hypothess H 0 ad of the alteatve H.. Choce of the test statstc Q g(,, ad detemato of the pobablty desty fcto of the f Q (q; θ. 3. Choce of the sgfcace level α of the test ad detemato of the ctcal ego R c. 4. Calclato of the vale q g(x,, x of Q fom the sample. 5. Applcato of the decso le ad ejecto o acceptace of H Calclato of the powe p of the test. The last step s sally omtted pactce, de to ts complexty. All emag steps ae clafed the followg secto Demostato of sgfcace testg fo the coelato coeffcet As a example of the above pocede we wll peset the sgfcace testg of the coelato coeffcet of two adom vaables ad Y, accodg to whch we ca decde whethe o ot the vaables ae lealy coelated. If the vaables ae ot lealy coelated the the coelato coeffcet wll be zeo. Based o ths obsevato, we poceed the followg steps of the statstcal testg.. The ll hypothess H 0 s ρ 0 ad the alteatve hypothess H s ρ 0. As a coseqece we wll poceed wth a two-sdes test. (If we wated to decde o the type of coelato, postve o egatve, the alteatve hypothess wold be fomlated as ρ > 0 o ρ < 0, ad we wold pefom a oe-sded test.. We choose the test statstc as + R Q l 3 Z 3 (3.77 R whee R s the sample coelato coeffcet ad Z s the axlay Fshe vaable (secto 3.3.6, whch, f H 0 s te, has appoxmately a omal dstbto wth mea 0 ad stadad devato / 3. Coseqetly, Q has stadad omal dstbto N(0,. 3. We choose a sgfcace level α If z α/ s the ( α/-qatle of the omal dstbto, the the coespodg ctcal ego R c s the q > z α/ o q > z 0.975, o fally q >.96, gve that

25 3.6 Hypothess testg 5 P( Q > z α/ P(Q < z α/ + P(Q > z α/ P(Q < z α/ α / α (We ecall that, becase of the symmety of the omal pobablty desty fcto, z z. I the case of the oe-sde test wth alteatve hypothess ρ > 0, the ctcal ego wold be q > z α. 4. The mecal vale of q s detemed fom the obseved sample by the followg eqatos + q l 5. The decso le wll be: ad fo α , eject H 0 f q > z α/ ( x x( y y ( x x ( y y + eject H 0 f q l 3 >. 96 (3.78 At the lmt of ths eqalty, solvg fo, we fd the ctcal vale c of the sample coelato coeffcet, that detemes the ctcal ego R c of the statstc R, that s, 3.9 / 3 e c ( / 3 e + A smple fomla easy to emembe that povdes a vey good appoxmato of (3.79 s: c (3.80 As a coseqece, we ca codct the hypothess testg a moe dect mae, by compag the absolte vale of wth the ctcal vale c. If > c the we coclde that thee s statstcally sgfcat coelato betwee the two vaables A mecal example of sgfcace testg fo the coelato coeffcet Fom a 8-yea-log ecod of measemets of cocet aal afall ad off at a catchmet, we have calclated the coelato coeffcet eqal to Is thee a lea coelato betwee the aal afall ad off? both: We calclate the ctcal vale c sg oe of (3.79 o (3.80. Hee fo compaso we se 0.470, / 5 c 3.9 / 5 c e e

26 6 3. Elemetay statstcal cocepts Ideed, the two eqatos gve pactcally the same eslt. ce 0.58 > 0.47 we coclde that thee s statstcally sgfcat coelato betwee the aal afall ad off. Ackowledgemet I thak mo Papalexo fo taslatg to Eglsh a eale Geek veso of ths text. Refeeces Bobée, B., ad R. Robtalle, Coecto of bas the estmato of the coeffcet of skewess, Wate Reso. Res., (6, , 975. Fed, J. E., F. J. Wllams, ad B. M. Peles, Elemetay Bsess tatstcs, The Mode Appoach, Petce-Hall, 988. Hsch, R. M., D. R. Helsel, T. A. Coh, ad E. J. Gloy, tatstcal aalyss of hydologc data, Hadbook of Hydology, edted by D. R. Madmet, McGaw-Hll, 993. Kedall, M. G., ad A. tat, The Advaced Theoy of tatstcs, Vol., Ifeece ad elatoshp, Thd edto, Chales Gff & Co., Lodo, 973. Kte, G. W., Feqecy ad Rsk Aalyses Hydology, Wate Resoces Pblcatos, Lttleto, Coloado, 988. Kby, W., Algebac bodess of sample statstcs, Wate Reso. Res., 0(, 0-, 974. Kjeldse, T.R., J.C. mthes ad R.E. chlze, Regoal flood feqecy aalyss povce, oth Afca, sg the dex-flood method, J. Hydol., 55, 94, 00. Koll, C.N., ad R.M. Vogel, Pobablty dstbto of low steamflow sees the Uted tates, J. Hydol. Eg., 7, 37 46, 00. Ladweh, J.M., N.C. Matalas ad J.R. Walls, Pobablty weghted momets wth some tadtoal techqes estmatg Gmbel paametes ad qatles. Wate Reso. Res., 5, , 979. Lm, Y.H., ad L.M. Lye, Regoal flood estmato fo gaged bass aawak, Malaysa, Hydol. c. J., 48, 79 94, 003. Papols, A., Pobablty ad tatstcs, Petce-Hall, New Jesey, 990. gh, V. P., ad A. K. Rajagopal, A ew method of paamete estmato fo hydologc feqecy aalyss, Hydologcal cece ad Techology, ( , 986. Walls, J. R., N. C. Matalas, ad J. R. lack, Jst a momet!, Wate Reso. Res., 0(, - 9, 974. Yevjevch, V., Pobablty ad tatstcs Hydology, Wate Resoces Pblcatos, Fot Colls, Coloado, 97. Zadma, M.D., V. Kelle, A.R. Yog ad D. Cadma, Flow-dato-feqecy behavo of Btsh ves based o aal mma data, J. Hydol., 77, 95 3, 003.