Improved Parameter Estimation in Rayleigh Model

Transcription

1 etodološ zvez, Vol. 3, No., 6, Impoved Paamete Etmato Raylegh odel Smal ahd Abtact I th pape we decbe ad peet eult o the paamete pot etmato fo the cale ad thehold paamete of the Raylegh dtbuto. Fve etmatg method have bee vetgated, amely, the mamum lelhood, the method of momet, the pobablty weghted momet method, the leat quae method ad the leat abolute devato method. odfed mamum lelhood etmato fo the paamete ae alo popoed. Smulato tude have how that the modfed lelhood etmato outpefom the etmato obtaed wth the othe method ecept the cae of vey mall ample. Itoducto The two-paamete Raylegh dtbuto a cotuou pobablty dtbuto whch uually ae whe a two dmeoal vecto ha t two othogoal compoet omally ad depedetly dtbuted. The Eucldea om of the vecto wll the have a Raylegh dtbuto. The dtbuto may alo ae the cae of adom comple umbe whoe eal ad magay compoet ae omally ad depedetly dtbuted. The modulu of thee umbe wll the be Raylegh dtbuted. The Raylegh vaable wth thehold paamete ad cale paamete chaactezed by the cumulatve fucto F e fo < ad >.. Th dtbuto play a mpotat eal lfe applcato ce t elate to a lage umbe of dtbuto uch a geealzed eteme value, Webull, chquae ad ce dtbuto. I th pape we vetgate the etmato of the cale ad thehold paamete ug a modfed mamum lelhood method L, the momet method, the pobablty weghted momet method Depatmet of Compute Scece, athematc ad Phyc, Uvety of the Wet Ide, Cave Hll Campu, Babado.

2 64 Smal ahd PW, the oday leat quae method OLS ad the leat abolute devato LAD method. The PW method a elatvely ecet techque whch well ued by hydologt fequecy aaly. The method togly advocated Hog et al. 985 ad accodg to Davo ad Smth 99 t cottute the mot eou competto to the L method, epecally, the cae of mall ample. The pefomace of th techque ha bee ecetly vetgated ahd ad Aha 4 ad Aha ad ahd 3 Webull ad Loglogtc model, epectvely. We ogae th pape a follow. I Secto, we have toduced the poblem ad Secto, we deve the paamete etmato ug the codeed method ad alo gve eult o the aymptotc vaace. Smulato eult ae dcued ad llutated Secto 3. Etmato method We deve ad peet below etmato fo the paamete ad by ug the fve codeed method. We tat wth the pobablty weghted momet method.. Pobablty weghted momet The pobablty weghted momet of ode,j, obtaed fom the vee cumulatve fucto F l F a j [ ] [ ] j, j, Ε F F F F F df Μ.. We ue the uual ode ad j ce th lead to a cla of lea L- momet, ee, Hog 986; 99, wth aymptotc omalty. We deote the coepodg pobablty weghted momet mplfcato, we obta,, α. Afte tegato ad α /.. Subttutg two dtct ode ad to equato. gve the pobablty weghted momet etmate fo a α α.3

3 Impoved Paamete Etmato Raylegh odel 65 whee α C C fo,, K,, wth the coveto C f < j, the ubaed etmato fo α, ee Hog 989. Thu, the etmato fo gve by α..4 j.. Aymptotc vaace of ad The aymptotc vaace of ad ae appomated by ug the aymptotc vaace of the PW etmate α. We wll ue eult 5.3, povded Hog 986, tatg that the vecto whoe th compoet α α, fo,,..., m, ha a Gaua lmtg dtbuto wth mea vecto ad m covaace mat Α Α I A I I ad, whee < y F F y F F d dy..5 Ug the appomato 6.. Abamowtz ad Stegu 97: 93, we get afte tegato ad mplfcato, I [ ] The ft ode appomato fo the vaace ad covaace of ad ae obtaed fom the equato.6 Cov a, GAG T whee the tem of the by mat G, deved fom the pobablty weghted momet equato of the fom.3 ad.4, ae gve by G α,.7

4 66 Smal ahd G,.8 α G α,.9 ad G α... amum lelhood Settg to zeo the ft devatve of the log-lelhood fucto wth epect to gve the L etmate fo fo a gve value a.. The mamum lelhood etmate fo the paamete whch o bouday of the dtbuto uppot gve by,,, m. Note that th etmato baed ce dtbuted a a Raylegh vaable wth thehold paamete ad cale paamete. To pove that, let u deote G the cumulatve fucto of ample K. The fucto G, evaluated at y,, G y Pm { P whch baed o the dtbuto adom, K, f y K P K, gve by y Pm Subttutg ow Fy fom equato., we get G y y / e, K, > y f y} [ F y].

5 Impoved Paamete Etmato Raylegh odel 67 whch ha the ame fom a Fy. Theefoe the Raylegh vaable ha the mea. We popoe the to ue the modfed mamum lelhood etmato ad that ae oluto of the ytem of equato.3. ad whch ae baed o the ubaed etmato. Squag equato.3 ad epeg a fucto of fom equato. yeld the ecod ode equato of,.4. The dcmat of the above equato potve ad gve by..5 Theefoe, equato.4 ha two dtct oluto. Futhemoe th equato admt a uque potve oluto ce the oot poduct. p Thu the modfed etmato fo ad ae epectvely gve by f.6 ad

6 68 Smal ahd.7 whee ad ae the ft ad ecod ample momet, epectvely ad the ft ode tattc baed o the adom ample,, K... Vaace of ad. The aymptotc vaace of appomately gve by ] l [ f E Va.8 obtaed fom the ample Fhe fomato o. O the othe had, we have that Va Va by ug the dtbuto of. Thu, a cotet etmato of..3 ethod of momet The momet about the og of ode gve by, d e E..9 Afte tegato we get [ ] [ ]. / / Γ Γ C C. Th ca be mplfed a [ ].. / Γ C

7 Impoved Paamete Etmato Raylegh odel 69 The ft ad ecod ode o cetal momet ca be evaluated fom ethe equato. o.. Ug fo tace equato., we get Γ 3/. ad 3/ Γ..3 Th yeld the followg momet method etmato fo ad, ad whee.5 a etmato of the populato tadad devato..3. Aymptotc vaace ad covaace of ad The aymptotc vaace ad covaace of ad ae etmated fom the vaace ad covaace of the ample geeal momet ad, ee fo tace, ahd ad Aha 4, a follow: Va Va Cov, Va Va Cov, l l.6 whee l l l l Va ; Va l ; Cov, l ; ; ; l ad I the codeed cae ad l, we have l.

8 7 Smal ahd 4 Va, Va,.8 [4 ] Cov,,.9,.3,.3.3 ad fally: Regeo method The paamete ad ca alo be etmated though the lea egeo techque fom the elato l F. Oday leat quae etmate a well a leat abolute devato etmate fo ad ae obtaed fom the ample pot {, y} whee the th ample value coepodg to the empcal quatle F ad y l F. Oday leat quae etmate fo ad ae obtaed fom the uual tecept ad lope lea egeo etmate, ee, fo tace Rce 995. The leat abolute devato o meda egeo etmate of ad ae obtaed a oluto to the mmzato poblem: y a wth epect to a ad b. The b oluto obtaed by applyg the mple method to the lea pogammg poblem: ude the cotat y a b whee ad ae, epectvely, the potve ad egatve edual aocated wth the obevato,,...,.

9 Impoved Paamete Etmato Raylegh odel 7.4. Vaace of OLS etmato The computato of the vaace of the leat abolute devato etmato etemely tedou. Howeve, we ca fd the vaace of the oday leat quae etmato ude the aumpto of the tadad tattcal model, ee, fo tace Rce 995. Let u deote Ol ad Ol the OLS etmato of ad, epectvely. The vaace of thee etmato ae, epectvely, gve by σ Va Ol.34 l F l F ad Va Ol σ l F l F l F whee σ 4.35 Va.36 obtaed fom equato. ad.3. 3 Dcuo We have aeed the pefomace of the codeed etmato method though mulato tude. Dffeet value of the paamete have bee codeed a well a dffeet ample ze. Ode,,,3 ad,3 ae ued the PW method. The ample pot wee geeated ug the vee cumulatve fucto techque. The pobablty weghted momet ae etmated wth the plottg method outled Hog 986, that,, u, v etmated by u v u v,, p p whee

10 7 Smal ahd γ p fo ' >γ >. ' We ued the value γ. 35 ad ' whch ae ecommeded Hog 986 fo the tudy of the geealzed eteme value dtbuto ce the Raylegh well elated to t. Seveal value fo, ad wee codeed, amely,,4, 6, 8, ;,,3, 4,5, 6,7,8,9,, ad, 3, 5, 7, 9 ad. Small ample ze fom to 9 wee alo codeed ad obtaed coepodg eult ae dplayed Table 3. The oot mea quae eo RSE fo the etmate wee the computed ad ued a pefomace de. Note that epeo fo the aymptotc vaace ae alo obtaed wheeve t poble. Thee vaace may be ued, fo tace, to compute appomate cofdece boud fo the udelyg paamete. Ft we have foud that the vaato of the value doe ot affect the RSE eult. Oe ca the et, wthout lo of geealty,. Howeve, the oot mea quae eo obtaed wth all method ceae a the value ceae, a llutated Table below. O the othe had, the tudy ha how that the PW ode ad povde bette RSE eult. Table : RSE of ad etmate obtaed wth the dffeet method combed by aveagg the ample ze,, 3, 4, 5, 6, 7, 8, 9 ad fo vaou value of. ad PW ode ad ae ued PW PW L L mm OLS OLS LAD LAD Ou vetgato ha alo how that the method of momet pefom pooly compao to the othe method. Table, dplayed below, gve the oot mea

11 Impoved Paamete Etmato Raylegh odel 73 quae eo a fucto of the ample ze. It how that the oot mea quae value ae mootocally deceag wth the ample ze. Oveall, all method have pefomed eaoably well ecept the method of momet. Howeve, the modfed mamum lelhood method povde bette etmate fo, wth ay ample ze, ad fo both ad paamete whe the ample ze ae ot mall, ay, a llutated Table ad 3. Note that the cae of mall ample, the PW method outpefom the mamum lelhood method fo the etmato of ad pefom almot a good a the mamum lelhood method fo the etmato of, ee, Table 3 eult. Coequetly, we ecommed ug the modfed mamum lelhood method fo the paamete etmato of the Raylegh dtbuto the cae of o mall ample. Howeve, we otce that thee a ga ug the PW method fo etmatg the Raylegh thehold paamete whe the ample ze mall; th cofm Davo ad Smth 99 tatemet. Table : RSE of ad etmate, obtaed wth the dffeet method, combed by aveagg ove the value, 4, 6, 8 ad fo vaou ample ze. ad PW ode ad ae ued PW PW L L mm OLS OLS LAD LAD Acowledgmet The facal uppot of UWI gatefully acowledged. The autho gateful to the edto ad efeee fo the valuable help ad uggeto that beefted ad mpoved th atcle.

12 74 Smal ahd Table 3: RSE of ad etmate, obtaed wth the dffeet method, combed by aveagg ove the value, 4, 6, 8 ad fo mall value 5, 6, 7, 8 ad 9. ad PW ode ad ae ued PW L L OLS LAD PW mm OLS LAD Note: The umecal tude have bee caed out wth Gau ad SPSS, Releae. Refeece [] Abamowtz,. ad Stegu, I. 97: Hadboo of athematcal Fucto. New Yo: Dove Publcato, Ic. [] Aha, F. ad ahd, S. 3: Compao of two fttg method fo the log-logtc dtbuto. Wate Reouce Reeach, 39, o. 8. [3] Davo, A.C. ad Smth, R.L. 99: odel fo eceedace ove hgh thehold. J. R. Stat. Soc. B, 5 3, [4] Hog, J.R.. 986: The Theoy of Pobablty Weghted omet. New Yo. Reeach Repot RC, IB Thoma J. Wato Reeach Cete. [5] Hog, J.R.. 99: L-omet: aaly ad etmato of dtbuto ug lea combato of ode tattc. J. R. Stat. Soc. B, 5, 5-4. [6] ahd, S. ad Aha, F. 4: Eplog Geealzed Pobablty Weghted omet, Geealzed omet ad amum Lelhood Etmatg ethod Two-Paamete Webull odel. Jou. of Hydology, 85, [7] Rce, J.A. 995: athematcal Stattc ad Data Aaly, d Ed. Dubuy Pe.