Rajesh gh & Floret maradache edtors & authors TUDIE I AMPLIG TECHIQUE AD TIME ERIE AALYI Comarso of forecasts Tme seres values 8 6 4 3 4 5 6 7 8 9 3 4 5 6 7 8 9 Perod Observed Data Forecast(sgle) Forecast(double) Z Publshg
TUDIE I AMPLIG TECHIQUE AD TIME ERIE AALYI Rajesh gh Deartmet of tatstcs, BHU, Varaas (U. P.), Ida Floret maradache Deartmet of Mathematcs, Uversty of ew Mexco, Gallu, UA edtors ad authors
Ths boo ca be ordered o aer or electroc formats from: Z Publshg 33 Chesaeae Aveue Columbus, Oho 43 UA Tel. (64) 485-7 E-mal: fo@zublshg.com ebste: www.zublshg.com Coyrght by Z Publshg ad the Authors Frot ad bac covers by Edtors Peer-revewers: Prof. Io Pătraşcu, Fraţ Buzeşt atal College, Craova, Romaa. Prof. Luge Vlădăreau, Isttude of old Mechacs, Romaa Academy, Bycharest, Romaa. Eg. Vctor Chrstato, Malag, Idoesa. Prof. H. P gh, chool of tudes tatstcs, Vram Uversty, Ujja, M.P., Ida. Dr. Jayat gh, Deartmet of tatstcs, Rajastha Uversty, Jaur, Ida. IB: 97859973599 Prted the Uted tates of Amerca
Cotets Preface: 4 Tme eres Aalyss of ater Qualty of Ramgarh Lae of Rajastha: 5 Estmatg the Poulato Mea tratfed Poulato usg Auxlary Iformato uder o-resose: 4 O ome ew Allocato chemes tratfed Radom amlg uder o- Resose: 4 A Famly of Estmators for Estmatg the Poulato Mea tratfed amlg: 55 A Famly of Estmators of Poulato Varace Usg Iformato o Auxlary Attrbute: 63 3
Preface Ths boo has bee desged for studets ad researchers who are worg the feld of tme seres aalyss ad estmato fte oulato. There are aers by Rajesh gh, Floret maradache, hweta Maurya, Ashsh K. gh, Maoj Kr. Chaudhary, V. K. gh, Muesh Kumar ad ach Mal. Frst chater deals wth the roblem of tme seres aalyss ad the rest of four chaters deal wth the roblems of estmato fte oulato. The boo s dvded fve chaters as follows: Chater. ater olluto s a major global roblem. I ths chater, tme seres aalyss s carred out to study the effect of certa ollutats o water of Ramgarh Lae of Rajastha, Ida. Chater. I ths chater famly of factor-tye estmators for estmatg oulato mea of stratfed oulato the resece of o-resose has bee dscussed. Choce of arorate estmator the famly order to get a desred level of accuracy resece of o-resose s wored out. Chater 3. I ths chater our am s to dscuss the exstg allocato schemes resece of o-resose ad to suggest some ew allocato schemes utlzg the owledge of resose ad o-resose rates of dfferet strata. Chater 4. I ths chater, we have suggested a mroved estmator for estmatg the oulato mea stratfed samlg resece of auxlary formato. Chater 5. I ths chater we have roosed some estmators for the oulato varace of the varable uder study, whch mae use of formato regardg the oulato roorto ossessg certa attrbute. 4 The Edtors
Tme eres Aalyss of ater Qualty of Ramgarh Lae of Rajastha Rajesh gh, hweta Maurya Deartmet of tatstcs, Baaras Hdu Uversty Varaas-5, IDIA Ashsh K. gh Raj Kumar Goel Isttute of Techology, Ghazabad, Ida. Floret maradache Deartmet of Mathematcs, Uversty of ew Mexco, Gallu, UA Abstract I ths chater a attemt has bee made to study the effect of certa ollutats o water of Ramgarh Lae of Rajastha. Tme seres aalyss of the observed data has bee doe usg tred, sgle exoetal smoothg ad double exoetal smoothg methods. Keywords: Pollutats, tred, sgle, double exoetal smoothg, tme seres.. Itroducto evety ercet of the earth s surface s covered by water. ater s udoubtedly the most recous atural resource that exsts o our laet. ater s a mortat comoet of the eco-system ad ay mbalace created ether terms of amout whch t s rereset or murtes added to t, Ca harm the whole eco-system. ater olluto occurs whe a body of water s adversely affected due to the addto of large amout of ollutat materals to the water. he t s uft for ts teded use, water s cosdered olluted. There are varous sources of water olluto (for detal refer to Ja ()) ome of the mortat water qualty factors are: ) Dssolved oxyge (D.O.) ) Bologcal oxyge demad (B.O.D.) 3) trate 5
4) Colform 5) P.H. Chemcal aalyss of ay samle of water gves us a comlete cture of ts hyscal ad chemcal costtuets. Ths wll gve us oly certa umercal value but for estmatg exact qualty of water a tme seres system has bee develoed ow as water qualty tred, whch gves us the dea of whole system for a log tme (see Ja()). I ths chater we are calculatg the tred values for fve water arameters of Ramgarh lae Rajastha for the year 995-6 ad for three arameters of Mah rver for the year 997-8.these methods vz. tred aalyss, sgle smoothg are used to aalyze the data.. Methodology After esurg the resece of tred the data, smoothg of the data s the ext requremet for tme seres aalyss. For smoothg the commo techques dscussed by Garder(985) are tred, smle exoetal smoothg (E), double exoetal smoothg (DE), trle exoetal smoothg (TE) ad adatve resose rate smle exoetal smoothg (ARRE). Ja () used tred method to aalyze the data. e have exteded the wor of Ja () ad aalyzed the data usg E ad DE ad comared these wth the hel of the avalable formato. The methods are descrbed below:. Fttg Of traght le The equato of the straght le s- U t a+b t where, u t observed value of the data, atercet value, bsloe of the straght le ad ttme ( years) Calculato for a ad b: 6
The ormal equatos for calculatg a ad b are U t a+bt t U t at+bt. gle Exoetal moothg The basc equato of exoetal smoothg s t α y t- + (-α) t-, α ad arameter α s called the smoothg costat. Here, stads for smoothed observato or EA ad y stads for the orgal observato. The subscrts refer to the tme erods,,3..,. The smoothed seres starts wth the smoothed verso of the secod observato. 3. Double Exoetal moothg gle smoothg does ot excel followg the data whe there s a tred. Ths stuato ca be mroved by the troducto of a secod equato wth a secod costat γ, whch must be chose cojucto wth α. t α y t + (-α) ( t- +b t- ), α b t γ ( t - t- ) + (- γ ) b t-, γ. For forecastg usg sgle ad double exoetal smoothg followg method s used- Forecastg wth sgle exoetal smoothg t αy t- +(-α) t- α The ew forecast s the old oe lus a adjustmet for the error that occurred the last forecast. 7
Boot strag of forecasts t+ α y org +(-α) t Ths formula wors whe last data ots ad o actual observato are avalable. Forecastg wth double exoetal smoothg The oe erod-ahead forecast s gve by: F t+ t +b t The m-erods-ahead forecast s gve by: F t+m t +mb t ( for detal of these methods refer to Garder (985)). 4. Results ad Dscusso Table shows the data o Dssolved Oxyge (D.O.) for Ramgardh Lae for the years 995-6 ad ftted values usg tred, sgle ad double exoetal smoothg. Table : Data o Dssolved Oxyge (D.O.) for Ramgardh Lae for the years 995-6 ad ftted values usg tred, sgle ad double exoetal smoothg Observed Data Tred values gle exoetal smoothg(alha.) Double exoetal smoothg(alha.9 ad gamma.) 995 5. 5.337 5. 996 5.75 5.336 5. 5.77 997 5.6 5.7 5.83 5.3857 998 5.7 5.364 5.97 5.698556 999 4.64 5.8 5.4363 4.765484 5.4 5.69 5.8367 5.884 3 4.3 5.356 5.7894 4.3944 4 6.6 5. 5.8446 5.858346 5 4.46 5.684 5.784 4.66965 6 5.5 5.348 5.647 5.437 Total 5.866 5.86 46.4684 5.96755 8
For tred, the ftted le s U t 5.866-.69 t wth mea squared error (ME).377. For sgle exoetal smoothg varous values of α are tred ad mmum ME.395 was obtaed for α.. For smoothg of the data, Holt s double exoetal smoothg was foud to be most arorate. Varous combatos of α ad γ both ragg betwee. ad.9 wth cremets of. were tred ad ME.94 was least for α.9 ad γ.. Fgure - Grah of observed data ad ftted values usg tred, sgle ad double exoetal smoothg of Dssolved Oxyge (D.O.) for Ramgardh Lae for the years 995-6 Dssolved oxyge 7 Tme seres values 6 5 4 3 995 996 997 998 999 3 4 5 6 Year Observed Data Tred values sgle exoetal smoothg(alha.) Double exoetal smoothg(alha.9 & gamma.) Adequatee dssolved oxyge s ecessary for good water qualty. Oxyge s a ecessary elemet to all forms of lfe. atural stream urfcato rocesses requre adequate oxyge levels order to rovde for aerobc lfe forms. As dssolved oxyge levels water dro below 5. mg/l, aquatc lfe s ut uder stress ( for detals see www.state.y.us) ). Form Table ad Fgure, we observe thatt level of D. O. Ramgarh Lae was above the requred stadard 5. mg/l, excet for the two years 999 ad 5. 9
Table : Comarso of forecasts Observed Perod Data 5. 5.75 3 5.6 4 5.7 5 4.64 6 5.4 7 4.3 8 6..6 9 4.46 5.5 3 4 5 6 7 8 9 Forecast(sgle) Forecast(double) 5.3 5.7437 6.384 5.593 5.7786965 5.7477 6.58348 4.6463633 5.34649 5.44367 5.85777 4.3948943 4.365563 6.658463 7.555859 4.467846 4.663535 5.5586475 6.686376834 5.4777573 5.666 5.8499858 5.744 5.849834 5.868 5.4864858 5.9794 5.75783657 6.97 5.359 6.46 5.38476 6.33 5.3496686 6.4498 5.359769657 6.5674 5.3757969 6.685 Fgure : Grah of forecasts Comarso of forecasts 8 Tme seres values 6 4 3 4 5 6 7 8 9 3 4 5 6 7 8 9 Perod Observed Data Forecast(sgle) Forecast(double)
Table 3 shows the data o trate for Ramgardh Lae for the years 995-6 ad ftted values usg tred, sgle ad double exoetal smoothg. Table 3: Data o trate for Ramgardh Lae for the years 995-6 ad ftted values usg tred, sgle ad double exoetal smoothg Year(x) Observed Data Tred values gle exoetal smoothg(alha.) Double exoetal smoothg(alha.9 ad gamma.) 995.3.588.3 996.8.546.3.76 997.38.54.46.4696 998.8.46.44.4883 999.4.4.3796.8797.9.378.34544.855 3.4.336.47.378 4.5.94.386448.4733 5.86.5.3783.84874 6.3..3543.843 Total 3.996 3.99 3.388897 3.989 For tred, the ftted le s U t.3996 -.6 t wth ME.378. For sgle exoetal smoothg varous values of α are tred ad mmum ME.34 was obtaed for α.. For smoothg of the data, Holt s double exoetal smoothg was foud to be most arorate. Varous combatos of α ad γ both ragg betwee. ad.9 wth cremets of. were tred ad ME.33 was least for α.9 ad γ.. trtes ca roduce a serous codto fsh called "brow blood dsease." trtes also react drectly wth hemoglob huma blood ad other warm-blooded amals to roduce methemoglob. Methemoglob destroys the ablty of red blood cells to trasort oxyge. Ths codto s esecally serous babes uder three moths of age. It causes a codto ow as methemoglobema or "blue baby" dsease. ater wth trte levels exceedg. mg/l should ot be used for feedg
babes. trte/troge levels below 9 mg/l ad trate levels below.5 mg/l seem to have o effect o warm water fsh (for detals see www.state.y.us). Form Table 3 ad Fgure 3, we observe that level of trate Ramgarh Lae was below the stadard. mg/l, excet for the year 996. Fgure 3: Grah of observed data ad ftted values usg tred, sgle ad double exoetal smoothg of trate for Ramgardh lae for the years 995-6 trate.4 tme seres values..8.6.4. 995 996 997 998 999 3 4 5 6 year Observed Data Tred values sgle exoetal smoothg(alha.) Double exoetal smoothg(alha.9 & gamma.) trtes ca roduce a serous codto fsh called "brow blood dsease." trtes also react drectly wth hemoglob huma blood ad other warm-blooded amals to roduce methemoglob. Methemoglob destroys the ablty of red blood cells to trasort oxyge. Ths codto s esecally serous babes uder three moths of age. It causes a codto ow as methemoglobema or "blue baby" dsease. ater wth trte levels exceedg. mg/l should ot be used for feedg babes. trte/troge levels below 9 mg/l ad trate levels below.5 mg/l seem to have o effect o warm water fsh ( for detals see www.state.y.us). Form Table 3 ad Fgure 3, we observe that level of trate Ramgarh Lae was below the stadard. mg/l, excet for the year 996.
Table 4: Comarso of forecasts Observed Perod Data.3.8 3.38 4.8 5.4 6.9 7.4 8.5 9..86.3 3 4 5 6 7 8 9 Forecast(sgle) Forecast(double).46.4.44.896.3796.3883.34544 -.73.47 -.795.386448.84785.3783.334.3543.7474.3487.439.3487.449.34384.446.339456.7.3355.6998.33959.384.38763.957.35887.5856.3398.4.3968 -.57.3887 -.586.36984 -.9 Fgure 4 : Grah of forecasts Comarso of forecastss.5 Tme seres values.5 -.5 3 4 5 6 7 8 9 3 4 5 6 7 8 9 erod Observed Data Forecast(sgle) Forecast(double) 3
Table 5 shows the data o B.O.D. for Ramgardh Lae for the years 995-6 ad ftted values usg tred, sgle ad double exoetal smoothg. Table 5: Data o Bologcal oxyge demad (B.O.D.) for Ramgardh Lae for the years 995-6 ad ftted values usg tred, sgle ad double exoetal smoothg Year(x) Observed Data Tred values sgle exoetal smoothg(alha.) Double exoetal smoothg(alha.9 & gamma.) 995.96 5..96 996 4.9 4.76.96.8767 997.84 4.4 3.73 3.47483 998.8 4.4 3.397.939 999.46 3.68.9573.49847.78 3.3.7757.6336 3.76.96.774.74563 4.976.6.777.49477 5.78.4.6964.69755 6 3.58.88.69956 3.489379 Total 35.6 35. 4.78994 34.98 Bochemcal oxyge demad s a measure of the quatty of oxyge used by mcroorgasms (e.g., aerobc bactera) the oxdato of orgac matter. atural sources of orgac matter clude lat decay ad leaf fall. However, lat growth ad decay may be uaturally accelerated whe utrets ad sulght are overly abudat due to huma fluece. Urba ruoff carres et wastes from streets ad sdewals; utrets from law fertlzers; leaves, grass clgs, ad chater from resdetal areas, whch crease oxyge demad. Oxyge cosumed the decomosto rocess robs other aquatc orgasms of the oxyge they eed to lve. Orgasms that are more tolerat of lower dssolved oxyge levels may relace a dversty of atural water systems cota bactera, whch eed oxyge (aerobc) to survve. Most of them feed o dead algae ad other dead orgasms ad are art of the decomosto cycle. Algae ad other roducers the water tae u orgac utrets ad use them the rocess of buldg u ther orgac tssues (for detals refer to www.freedrgwater.com). 4
For tred, the ftted le s U t 3.56+.894 t wth ME.74366. For sgle exoetal smoothg varous values of α are tred ad mmum ME 7.93 was obtaed for α.. For smoothg of the data, Holt s double exoetal smoothg was foud to be most arorate. Varous combatos of α ad γ both ragg betwee. ad.9 wth cremets of. were tred ad ME.3 was least for α..9 ad γ.. Fgure 5: Grah of observed data ad ftted values usg tred, sgle ad double exoetal smoothg of B.O.D. for Ramgardh Lae for the years 995-6 Bologcal oxyge demadd Tme seres values 6 4 8 6 4 995 996 997 998 999 3 4 5 6 Year Observed Data Tred values sgle exoetal smoothg(alha.) Double exoetal smoothg(alha.9 & gamma.) 5
Table 6: Comarso of forecasts Observed Perod Data.96 4.9 3.84 4.8 5.46 6.78 7.76 8..976 9.78 3.58 3 4 5 6 7 8 9 Forecast(sgle) Forecast(double).96667 3.73 3.9483 3.397 3.395.9573.76847.7757.364.774.58569.777.7768.6964.95553.69956.67379.78766 3.547575.787 3.5474.86639 3.655.93775 3.6636 3.976 3.77 3.59778 3.7798 3.8 3.8379 3.586 3..896 3.758 3.954 3.38683 4. 3.784 4.73 Fgure 6 : Grah of forecasts Comarso of forecasts Tme seres values 5 5 3 4 5 6 7 8 9 3 4 5 6 7 8 9 Perod Observed Data Forecast(sgle) Forecast(double) 6
Table 7 shows the data o Total Colfrm for Ramgardh Lae for the years 995-6 ad ftted values usg tred, sgle ad double exoetal smoothg. Table 7 : Data o Total Colfrm for Ramgardh lae for the years 995-6 ad ftted values usg tred, sgle ad double exoetal smoothg Year(x) Observed Data Tred values sgle exoetal smoothg(alha.6) Double exoetal smoothg(alha.9 & gamma.) 995 69 687.894 69 996 85 65.34 69 339.333 997 84.75 54.734 638.6 75.557 998 44 47.54 759.89 73.7353 999 65.33 397.574 39.356 4.47463.5 34.994 95.344 8.95854 3 9 5.44 5.36 87.566 4 7.6 79.834 3.845 63.4 5 86 7.54 485.858 3.3548 6 6.66 34.674 45.6343 47.87 Total 36.84 36.84 466.757 3447.783 Total colform bactera are a collecto of relatvely harmless mcroorgasms that lve large umbers the testes of ma ad warm- ad cold-blooded amals. They ad the dgesto of food. A secfc subgrou of ths collecto s the fecal colform bactera, the most commo member beg Eschercha col. These orgasms may be searated from the total colform grou by ther ablty to grow at elevated temeratures ad are assocated oly wth the fecal materal of warm-blooded amals. The resece of fecal colform bactera aquatc evromets dcates that the water has bee cotamated wth the fecal materal of ma or other amals. The resece of fecal cotamato s a dcator that a otetal health rs exsts for dvduals exosed to ths water (for detals see www.state.y.us). 7
For tred, the ftted le s U t 36.84-36.989 t wth ME 99896.33. For sgle exoetal smoothg varous values of α are tred ad mmum ME 5949.6 was obtaed for α.6. For smoothg of the data, Holt s double exoetal smoothg was foud to be most arorate. Varous combatos of α ad γ both ragg betwee. ad.9 wth cremets of. were tred ad ME 43.458 was least for α.9 ad γ.. Fgure 7: Grah of observed data ad ftted values usg tred, sgle ad double exoetal smoothg of Total Colform for Ramgardh Lae for the years 995-6. Tme seres values 4 8 6 4 Total colform 9959969979989993456 Year Observed Data Tred values sgle exoetal smoothg(alha.6) Double exoetal smoothg(alha.9 & gamma.) 8
Table 8: Comarso of forecasts Perod Observed Data Forecast(sgle) Forecast(double) 69 87.3333 85 638.6-5.433 3 84.75 759.89 44.3534 4 44 39.356-63.4 5 65.33 95.344-73.95 6.5 5.36-98.843 7 9 3.845-64.98 8 7.6 485.858 459.548 9 86 45.6343-8.7583 6.66 35.497-45.897 35.48-45.897 9.95-339. 3 73.4348-53.7 4 66.36963-75.4 5 63.54385-98.357 6 6.4354 -.47 7 6.964-34.59 8 6.7857-497.7 9 6.783-69.8 6.6799-883.93 Fgure 8: Comarso of forecasts 9
Comarso of forecasts Tme seres values - - 3 4 5 6 7 8 9 3 4 5 6 7 8 9-3 Perod Observed Data Forecast(sgle) Forecast(double) Table 9 shows the data o H for Ramgardh Lae for the years 995-6 ad ftted values usg tred, sgle ad double exoetal smoothg. Table 9: Data o H for Ramgardh Lae for the years 995-6 ad ftted values usg tred, sgle ad double exoetal smoothg Observed Year(x) Data 995 7.64 996 7.48 997 7.87 998 8.5 999 8.44 8.3 3 7.5 4 8.8 5 7.87 6 8.6 Total 79.86 Tred values 6.9 6.65 7. 7.37 7.73 8.9 8.45 8.8 9.7 9.53 79. sgle exoetal smoothg(alha.) 7.64 7.68 7.664 7.7383 7.8786566 7.9695 7.834 7.97 7.93877 7.473 Double exoetal smoothg(alha.9 & gamma.) 7.64 7.59667 7.844963 8.4746 8.4477 8.37645 7.3753 8.9954 7.993 8.3557 79.594 H s a measure of the acdc or basc ( alale) ature of a soluto. The cocetrato of the hydroge o [H+] actvty a soluto determes the H. A H rage of 6. to 9.. aears to rovde rotecto for the lfe of freshwater fsh ad bottom
dwellg vertebrates. The most sgfcat evrometal mact of H volves syergstc effects. yergy volves the combato of two or more substaces whch roduce effects greater tha ther sum. Ths rocess s mortat surface waters. Ruoff from agrcultural, domestc, ad dustral areas may cota ro, alumum, ammoa, mercury or other elemets. The H of the water wll determe the toxc effects, f ay, of these substaces. For examle, 4 mg/l of ro would ot reset a toxc effect at a H of 4.8. However, as lttle as.9 mg/l of ro at a H of 5.5 ca cause fsh to de (for detals see www.state.y.us). For tred, the ftted le s U t 7.986 +.845 t wth ME.9995. For sgle exoetal smoothg varous values of α are tred ad mmum ME.83 was obtaed for α.. For smoothg of the data, Holt s double exoetal smoothg was foud to be most arorate. Varous combatos of α ad γ both ragg betwee. ad.9 wth cremets of. were tred ad ME.735was least for α.9 ad γ.. Fgure 9: Grah of observed data ad ftted values usg tred, sgle ad double exoetal smoothg of H for Ramgardh Lae for the years 995-6. PH Observed Data Tme seres values 8 6 4 995 996 997 998 999 3 4 5 6 Tred values sgle exoetal smoothg(alha.) Double exoetal smoothg(alha.9 & gamma.) Year
Table : Comarso of forecasts Observed Perod Data 7.64 7.48 3 7.87 4 8.5 5 8.44 6 8.3 7 7.5 8 8.8 9 7.87 8..6 3 4 5 6 7 8 9 Forecast(sgle) Forecast(double) 7.776667 6.73 7.69633 7.83 7.977463 7.45 8.8774 7.596 8.576446 7.47 8.46533 6.55 7.399539 7.45 8.993 7.83 7.98569 7.54 8.744 7.8368 8.74395 7.8537 8.454 7.868948 8.685 7.88653 8.8693 7.894988 8.357775 7.9689 8.486 7.968 8.499465 7.957 8.573 7.93365 8.6455 7.94448 8..7 Fgure : Grah of forecasts Comarso of forecasts Tme seres values 8 6 4 3 4 5 6 7 8 9 3 4 5 6 7 8 9 Perod Observed Data Forecast(sgle) Forecast(double)
e observe from the calculatos for the dfferet arameters of ollutats that double smoothg follows the data much closer tha sgle smoothg. Furthermore, for forecastg sgle smoothg caot do better tha rojectg a straght horzotal le, whch s ot very lely to occur realty. o for forecastg uroses for our data double exoetal smoothg s more referable. Cocluso From the above dscussos, we observe that the varous ollutats cosdered the artcle may have very hazardous effect o qualty of water. Icrease of ollutats water beyod a certa lmt may be dagerous for aquatc amals. Also, accordg to recet reorts, most of the ta ad well water the Ida are ot safe for drg due to resece of varous ollutats arorate ercetage. ow, we have reached the ot where all sources of our drg water, cludg mucal water systems, wells, laes, rvers, ad eve glacers, cota some level of cotamato. o, we eed to ee a route chec of the qualty of water so that we ca lead a healthy lfe. Refereces Garder, E.. (985) : Exoetal smoothg- The state of the art. Jour. Of Forecastg,4, -8. Ja, mta (): A tatstcal study of effect of dfferet ollutats o water (wth secfc referece to Rajastha). Uublshed thess submtted to Uversty of Rajastha, Ida. www.state.y.us www.freedrgwater.com 3
Estmatg the Poulato Mea tratfed Poulato usg Auxlary Iformato uder o-resose Maoj Kr. Chaudhary, V. K. gh ad Rajesh gh Deartmet of tatstcs, Baaras Hdu Uversty Varaas-5, IDIA Floret maradache Deartmet of Mathematcs, Uversty of ew Mexco, Gallu, UA Abstract The reset chater deals wth the study of geeral famly of factor-tye estmators for estmatg oulato mea of stratfed oulato the resece of oresose wheever formato o a auxlary varable are avalable. The roosed famly cludes searate rato, roduct, dual to rato ad usual samle mea estmators as ts artcular cases ad exhbts some ce roertes as regards to locate the otmum estmator belogg to the famly. Choce of arorate estmator the famly order to get a desred level of accuracy eve f o-resose s hgh, s also dscussed. The emrcal study has bee carred out suort of the results. Keywords: Factor-tye estmators, tratfed oulato, o-resose, Otmum estmator, Emrcal study.. Itroducto I samlg theory the use of sutable auxlary formato results cosderable reducto varace of the estmator. For ths reaso, may authors used the auxlary formato at the estmato stage. Cochra (94) was the frst who used the auxlary formato at the estmato stage estmatg the oulato arameters. He roosed the rato estmator to estmate the oulato mea or total of a character uder study. Hase et. al. (953) suggested the dfferece estmator whch was subsequetly modfed to gve the lear regresso estmator for the oulato mea or ts total. Murthy (964) have studed the roduct estmator to estmate the oulato mea or total whe the character uder study ad the auxlary character are egatvely 4
correlated. These estmators ca be used more effcetly tha the mea er ut estmator. There are several authors who have suggested estmators usg some ow oulato arameters of a auxlary varable. Uadhyaya ad gh (999) have suggested the class of estmators smle radom samlg. Kadlar ad Cg (3) ad habbr ad Guta (5) exteded these estmators for the stratfed radom samlg. gh et. al. (8) suggested class of estmators usg ower trasformato based o the estmators develoed by Kadlar ad Cg (3). Kadlar ad Cg (5) ad habbr ad Guta (6) have suggested ew rato estmators stratfed samlg to mrove the effcecy of the estmators. Koyucu ad Kadlar (8) have roosed famles of estmators for estmatg oulato mea stratfed radom samlg by cosderg the estmators roosed earls (964) ad Khoshevsa et. al. (7). gh ad Vshwaarma (8) have suggested a famly of estmators usg trasformato the stratfed radom samlg. Recetly, Koyucu ad Kadlar (9) have roosed a geeral famly of estmators, whch uses the formato of two auxlary varables the stratfed radom samlg to estmate the oulato mea of the varable uder study. The wors whch have bee metoed above are based o the assumto that both the study ad auxlary varables are free from ay d of o-samlg error. But, ractce, however the roblem of o-resose ofte arses samle surveys. I such stuatos whle sgle survey varable s uder vestgato, the roblem of estmatg oulato mea usg sub-samlg scheme was frst cosdered by Hase ad Hurwtz (946). If we have comlete formato o study varable X ad comlete formato o auxlary varable X, other words f the study varable s affected by o-resose error but the auxlary varable s free from o-resose. The utlzg the Hase-Hurwtz (946) techque of sub-samlg of the o-resodets, the covetoal rato ad roduct estmators the resece of o-resose are resectvely gve by ( T x) T R HH / X (.) 5
T P T HH. x / X. (.) ad ( ) The urose of the reset chater s to suggest searate-tye estmators stratfed oulato for estmatg oulato mea usg the cocet of sub-samlg of o-resodets the resece of o-resose study varable the oulato. I ths cotext, the formato o a auxlary characterstc closely related to the study varable, has bee utlzed assumg that t s free from o-resose. I order to suggest searate-tye estmators, we have made use of Factor-Tye Estmators (FTE) roosed by gh ad hula (987). FTE defe a class of estmators volvg usual samle mea estmator, usual rato ad roduct estmators ad some other estmators exstg lterature. Ths class of estmators exhbts some ce roertes whch have bee dscussed subsequet sectos.. amlg trategy ad Estmato Procedure Let us cosder a oulato cosstg of uts dvded to strata. Let the sze of th stratum s, (,,..., ) ad we decde to select a samle of sze from the etre oulato such a way that uts are selected from the th stratum. Thus, we have. Let the o-resose occurs each stratum. The usg Hase ad Hurwtz (946) rocedure we select a samle of sze m uts out of o-resodet uts the th (ROR) scheme such that the stratum wth the hel of smle radom samlg wthout relacemet m uts by tervew method. L m, L ad the formato are observed o all The Hase-Hurwtz estmator of oulato mea X of study varable X for the th stratum wll be T x + x m, (,,..., ) (.) 6
where x ad x m are the samle meas based o resodet uts ad m oresodet uts resectvely the th stratum for the study varable. Obvously T s a ubased estmator of X. Combg the estmators over all the strata we get the estmator of oulato mea X of study varable X, gve by T st T (.) where. whch s a ubased estmator of X. ow, we defe the estmator of oulato mea X of auxlary varable X as T st x (.3) where x s the samle mea based o uts the th stratum for the auxlary varable. It ca easly be see that estmates of the oulato mea 3. uggested Famly of Estmators T st s a ubased estmator of X because X of auxlary varable for the th stratum. x gves ubased Let us ow cosder the stuato whch the study varable s subjected to oresose ad the auxlary varable s free from o-resose. Motvated by gh ad hula (987), we defe the searate-tye famly of estmators of oulato mea X usg factor-tye estmators as ( α ) T ( α ) T (3.) F F 7
where T ( ) ( A + C) ( A + fb) X + fbx F α T (3.) X + Cx ad C ; α >. f, A ( α )( α ), B ( α )( α 4), ( α )( α 3)( α 4) 3. Partcular Cases of T ( α ) Case-: If α the A B, C 6 F so that T () T F X x X ad hece TF () T. (3.3) x Thus, T () s the usual searate rato estmator uder o-resose. F Case-: If α the A C, B so that T ( ) T F x X ad hece T ( ) F x T (3.4) X whch s the usual searate roduct estmator uder o-resose. Case-3: If α 3 the A, B, C so that T () 3 T F X f x ( f ) X ad hece T () 3 T () 3 F (3.5) F 8
whch s the searate dual to rato-tye estmator uder o-resose. The dual to rato tye estmator was roosed by rveataramaa (98). Case-4: If α 4 the A 6, B, C T T so that ( ) F 4 F 4 st ad hece T ( ) T T (3.6) whch s usual mea estmator defed stratfed oulato uder o-resose. 3. Proertes of T ( α ) F Usg large samle aroxmato, the bas of the estmator T ( α ) order of aroxmato was obtaed followg gh ad hula (987) as B [ X ] [ T ( α )] E T ( α ) F F F, u to the frst C φ ( α ) X C ρ CC (3.7) A + fb + C where ( α ) C fb φ, A + fb + C C, X C, X ad are the oulato mea squares of study ad auxlary varables resectvely the th stratum. ρ s the oulato correlato coeffcet betwee X ad X the th stratum. The Mea quare Error (ME) u to the frst order of aroxmato was derved as M [ X ] [ T ( α )] E T ( α ) F F ME T F [ ( α )] 9
3 ( ) ( ) ( ) ( ) ( ) + X X x Cov T X x V X T V X, α φ α φ. ce ( ) L T V +, ( ) x V ad ( ) x T Cov, ρ [ due to gh (998)]. where s the oulato mea square of the o-resose grou the th stratum ad s the o-resose rate of the th stratum the oulato. Therefore, we have ( ) [ ] ( ) ( ) [ ] + F R R M T ρ α ϕ α ϕ α + L (3.8) where X X R. 3.3 Otmum Choce of α I order to obta mmum ME of ( ) α F T, we dfferetate the ME wth resect to α ad equate the dervatve to zero ( ) ( ) ( ) [ ] R R ' ' ρ α φ α φ α φ, (3.9) where ( ) α φ' stads for frst dervatve of ( ) α φ.from the above exresso, we have
R ρ φ ( α ) V (say). (3.) R It s easy to observe that φ ( α ) s a cubc equato the arameterα. Therefore, the equato (3.) wll have at the most three real roots at whch the ME of the estmator ( α ) T attas ts mmum. F Let the equato (3.) yelds solutos asα, α ad [ ] α such that ( α ) M s same. A crtero of mag a choce betweeα, α ad α s that comute the bas of the estmator at α α, α ad α ad select ovel roerty of the FTE. 3.4 Reducg ME through Arorate Choce of α T F α ot at whch bas s the least. Ths s a By usg FTE for defg the searate-tye estmators ths chater, we have a advatage terms of the reducto of the value of ME of the estmator to a desred extet by a arorate choce of the arameter α eve f the o-resose rate s hgh the oulato. The rocedure s descrbed below: ce ME s of the roosed strateges are fuctos of the uow arameter α as well as fuctos of o-resose rates, t s obvous that f α s tae to be costat, ME s crease wth creasg o-resose rate, f other characterstcs of the oulato rema uchaged, alog wth the rato to be sub samled the o-resose class, that s, L. It s also true that more the o-resose rate, greater would be the sze of the o-resose grou the samle ad, therefore, order to lowerg dow the ME of the estmator, the sze of sub samled uts should be creased so as to ee the value of L the vcty of ; but ths would, term, cost more because more effort ad moey would be requred to obta formato o sub samled uts through ersoal tervew method. Thus, creasg the sze of the sub samled uts order to 3
reduce the ME s ot a feasble soluto f o-resose rate s suosed to be large eough. The classcal estmators such as T HH, T R, T P, dscussed earler lterature resece of o-resose are ot helful the reducto of ME to a desred level. I all these estmators, the oly cotrollg factor for lowerg dow the ME s L, f oe desres so. By utlzg FTE order to roose searate- tye estmators the reset wor, we are able to cotrol the recso of the estmator to a desred level oly by mag a arorate choce of α. Let the o-resose rate ad mea-square of the o-resose grou the th stratum at a tme be ad ME of the estmator would be resectvely. The, for a choce of α α, the [ R ] [ ( α ) ] + φ( α ) R φ( α ) M T F / ρ + L (3.) Let us ow suose that the o-resose rate creased over tme ad t s ' ' ' such that >. Obvously, wth chage o-resose rate, oly the arameter wll chage. Let t becomes '. The we have [ R ] ' [ ( α ) ] + φ( α ) R φ( α ) M T F / ρ + L ' ' (3.) 3
33 Clearly, f α α ad ' > the ( ) [ ] ( ) [ ] ' F F M T T M α α >. Therefore, we have to select a sutable value α, such that eve f ' > ad ' >, exresso (3.) becomes equal to equato (3.) that s, the ME of ( ) α F T s reduced to a desred level gve by (3.). Equatg (3.) to (3.) ad solvg for ( ) α φ, we get ( ) ( ) R R ρ α φ α φ ( ) ( ) { } R R ρ α φ α φ ( ) ' ' + L, (3.3) whch s quadratc equato ( ) α φ. O solvg the above equato, the roots are obtaed as ( ) ± R R ρ α φ + R R ρ ( ) ( ) { } ( ) ' ' R L R R ρ α φ α φ (3.4) The above equato rovdes the value ofα o whch oe ca obta the recso to a desred level. ometmes the roots gve by the above equato may be magary. o, order that the roots are real, the codtos o the value of α are gve by
34 ( ) ( ) ' ' + > R L R R ρ α φ (3.5) ad ( ) ( ) ' ' < R L R R ρ α φ (3.6) 4. Emrcal tudy I ths secto, therefore, we have llustrated the results, derved above, o the bass of some emrcal data. For ths urose, a data set has bee tae to cosderato. Here the oulato s MU84 oulato avalable ardal et. al. (99, age 65, Aedx B). e have cosdered the oulato the year 985 as study varable ad that the year 975 as auxlary varable. There are 84 mucaltes whch have bee dvded radomly to four strata havg szes 73, 7, 97 ad 44. Table shows the values of the arameters of the oulato uder cosderato for the four strata whch are eeded comutatoal rocedure. Table : Parameters of the Poulato tratum () tratum ze ( ) Mea ( ) X Mea ( ) X ( ) ( ) ρ ( ) 73 4.85 39.56 6369.999 664.4398 79.866 8.397.999 68.8844 7 7.83 7.57 5.75 47. 3.4 33.8676.998 4.95 3 97 5.79 5.44 4.965 5.4 44.8884 46.967.999 65.5 4 44.64.36 538.4749 485.655 3.5.87.997 83.6944
The value of R X / X comes out to be.9. e fx the samle sze to be 6. The the allocato of samles to dfferet strata uder roortoal ad eyma allocatos are show the followg table Table : Allocato of amle tratum () Proortoal Allocato ze of amles uder eyma Allocato 5 6 5 3 9 4 9 5 O the bass of the equato (3.), we obtaed the otmum values of α : Uder Proortoal Allocato ( α ) φ.949, α ot (3.9975,.68,.) ad Uder eyma Allocato ot φ ( α).957, α ot (34.435,.64,.3). The followg table dects the values of the ME s of the estmators ( α ) T for α, α ad 4 uder roortoal ad eyma allocatos. A comarso of ME of F ( α ) T wth α ot ad α wth that at α 4 reveals the fact that the utlzato of auxlary formato at the estmato stage certaly mroves the effcecy of the estmator as comared to the usual mea estmatort. st F 35
Table 3: ME Comarso ( L, % for all ) ME Allocato Proortoal eyma M [ ( α) ] T F ot.664.65 [ () ] M.77.675 T F M [ ( 4) ] [ ] T F V 35.669 8.68 T st e shall ow llustrate how by a arorate choce of α, the ME of the estmators ( α ) creased. T ca be reduced to a desred level eve f the o-resose rate s F Let us tae L,. ',. 3 Uder Proortoal Allocato as ' 4 ad ( ) for all 3 From the codto (3.5) ad (3.6), we have codtos for real roots of φ ( α ) φ ( α ) >.57 ad ( α ) φ <.7454. Therefore, f we tae φ ( α )., the for ths choce of φ ( α ) ' M [ ( α ) ] 3.7 ad M ( α ) 4.688. T F [ ] T F, we get Thus, there s about 5 ercet crease the ME of the estmator f oresose rate s trled. ow usg (3.4), we get φ ( α ).957 ad.85. At ths value of φ ( α ), M [ T F ( α )] reduces to 3.7 eve f o-resose rate s 3 ercet. Thus a ossble choce of α may be made order to reduce the ME to a desred level. 36
Uder eyma Allocato Codtos for real roots of φ ( α ) φ ( α ) >.746 ad ( α ) φ <.739. If φ ( α ). the we have M [ ( α ) ].4885 ad ( ) T F ' [ ] M T F α 4.7. Further, we get from (3.4), φ ( α ).6 ad.8435, so that ' M [ ( α ) ].4885 for φ ( ).6. 5. Cocluso T F α e have suggested a geeral famly of factor-tye estmators for estmatg the oulato mea stratfed radom samlg uder o-resose usg a auxlary varable. The otmum roerty of the famly has bee dscussed. It has also bee dscussed about the choce of arorate estmator of the famly order to get a desred level of accuracy eve f o-resose s hgh. The Table 3 reveals that the otmum estmator of the suggested famly has greater recso tha searate rato ad samle mea estmators. Besdes t, the reducto of ME of the estmators ( α ) 37 T to a desred extet by a arorate choce of the arameter α eve f the o-resose rate s hgh the oulato, has also bee llustrated. Refereces Cochra,. G. (94): The estmato of the yelds of cereal exermets by samlg for the rato of gra total roduce. Jour. of The Agr. c., 3, 6-75. Hase, M. H. ad Hurwtz,.. (946): The roblem of o-resose samle surveys. Jour. of The Amer. tats. Assoc., 4, 57-59. Hase, M. H., Hurwtz,.. ad Madow,,. G. (953): amle urvey Methods ad Theory, Volume I, Joh ley ad os, Ic., ew Yor. Kadlar, C. ad Cg, H. (3): Rato estmators stratfed radom samlg, Bom. Jour. 45 (), 8 5. F
Kadlar, C. ad Cg, H. (5): A ew rato estmator stratfed samlg, Comm. tat. Theor. ad Meth., 34, 597 6. Khoshevsa, M., gh, R., Chauha, P., awa,., maradache, F. (7): A geeral famly of estmators for estmatg oulato mea usg ow value of some oulato arameter(s), Far East Jour. of Theor. tats.,, 8 9. Koyucu,. ad Kadlar, C. (8): Rato ad roduct estmators stratfed radom samlg. Jour. of tats. Pla. ad If., 38, -7. Koyucu,. ad Kadlar, C. (9): Famly of estmators of oulato mea usg two auxlary varables stratfed radom samlg. Comm. tat. Theor. ad Meth., 38, 398 47. Murthy, M.. (964): Product method of estmato. ahya, 6A, 69-74. ardal, C. E., wesso, B. ad retma, J. (99): Model Asssted urvey amlg, rger-verlag, ew Yor, Ic. earls, D. T. (964): The utlzato of a ow coeffcet of varato the estmato rocedure. Jour. of The Amer. tats. Assoc., 59, 5-6. habbr, J. ad Guta,. (5): Imroved rato estmators stratfed samlg. Amer. Jour. of Math. ad Maag. c., 5 (3-4), 93-3. habbr, J. ad Guta,. (6): A ew estmator of oulato mea stratfed samlg. Comm. tat. Theor. ad Meth., 35, 9. gh, H. P., Talor, R., gh,. ad Km, J. (8): A modfed estmator of oulato mea usg ower trasformato. tat. Paer., 49, 37-58. gh, H. P. ad Vshwaarma, G. K. (8): A Famly of Estmators of Poulato Mea Usg Auxlary Iformato tratfed amlg, Comm. tat. Theor. ad Meth., 37, 38 5. gh, L. B. (998): ome Classes of Estmators for Fte Poulato Mea Presece of o-resose. Uublshed Ph. D. Thess submtted to Baaras Hdu Uversty, Varaas, Ida. 38
gh, V. K. ad hula, D. (987): Oe arameter famly of factor-tye rato estmators. Metro, 45 (-), 73-83. rveataramaa, T. (98): A dual to rato estmator samle surveys, Bometra, 67 (), 99-4. Uadhyaya, L.. ad gh, H. P. (999): Use of trasformed auxlary varable estmatg the fte oulato mea, Bom. Jour., 4 (5), 67-636. 39
O ome ew Allocato chemes tratfed Radom amlg uder o-resose Maoj Kr. Chaudhary, V. K. gh ad Rajesh gh Deartmet of tatstcs, Baaras Hdu Uversty Varaas-5, IDIA Floret maradache Deartmet of Mathematcs, Uversty of ew Mexco, Gallu, UA Abstract Ths chater resets the detaled dscusso o the effect of o-resose o the estmator of oulato mea a frequetly used desg, amely, stratfed radom samlg. I ths chater, our am s to dscuss the exstg allocato schemes resece of o-resose ad to suggest some ew allocato schemes utlzg the owledge of resose ad o-resose rates of dfferet strata. The effects of roosed schemes o the samlg varace of the estmator have bee dscussed ad comared wth the usual allocato schemes, amely, roortoal allocato ad eyma allocato resece of o-resose. The emrcal study has also bee carred out suort of the results. Keywords: tratfed radom samlg, Allocato schemes, o-resose, Mea squares, Emrcal tudy.. Itroducto uhatme (935) has show that by effectvely usg the otmum allocato stratfed samlg, estmates of the strata varaces obtaed a revous survey or a secally laed lot survey based eve o samles of moderate samle sze would be adequate for creasg the recso of the estmator. Evas (95) has also cosdered the roblem of allocato based o estmates of strata varaces obtaed earler survey. Accordg to lterature of samlg theory, varous efforts have bee made to reduce the error whch arses because of tag a art of the oulato,.e., samlg 4
error. Besdes the samlg error there are also several o-samlg errors whch tae lace from tme to tme due to a umber of factors such as faulty method of selecto ad estmato, comlete coverage, dfferece tervewers, lac of roer suervso, etc. Icomleteess or o-resose the form of absece, cesorg or groug s a troublg ssue of may data sets. I choosg the samle szes from the dfferet strata stratfed radom samlg oe ca select t such a way that t s ether exclusvely roortoal to the strata szes or roortoal to strata szes alog wth the varato the strata uder roortoal allocato or eyma allocato resectvely. If o-resose s heret the etre oulato ad so are all the strata, obvously t would be qute mossble to adot eyma allocato because the the owledge of stratum varablty wll ot be avalable, rather the owledge of resose rate of dfferet strata mght be easly avalable or mght be easly estmated from the samle selected from each stratum. Thus, t s qute reasoable to utlze the resose rate (or o-resose rate) whle allocatg samles to stratum stead of eyma allocato resece of o-resose error. I the reset chater, we have roosed some ew allocato schemes selectg the samles from dfferet strata based o resose (o-resose) rates of the strata resece of o-resose. e have comared them wth eyma ad roortoal allocatos. The results have bee show wth a umercal examle.. amlg trategy ad Estmato Procedure I the study of o-resose, accordg to oe determstc resose model, t s geerally assumed that the oulato s dchotomzed two strata; a resose stratum cosderg of all uts for whch measuremets would be obtaed f the uts haeed to fall the samle ad a o-resose stratum of uts for whch o measuremet would be obtaed. However, ths dvso to two strata s, of course, a oversmlfcato of the roblem. The theory volved HH techque, s as gve below: Let us cosder a samle of sze s draw from a fte oulato of sze. Let uts the samle resoded ad uts dd ot resod, so that + 4.
The uts may be regarded as a samle from the resose class ad uts as a samle from the o-resose class belogg to the oulato. Let us assume that ad be the umber of uts the resose stratum ad o-resose stratum resectvely the oulato. Obvously, ad are ot ow but ther ubased estmates ca be obtaed from the samle as ˆ / ; ˆ /. Let m be the sze of the sub-samle from o-resodets to be tervewed. Hase ad Hurwtz (946) roosed a estmator to estmate the oulato mea X of the study varable X as T HH x + x m, (.) whch s ubased for X, whereas x ad x m are samle meas based o samles of szes ad m resectvely for the study varable X. The varace of T HH s gve by L V ( T HH ) +, (.) where L, m, ad are the mea squares of etre grou ad oresose grou resectvely the oulato. Let us cosder a oulato cosstg of uts dvded to strata. Let the sze of th stratum s, (,,..., ) ad we decde to select a samle of sze from the etre oulato such a way that uts are selected from the th stratum. Thus, we have. 4
Let the o-resose occurs each stratum. The usg Hase ad Hurwtz rocedure we select a samle of sze m uts out of o-resodet uts the th stratum wth the hel of smle radom samlg wthout relacemet (ROR) such that method. L m, L ad the formato are observed o all the m uts by tervew The Hase-Hurwtz estmator of oulato mea X for the th stratum wll be T x + x m, (,,..., ) (.3) where x ad x m are the samle meas based o resodet uts ad m oresodet uts resectvely the th stratum. Obvously T s a ubased estmator of X. Combg the estmators over all strata we get the estmator of oulato mea X, gve by T st T (.4) where. Obvously, we have E[ ] T X. (.5) st The varace of T st s gve by [ ] V T st + ( L ) (.6) 43
where, ad are the mea squares of etre grou ad o-resose grou resectvely the th stratum. for It s easy to see that uder roortoal allocato (PA), that s, whe,,...,, V [ ] s obtaed as T st [ T st ] PA V + ( L ), (.7) whereas uder the eyma allocato (A), wth equal to (,,..., ), t s [ T st ] A V + ( L ). (.8) It s mortat to meto here that the last terms the exressos (.7) ad (.8) arse due to o-resose the oulato. Further, resece of o-resose the oulato, eyma allocato may or may ot be effcet tha the roortoal allocato, a stuato whch s qute cotrary to the usual case whe oulato s free from o-resose. Ths ca be uderstood from the followg: e have V [ ] [ ] ( ) ( ) + w T st V T PA st A w L (.9) w. 44
hole the frst term the above exresso s ecessarly ostve, the secod term may be egatve ad greater tha the frst term magtude deedg uo the w sg ad magtude of the term for all. Thus, resece of o-resose the stratfed oulato, eyma allocato does ot always guaratee a better result as t s case whe the oulato s free from o-resose error. 3. ome ew Allocato chemes It s a well ow fact that case the stratfed oulato does ot have oresose error ad strata mea squares, (,,..., ), are ow, t s always advsable to refer eyma allocato scheme as comared to roortoal allocato scheme order to crease the recso of the estmator. But, f the oulato s affected by o-resose, eyma allocato s ot always a better roosto. Ths has bee hghlghted uder the secto above. Moreover, case o-resose s reset strata, owledge o strata mea squares,, are mossble to collect, rather drect estmates of ad may be had from the samle. Uder these crcumstaces, t s, therefore, ractcally dffcult to adot eyma allocato f o-resose s heret the oulato. However, roortoal allocato does ot demad the owledge of strata mea squares ad rests oly uo the strata szes, hece t s well alcable eve the resece of o-resose. As dscussed the secto, ubased estmates of resose ad o-resose rates the oulato are readly avalable ad hece t seems qute reasoable to th for develog allocato schemes whch volve the owledge of oulato resose (o-resose) rates each stratum. If such allocato schemes yeld récsed estmates as comared to roortoal allocato, these would be advsable to adot stead of eyma allocato due to the reasos metoed above. I ths secto, we have, therefore, roosed some ew allocato schemes whch utlze the owledge of resose (o-resose) rates suboulatos. hle some of the roosed schemes do ot utlze the owledge of, some others are roosed 45
based o the owledge of just order to mae a comarso of them wth eyma allocato uder the resece of o-resose. I addto to the assumtos of roortoal ad eyma allocatos, we have further assume t logcal to allocate larger samle from a stratum havg larger umber of resodets ad vce-versa whe roosg the ew schemes of allocatos. cheme-[oa ()]: Let us assume that larger sze samle s selected from a larger sze stratum ad wth larger resose rate, that s, for,,...,. The we have K where K s a costat. The value of K wll be K. Thus we have. (3.) Puttg ths value of exresso (.6), we get V ( L ) [ Tst ] + (3.) 46
47 cheme-[oa ()]: Let us assume that. The, we have (3.3) ad hece the exresso (.6) becomes [ ] + st L T V ) (. (3.4) cheme-3[oa (3)]: Let us select larger sze samle from a larger sze stratum but smaller sze samle f the o-resose rate s hgh. That s,. The (3.5) ad the exresso of [ ] T st V reduces to [ ] ( ) { } + st L T V 3. (3.6)
48 cheme-4[oa (4)]: Let, the. (3.7) The corresodg exresso of [ ] T st V s [ ] ( ) + st L T V 4. (3.8) cheme-5[oa (5)]: Let, the. (3.9) The exresso (.6) gves [ ] ( ) + st L T V 5. (3.)
cheme-6[oa (6)]: If, the, we have. (3.) T st I ths case, [ ] V becomes V [ T ] st 6 ( L ) +. (3.) Remar : It s to be metoed here that f resose rate assumes same value all the strata, that s (say), the schemes, 3 ad 5 reduces to roortoal allocato, whle the schemes, 4 ad 6 reduces to eyma allocato. The corresodg V T exressos, [ st ] r reduce to V [ T st ] A, (,3,5 ). V T V T r are the smlar to [ st ] PA ad [ st ] r Remar : Although the theoretcal comarso of exressos of V [ T st ] r ad V [ T st ] r, (,4,6) V T r wth [ st ] PA ad [ st ] A, ( r,4,6), ( r,3,5 ) V T resectvely s requred order to uderstad the sutablty of the roosed schemes, but such comarsos do ot yeld exlct solutos geeral. The sutablty of a scheme does deed uo the arametrc values of the oulato. e have, therefore, llustrated the results wth the hel of some emrcal data. 49
4. Emrcal tudy I order to vestgate the effcecy of the estmator T st uder roosed allocato schemes, based o resose (o-resose) rates, we have cosdered here a emrcal data set. e have tae the data avalable ardal et. al. (99) gve Aedx B. The data refer to 84 mucaltes wede, varyg cosderably sze ad other characterstcs. The oulato cosstg of the 84 mucaltes s referred to as the MU84 oulato. For the urose of llustrato, we have radomly dvded the 84 mucaltes to four strata cosstg of 73, 7, 97 ad 44 mucaltes. The 985 oulato ( thousads) has bee cosdered as the study varable, X. O the bass of the data, the followg values of arameters were obtaed: Table : Partculars of the Data ( 84) tratum () ze ( ) tratum Mea ( X ) tratum Mea quare ( ) Mea quare of the oresose Grou 4 ( ) 5 73 4.85 6369. 595.8 7 7.83 5.7 84.86 3 97 5.78 4.97 6.97 4 44.64 538.47 43.78 e have tae samle sze, 6. 5
Tables dects the values of samle szes, (,,3,4 ) ad values of V [ ] uder PA, A ad roosed schemes OA() to OA(6) for dfferet selectos of the (,,3,4 ) values of L ad. T st 5
tratum oresose Rate ( ) (Percet) Table amle zes ad Varace of T st uder Dfferet Allocato chemes L.,.5,.5, 3.5 for,, 3, 4 resectvely) ( amle ze ( ) ad [ ] V uder PA A OA() OA() OA(3) OA(4) OA(5) OA(6) V [ T st ] [ ] V T st [ ] V T st [ ] V T st T st [ ] V T st [ ] V T st [ ] V T st V [ ] T st 5 43.8 6 36.4 7 4. 8 6.59 38.43 3 38.43 37.85 33 4.5 3 4 5 3 35 5 9 9 5 5 8 8 4 5 8 7 6 3 5 7 6 4 3 3 4 35 3 5 5 5 9 45.97 6 9 5 37.7 4 4 49.7 4 5 7.37 3 3 55.4 7 39.7 3 3 4 6.76 9 4 7 4.7 3 4 5 3 35 5 5 9 43.9 6 9 5 36.3 6 6 8 43.4 7 8 4 6.54 6 9 8 7 44.5 7 3 7 3 37.76 6 7 6 44.69 7 4 6 3 38.94 3 4 5 35 3 5 5 9 43.7 6 9 5 35.99 7 5 9 9 4.3 8 7 5 5.4 6 6 8 39.45 3 4 4 38.8 6 4 8 39.73 34 4 4.3 5
5. Cocludg Remars I the reset chater, our am was to accommodate the o-resose error heret the stratfed oulato durg the estmato rocedure ad hece to suggest some ew allocato schemes whch utlze the owledge of resose (o-resose) rates of strata. As dscussed dfferet sub-sectos, eyma allocato may sometmes roduce less récsed estmates of oulato mea comarso to roortoal allocato f o-resose s reset the oulato. Moreover, eyma allocato s sometmes mractcal such stuato, sce the ether the owledge of (,,3,4 ), the mea squares of the strata, wll be avalable, or these could be estmated easly from the samle. I cotrast to ths, what mght be easly ow or could be estmated from the samle are resose (o-resose) rates of dfferet strata. It was, therefore, thought to roose some ew allocato schemes deedg uo resose (o-resose) rates. A loo of Table reveals that most of the stuatos (uder dfferet combatos of ad L ), allocato schemes OA (), OA (3) ad OA (5), deedg solely uo the owledge of ad (or ), roduce more récsed estmates as comared to PA. Further, as for as a comaratve study of schemes OA (), OA (3) ad OA (5) s cocered, o doubt, all these schemes are more or less smlar terms of ther effcecy. Thus, addto to the owledge of strata szes,, the owledge of resose (o-resose) rates, (or ), whle allocatg samle to dfferet strata; certaly adds to the recso of the estmate. It s also evdet from the table that the addtoal formato o the mea squares of strata certaly adds to the recso of the estmate, but ths cotrbuto s ot very much sgfcat comarso to A. cheme OA () s throughout worse tha ay other scheme. 53
Refereces Evas,. D. (95) : O stratfcato ad otmum allocato. Jour. of The Amer. tat. Assoc., 46, 95-4. Hase, M. H. ad Hurwtz,.. (946) : The roblem of o-resose samle urveys. Jour. of The Amer. tat. Assoc., 4, 57-59 ardal, C. E., wesso, B. ad retma, J. (99) : Model Asssted urvey amlg. rger-verlag, ew Yor, Ic. uhatme, P. V. (935) : Cotrbutos to the theory of the reresetatve Method. Jour. of the Royal tat. oc.,, 53-68. 54
A Famly Of Estmators For Estmatg The Poulato Mea I tratfed amlg Rajesh gh, Muesh Kumar, ach Mal ad Maoj K. Chaudhary Deartmet of tatstcs, B.H.U., Varaas (U.P.)-Ida Floret maradache Deartmet of Mathematcs, Uversty of ew Mexco, Gallu, UA Abstract I ths chater, we have suggested a mroved estmator for estmatg the oulato mea stratfed samlg resece of auxlary formato. The mea square error (ME) of the roosed estmator have bee derved uder large samle aroxmato. Besdes, cosderg the mmum case of the ME equato, the effcet codtos betwee the roosed ad exstg estmators are obtaed. These theoretcal fdgs are suorted by a umercal examle. Keywords : Auxlary varable, mea square errors; exoetal rato tye Estmates; stratfed radom samlg.. Itroducto I lag surveys, stratfed samlg has ofte roved useful mrovg the recso of other ustratfed samlg strateges to estmate the fte oulato mea Y y / Cosder a fte oulato of sze. Let y ad x resectvely, be the study ad auxlary varates o each ut U (j,,3 ) of the oulato U. Let the oulato be dvded to L strata wth the h stratum cotag uts, h,,3,l so that 55
. uose that a smle radom samle of sze s draw wthout relacemet (ROR) from the h stratum such that. he the oulato mea X of the auxlary varable x s ow, Hase et. al. (946) suggested a combed rato estmator y y ( ) (.) where, y w y, x w x y y ad x. w ad X w X x The combed roduct estmator for Y s defed by y y ( ) (.) To the frst degree of aroxmato, the mea square error (ME) of y ad y are resectvely gve by ME(y ) w θ +R R (.3) ME(y ) w θ +R +R (.4) where θ, R s the oulato rato, s the oulato varace of varate of terest stratum h, s the oulato varace of auxlary varate stratum h ad s the oulato covarace betwee auxlary varate ad varate of terest stratum h. Followg Bahl ad Tuteja (99), gh et. al. (9) roosed followg estmator stratfed radom samlg - y y ex (.5) 56
The ME of y, to the frst degree of aroxmato s gve by + ME(y ) w θ R (.6) Usg the estmator y ad y, gh ad Vshwaarma (5) suggested the combed rato-roduct estmator for estmatg Y as y y α +(α) (.7) For mmum value of α (+C ) α (say), the mmum ME of the estmator y s gve by ME (y ) w θ (ρ ) (.8) where C (, ) ( ), ρ (, ) ( )( ), R.. Proosed estmator estmators - Followg gh ad Vshwaarma (5), we roose a ew famly of tλ y ex α β + ( λ)y ex α β (.) where s real costat to be determed such that the ME of t s a mmum ad α, β are real costats such that - α. Remar.: For ad α the estmator t teds to gh et. al. (9) estmator. For ad α the estmator t taes the form of Hase et. al. (946) estmator y. For ad α the estmator t teds to gh et. al. (9) estmator. For ad α the estmator t taes the form of the estmator y. To obta the ME of t to the frst degree of aroxmato, we wrte y w y Y(+e ) ad 57
x w x X(+e ) uch that, E(e ) E(e ). Uder ROR, we have E(e ) E(e ) E(e e ) w θ w θ w θ Exressg equato (.) terms of e s we have ty( + e )λ ex + α (+e ) α + ( λ)ex + α (+e ) (α) (.) e ow assume that e so that we may exad (+e ) as a seres owers of e. Exadg the rght had sde of (.) to the frst order of aroxmato, we obta (t Y) Y e +e ( + αλ α λ) (.3) quarg both sdes of (.3) ad the tag exectatos, we get the ME of the estmator t, to the frst order of aroxmato, as ME(t) V(y ) +R (λ) (λ)a +C A (.4) where A α. Mmsato of (.4) wth resect to yelds ts otmum values as λ + λ (say) (.5) Puttg λ λ (.4) we get the mmum ME of the estmator t as 58
m ME (t) V(y ) ( ρ ) w θ ( ρ ). (.6) 3. Effcecy comarsos I ths secto we have comared roosed estmator wth dfferet already roosed estmators, obtaed the codtos uder whch our roosed estmator erforms better tha other estmators. samlg. Frst we have comared roosed estmator wth smle mea stratfed radom ME(t) ME(y ), f V(y ) +R (λ) (λ)a +C A V(y ) m + λ max + ext we comare roosed estmator wth combed rato estmator ME(t) ME(y ), f V(y ) + w θ R (λ) (λ)a +C A w θ or, f (C ) (λ)(λ)a +C A or, f λ. +R R ext we comare effcecy of roosed estmator wth roduct estmator ME(t) ME(y ), f V(y ) + w θ R (λ) (λ)a +C A w θ +R +R 59
or, f (+C ) (λ)((λ)a +C A) or, f λ. ext we comare effcecy of roosed estmator ad exoetal rato estmator stratfed samlg ME(t) ME(y ), f V(y ) + w θ R (λ) (λ)a +C A w θ 6 + R or, f (4C ) 4(λ)((λ)A +C A) or, f λ Fally we comare effcecy of roosed estmator wth exoetal roduct estmator stratfed radom samlg ME(t) ME(y ), f or, f V(y ) + w θ R (λ) (λ)a +C A w θ or, f (+4C ) 4(λ)((λ)A +C A) or, f λ + +R heever above codtos are satsfed the roosed estmator erforms better tha other metoed estmators. 4. umercal llustrato All the theoretcal results are suorted by usg the data gve gh ad Vshwaarma (5). Data statstcs:
tratum.57.454 3.66 593.7 579.6 3.48.89 38.44 65885.6 44.7 R49.3 ad λ.94(α ) ad.38455 (α ) Usg the above data ercetage relatve effceces of dfferet estmators y, y, y, y ad roosed estmator t w.r.t y have bee calculated. Table 4.: PRE of dfferet estmators of Y Estmator y y y y y y () y () PRE 48.56 3.36 45. 4.6 43.37 43.37 e have also show the rage of for whch roosed estmator erforms better tha y. Table 4.: Rage of for whch roosed estmator erforms better tha y Value of costat Form of roosed estmator Rage of α y (.5,.3) α y (.5,.) 5. Cocluso From the theoretcal dscusso ad emrcal study we coclude that the roosed estmator uder otmum codtos erforms better tha other estmators cosdered the artcle. The relatve effcecy of varous estmators are lsted Table 6
4. ad the rage of for whch roosed estmator erforms better tha y s wrtte Table 4.. Refereces Bahl.. ad Tuteja, R.K. (99): Rato ad Product Tye Exoetal Estmator. Ifrm. ad Otm. c., XIII, 59-63. Hase, M.H. ad Hurwtz,.. (946): The roblem of o-resose samle surveys. J. Am. tat. Assoc. 4:57 59. gh, H. P. ad Vshwaarma, G. K. (5): Combed Rato-Product Estmator of Fte Poulato Mea tratfed amlg. Metodologa de Ecuestas 8: 35-44. gh, R., Kumar, M., Chaudhary, M.K., Kadlar, C. (9) : Imroved Exoetal Estmator tratfed Radom amlg. Pa. J. tat. Oer. Res. 5(), 67-8. 6
A Famly Of Estmators Of Poulato Varace Usg Iformato O Auxlary Attrbute Rajesh gh ad Muesh Kumar Deartmet of tatstcs, B.H.U., Varaas (U.P.)-Ida Ashsh K. gh College of Maagemet tudes, Raj Kumar Goel Isttute of Techology Floret maradache Deartmet of Mathematcs, Uversty of ew Mexco, Gallu, UA Abstract Ths chater rooses some estmators for the oulato varace of the varable uder study, whch mae use of formato regardg the oulato roorto ossessg certa attrbute. Uder smle radom samlg wthout relacemet (ROR) scheme, the mea squared error (ME) u to the frst order of aroxmato s derved. The results have bee llustrated umercally by tag some emrcal oulato cosdered the lterature. Keywords: Auxlary attrbute, exoetal rato-tye estmates, smle radom samlg, mea square error, effcecy.. Itroducto It s well ow that the auxlary formato the theory of samlg s used to crease the effcecy of estmator of oulato arameters. Out of may rato, regresso ad roduct methods of estmato are good examles ths cotext. There exst stuatos whe formato s avalable the form of attrbute whch s hghly correlated wth y. Tag to cosderato the ot bseral correlato coeffcet betwee auxlary attrbute ad study varable, several authors cludg a ad 63
Guta (996), Jhajj et. al. (6), habbr ad Guta (7), gh et. al. (7, 8) ad Abd-Elfattah et. al. () defed rato estmators of oulato mea whe the ror formato of oulato roorto of uts, ossessg the same attrbute s avalable. I may stuatos, the roblem of estmatg the oulato varace σ of study varable y assumes mortace. he the ror formato o arameters of auxlary varable(s) s avalable, Das ad Trath (978), Isa (983), Prasad ad gh (99), Kadlar ad Cg (6, 7) ad gh et. al. (7) have suggested varous estmators of y. I ths chater we have roosed famly of estmators for the oulato varace y whe oe of the varables s the form of attrbute. For ma results we cofe ourselves to samlg scheme ROR gorg the fte oulato correcto.. The roosed estmators ad ther roertes Followg Isa (983), we roose a rato estmator φ s y (.) sφ t ext we roose regresso estmator for the oulato varace ( ) t s y + b φ sφ (.) Ad followg gh et. al. (9), we roose aother estmator 64
φ sφ t 3 s y ex (.3) φ s φ wheres y ad s φ are ubased estmator of oulato varaces y ad φ resectvely ad b s a costat, whch maes the ME of the estmator mmum. To obta the bas ad ME, we wrte- ( ), s ( + e ) s y y + e φ φ E uch that ( e ) E( e ) δ4 ( ) ( ) ad E( e ), E( e ), δ4 E( e e ) ( δ ), where μq δ, μ q ( μ / μ q / ) 4 β δ ad β φ μ μ μ μ q 4 δ ( y) 4 ( ) 4 ( y Y) ( φ P) ( ) q. Let ( ) β ( ), β ( ) β ( ), ad δ δ β y y φ x q q P s the roortos of uts the oulato. ow the estmator t defed (.) ca be wrtte as ( t ) ( e e + e e ) (.4) y y e mlarly, the estmator t ca be wrtte as ( t y ) y e bφe (.5) Ad the estmator t 3 ca be wrtte as ( ) e + 3e ee t 3 y y e (.6) 8 The ME of t, t 3 ad varace of t are gve, resectvely, as 65
4 y ( t ) [ β( y) + β( ) δ ] ME φ (.7) 4 ( ) y β φ ME ( t 3 ) β ( y) + δ (.8) 4 The varace of t s gve as [ ] 4 y 4 φ 4 y x (.9) ( t ) ( λ ) + b ( λ ) b ( λ ) V O dfferetatg (.9) wth resect to b ad equatg to zero we obta ( δ ) ( δ ) y b (.) x 4 ubsttutg the otmum value of b (.9), we get the mmum varace of the estmator t, as m.v t 4 y ( ) β( y) β ( y) ( φ) ) Var ( ) ( ) ρ(, ) y φ δ β (.) 3. Adated estmator e adat the habbr ad Guta (7) ad Grover () estmator, to the case whe oe of the varables s the form of attrbute ad roose the estmator t 4 [ ( )] φ sφ s y + φ sφ ex φ + φ t 4 (3.) s where ad are sutably chose costats. Exressg equato (3.) terms of e s ad retag oly terms u to secod degree of e s, we have: e [ ( + ) ] 3 e s e + y t 4 φ e (3.) 8 66
U to frst order of aroxmato, the mea square error of t 4 s ME where, ( t ) E( t ) 4 4 y 3 [( ) + λ ( β ( y) + β ( φ) δ ) + λ δ β ( φ) 4 y 4 4 + φ λβ ( φ) + λyx ( β ( x) δ ) β ( x) λ O artally dfferetatg (3.3) wth resect to (, ) ad, resectvely as (3.3), we get otmum values of λ β ( φ) β ( φ) 4 ( β ( φ)( λa + ) λb ) (3.4) ad ( φ)( λa + ) λb B β ( φ) 4 x ( β ( φ)( λa + ) λ ) λ y β (3.5) x B where, A β ( y) + β ( φ) δ ad B β ( φ). δ O substtutg these otmum values of ad (3.3), we get the mmum value of ME of t 4 as 4 ( ) λ y β λβ x ME(t ) + ME(t ) 6 ME (t 4 ) (3.6) ME(t ( ) + ) 4 ME t 4 + y 4 y ( φ) 67
4. Effcecy Comarso Frst we have comared the effcecy of roosed estmator uder otmum codto wth the usual estmator as - ( ) ME( Ŝ ) V Ŝ y ot 4 λyδ β ( x ) + λβ ME(t ) ME(t ) + 4 ( x) y 4 λyβ ME(t ) + 6 ME( t ) 4 + 4 y ( φ) always. (4.) ext we have comared the effcecy of roosed estmator uder otmum codto wth the rato estmator as - From (.) ad (3.6) we have ME 4 ( t ) ME( Ŝ ) λ y β( x ) ot δ β ( x ) ME(t ) ME(t ) + 4 4 λyβ λβ ( x) ME(t ) + 6 + always. (4.) ME( t ) 4 + 4 y ext we have comared the effcecy of roosed estmator uder otmum codto wth the exoetal rato estmator as - From (.3) ad (3.6) we have ( φ) y 68
4 ME(t ) ME( t ) ME( Ŝ ) δ 3 λ y β( x ) ot ME(t ) β( x ) + 4 y 4 λyβ ( φ) λβ ( x) ME(t ) + 6 + always. (4.3) ME( t ) 4 + 4 y Fally we have comared the effcecy of roosed estmator uder otmum codto wth the Regresso estmator as - 4 ( ) ( ) y x λ β φ λβ ME(t ) + ME(t ) 6 ME( t ) ME(t ) 4 > always. ME(t ) ME( t ) + 4 4 + y 4 y (4.4) 5. Emrcal study e have used the data gve uhatme ad uhatme (97),. 56. here, Yumber of vllages the crcle, ad φ Rereset a crcle cosstg more tha fve vllages. y λ 4 λ 4 λ 3 89 4.74. 3.8 6.6 3.996 The followg table shows PRE of dfferet estmator s w. r. t. to usual estmator. Table : PRE of dfferet estmators Estmators t t t t 3 t 4 PRE 4.898 6.87 54.74 96.6 Cocluso uerorty of the roosed estmator s establshed theoretcally by the uversally true codtos derved ectos 4. Results Table cofrms ths suerorty umercally usg the revously used data set. 69
Refereces Abd-Elfattah, A.M. El-herey, E.A. Mohamed,.M. Abdou, O. F. (): Imrovemet estmatg the oulato mea smle radom samlg usg formato o auxlary attrbute. Al. Mathe. ad Comt. Das, A. K., Trath, T. P. (978). Use of auxlary formato estmatg the fte oulato varace. ahya 4:39 48. Grover, L.K. (): A Correcto ote o Imrovemet Varace Estmato Usg Auxlary Iformato. Commucatos tatstcs Theory ad Methods, 39: 753 764, Kadlar, C., Cg, H. (6). Imrovemet varace estmato usg auxlary formato. Hacettee J. Math. tatst. 35(): 5. Kadlar, C., Cg, H. (7). Imrovemet varace estmato smle radom samlg. Commu. tatst. Theor. Meth. 36:75 8. Isa, C. T. (983). Varace estmato usg auxlary formato, Jour. of Amer. tatst.asso.78, 7 3, 983. Jhajj, H.., harma, M.K. ad Grover, L.K. (6). A famly of estmators of Poulato mea usg formato o auxlary attrbute. Pa. J. tatst., (),43-5. a, V.D., Guta, P.C. (996): A ote o estmato of mea wth ow oulato of a auxlary character, Joural of Id. oc. Agr. tatst. 48() 5 58. Prasad, B., gh, H. P. (99). ome mroved rato-tye estmators of fte oulato varace samle surveys. Commu. tatst. Theor. Meth. 9:7 39 gh, R. Chauha, P. awa,. maradache, F. (7): A geeral famly of estmators for estmatg oulato varace usg ow value of some oulato arameter(s). Reassace Hgh Press. gh, R. Chauha, P. awa,. maradache, F. (8): Rato estmators smle radom samlg usg formato o auxlary attrbute. Pa. J. tat. Oer. Res. 4() 47 53 habbr, J., Guta,. (7): O estmatg the fte oulato mea wth ow oulato roorto of a auxlary varable. Pa. Jour. of tatst. 3 () 9. habbr, J., Guta,. (7). O mrovemet varace estmato usg auxlary formato. Commu. tatst. Theor. Meth. 36():77 85. 7
Double exoetal Observed Data Tred values gle exoetal smoothg (alha.) smoothg (alha.9 ad gamma.) 995 5. 5.337 5. 996 5.75 5.336 5. 5.77 997 5.6 5.7 5.83 5.3857 998 5.7 5.364 5.97 5.698556 999 4.64 5.8 5.4363 4.765484 5.4 5.69 5.8367 5.884 3 4.3 5.356 5.7894 4.3944 4 6.6 5. 5.8446 5.858346 5 4.46 5.684 5.784 4.66965 6 5.5 5.348 5.647 5.437 Total 5.866 5.86 46.4684 5.96755 Ths boo has bee desged for studets ad researchers who are worg the feld of tme seres aalyss ad estmato fte oulato. There are aers by Rajesh gh, Floret maradache, hweta Maurya, Ashsh K. gh, Maoj Kr. Chaudhary, V. K. gh, Muesh Kumar ad ach Mal. Frst chater deals wth the roblem of tme seres aalyss ad the rest of four chaters deal wth the roblems of estmato fte oulato.