Table 8 Simulation results of eight different estimators for \(\beta =2.72, \lambda =0.8, \theta =1, \alpha =1.57\).
From: Properties, estimation, and applications of the extended log-logistic distribution
n | Measures | Parameters | MLEs | MPSEs | OLSEs | WLSEs | ADEs | CVMEs | PCEs | RADEs |
|---|---|---|---|---|---|---|---|---|---|---|
20 | AEs | \(\beta \) | \(3.83317^7\) | \(2.69465^5\) | \(2.66900^4\) | \(2.01021^3\) | \(1.99556^2\) | \(2.81901^6\) | \(0.84400^1\) | \(3.04477^7\) |
\(\lambda \) | \(1.39140^8\) | \(0.89289^3\) | \(0.80380^2\) | \(1.08647^7\) | \(0.89690^4\) | \(0.90696^5\) | \(0.29584^1\) | \(0.93964^6\) | ||
\(\theta \) | \(1.05487^6\) | \(0.94697^1\) | \(1.00712^3\) | \(1.00615^2\) | \(1.03610^5\) | \(1.06801^7\) | \(1.00721^4\) | \(1.06808^8 \) | ||
MSEs | \(\alpha \) | \(1.00052 ^1\) | \(1.56094^6\) | \(1.59519^8\) | \(1.21050^2\) | \(1.35700^3\) | \(1.59161^7\) | \(1.40976^4\) | \(1.49233^5\) | |
\(\beta \) | \(2.95806^1\) | \(2.95840^4\) | \(2.95840^4\) | \(2.95840^4\) | \(2.95840^4\) | \(2.95840^4\) | \(5.20635^7\) | \(6.07616^8\) | ||
\(\lambda \) | \(0.47867^6\) | \(0.38595^1\) | \(0.40950^3\) | \(0.41542^4\) | \(0.39740^2\) | \(0.42329^5\) | \(0.56572^8\) | \(0.56146^7\) | ||
\(\theta \) | \(0.02600^1\) | \(0.02821^3\) | \(0.03196^5\) | \(0.03142^4\) | \( 0.02806^2\) | \(0.03497^6\) | \(0.17730^8\) | \(0.04975^7\) | ||
ABs | \(\alpha \) | \(0.32489^{4.5}\) | \(0.03087^1\) | \(0.07432^3\) | \(0.32489^6\) | \(0.32490^{4.5}\) | \(0.06911^2\) | \(0.88049^8\) | \(0.32644^7\) | |
\(\beta \) | \(1.71990^1\) | \(1.72000^4\) | \(1.72000^4\) | \(1.72000^4\) | \(1.72000^4\) | \(1.72000^4\) | \(2.28174^7\) | \(2.46499^8\) | ||
\(\lambda \) | \(0.69186^6\) | \(0.62125^1\) | \(0.63992^3\) | \(0.64453^4\) | \(0.63039^2\) | \(0.65061^5\) | \(0.75214^8\) | \(0.74931^7\) | ||
\(\theta \) | \(0.16125^1\) | \(0.16795^3\) | \(0.17878^5\) | \(0.17726^4\) | \(0.16753^2\) | \(0.18699^6\) | \(0.42107^8\) | \(0.22305^7\) | ||
| Â | \(\alpha \) | \(0.56999^{4.5}\) | \(0.17571^1\) | \(0.27262^3\) | \(0.56999^6\) | \(0.56999^{4.5}\) | \(0.26289^2\) | \(0.93834^8\) | \(0.57135^7\) | |
\(\sum Ranks\) | \(47^{3.5}\) | \(33^1\) | \(47^{3.5}\) | \(50^5\) | \(39^2\) | \(59^6\) | \(72^7\) | \(84^8\) | ||
50 | Â | \(\beta \) | \(3.01269^7\) | \(2.68265^4\) | \(2.74437^6\) | \(2.08949^2\) | \(2.22550^3\) | \(2.71746^5\) | \(0.71635^1\) | \(3.18805^8\) |
| Â | \(\lambda \) | \(1.14052^7\) | \(0.81202^3\) | \(0.87854^4\) | \(1.02755^5\) | \(1.16089^8\) | \(0.78891^2\) | \(0.32478^1\) | \(1.04723^6\) | |
AEs | \(\theta \) | \(1.01559^7\) | \(0.96681^1\) | \(1.00384^5\) | \(1.00004^3\) | \(1.00269^4\) | \(1.03366^8\) | \(0.97802^2\) | \(1.00460^6 \) | |
| Â | \(\alpha \) | \(1.00119^1\) | \(1.57261^8\) | \(1.55631^7\) | \(1.17364^2\) | \(1.20999^4\) | \(1.53681^6\) | \(1.33058^5\) | \(1.19198^3\) | |
| Â | \(\beta \) | \(2.95806^2\) | \(1.90770^1\) | \(2.95840^{4.5}\) | \(2.95840^{4.5}\) | \(2.95840^{4.5}\) | \(2.95840^{4.5}\) | \(5.30117^8\) | \(4.61663^7\) | |
| Â | \(\lambda \) | \(0.37584^5\) | \(0.25979^1\) | \(0.35982^3\) | \(0.34376^2\) | \(0.38376^6\) | \(0.36577^4\) | \(0.55806^8\) | \(0.47437^7\) | |
MSEs | \(\theta \) | \(0.01240^1\) | \(0.01354^2\) | \(0.01843^6\) | \(0.01498^4\) | \(0.01490^3\) | \(0.01692^5\) | \(0.12654^8\) | \(0.02001^7\) | |
| Â | \(\alpha \) | \(0.32485^5\) | \(0.00773^1\) | \(0.10265^3\) | \(0.32488^{6.5} \) | \(0.32488^{6.5}\) | \(0.08529^2\) | \(0.36917^8\) | \(0.32346^4\) | |
| Â | \(\beta \) | \(1.71990^2\) | \(1.38120 ^1\) | \(1.72000^{4.5}\) | \(1.72000^{4.5}\) | \(1.72000^{4.5}\) | \(1.72000^{4.5}\) | \(2.30243^8\) | \(2.14864^7\) | |
| Â | \(\lambda \) | \(0.61306^5\) | \(0.50970^1\) | \(0.59985^3\) | \(0.58631^2\) | \(0.61948^6\) | \(0.60479^4\) | \(0.74703^8\) | \(0.68874^7\) | |
ABs | \(\theta \) | \(0.11133^1\) | \(0.11638^2\) | \(0.13576^6\) | \(0.12240^4\) | \(0.12208^3\) | \(0.13008^5\) | \(0.35573^8 \) | \(0.14147^7\) | |
| Â | \(\alpha \) | \(0.56996^5\) | \(0.08790^1\) | \(0.32039^3\) | \(0.56999^{6.5}\) | \(0.56999^{6.5}\) | \(0.29205^2\) | \(0.60759^8\) | \(0.56873^4\) | |
\(\sum Ranks\) | \(48^3\) | \(26^1\) | \(55^5\) | \(46^2\) | \(59^6\) | \(52^4\) | \(73^{7.5}\) | \( 73^{7.5}\) | ||
100 | Â | \(\beta \) | \(2.67797^4\) | \(2.69462^5\) | \(2.69664^6\) | \(2.22696^3\) | \(2.08596^2\) | \(2.76898^8\) | \(0.93592^1\) | \(2.76245^7\) |
| Â | \(\lambda \) | \(1.09874^7\) | \(0.80411^2\) | \(0.88255^4\) | \(1.04604^6\) | \(1.13597^8\) | \(0.86722^3\) | \(0.58790^1\) | \(0.91953^5\) | |
AEs | \(\theta \) | \(0.99736^5\) | \(0.97975^2\) | \(0.99701^3\) | \(0.99728^4\) | \(1.00156^7\) | \(1.01634^8\) | \(0.94139^1\) | \(1.00139^6 \) | |
| Â | \(\alpha \) | \(1.00503^1\) | \(1.56955^8\) | \(1.51926^6\) | \(1.20729^2\) | \(1.32401^4\) | \(1.52504^7\) | \(1.32438^5\) | \(1.27476^3 \) | |
| Â | \(\beta \) | \(1.88643^2\) | \(0.98469^1\) | \(2.62116^5\) | \(2.50947^3\) | \(2.52599^4\) | \(2.62966^6\) | \(5.22572^8\) | \(3.29159^7\) | |
| Â | \(\lambda \) | \(0.27710^2\) | \( 0.12979^1\) | \(0.29357^5\) | \(0.27942^3\) | \(0.30368^6\) | \(0.28689^4\) | \(0.55007^8\) | \(0.36429^7\) | |
MSEs | \(\theta \) | \(0.00747^2\) | \(0.00744^1\) | \(0.01059^5\) | \(0.00853^4\) | \(0.00792^3\) | \(0.01063^6\) | \(0.08866^8\) | \(0.01298^7\) | |
| Â | \(\alpha \) | \(0.32478^8\) | \(0.00257^1\) | \(0.10683^3\) | \(0.32448^6\) | \(0.32459^7\) | \(0.09525^2\) | \(0.32319^5\) | \(0.32114^4\) | |
| Â | \(\beta \) | \(1.37347^2\) | \(0.99232^1\) | \(1.61900^5\) | \(1.58413^3\) | \(1.58934^4\) | \(1.62162^6\) | \(2.28598^8\) | \(1.81427^7\) | |
| Â | \(\lambda \) | \(0.52641^2\) | \(0.36026^1\) | \(0.54182^5\) | \(0.52860^3\) | \(0.55107^6\) | \(0.53562 ^4\) | \(0.74167^8 \) | \(0.60356^7\) | |
ABs | \(\theta \) | \(0.08642^2\) | \(0.08628^1\) | \(0.10293^5\) | \(0.09235^4\) | \(0.08899^3\) | \(0.10310^6\) | \(0.29775^8\) | \(0.11391^7\) | |
\(\sum Ranks\) | \(\alpha \) | \(0.56989^8\) | \(0.05071^1\) | \(0.32684^3\) | \(0.56963^6\) | \(0.56973^7\) | \(0.30863^2\) | \(0.56850^5\) | \(0.56669^4\) | |
| Â | \(45^2\) | \(25^1\) | \(55^4\) | \(47^3\) | \(61^5\) | \(62^6\) | \(66^7\) | \(71^8\) | ||
300 | AEs | \(\beta \) | \(2.51671^4\) | \(2.70958^8\) | \(2.53953^5\) | \(2.31255^3\) | \(2.24721^2\) | \(2.66517^7\) | \(0.91569^1\) | \(2.58886^6\) |
\(\lambda \) | \(0.99776^7\) | \(0.79787^2\) | \( 0.92055^3\) | \(0.96069^6\) | \(1.06673^8\) | \(0.95149^5\) | \(0.56696^1\) | \(0.94255^4 \) | ||
\(\theta \) | \(0.99376^2\) | \(0.99395^3\) | \(0.99989^8\) | \(0.99984^7\) | \(0.99931^5\) | \(0.99697^4 \) | \(0.94941^1\) | \( 0.99951^6\) | ||
MSEs | \(\alpha \) | \(1.21193^1\) | \(1.57148^8\) | \(1.51225^7\) | \(1.22915^2\) | \(1.32601^3\) | \(1.48470^6\) | \(1.35326^5\) | \(1.34156^4\) | |
\(\beta \) | \(0.72901^2\) | \(0.00114^1\) | \(1.41257^6\) | \(0.93645^3\) | \(1.22649^4\) | \(1.28174^5\) | \(4.92350^8\) | \(1.61073^7\) | ||
\(\lambda \) | \(0.14349^2\) | \(0.00507^1\) | \(0.17120^4\) | \(0.16905^3\) | \(0.19757^6\) | \(0.17164^5\) | \(0.55873^8\) | \(0.22582^7\) | ||
\(\theta \) | \(0.00233^2\) | \(0.00210^1\) | \(0.00446^6\) | \(0.00307^3\) | \(0.00315^4\) | \(0.00453^7\) | \(0.07117^8\) | \(0.00442^5\) | ||
ABs | \(\alpha \) | \(0.32371^7\) | \(0.00027^1\) | \(0.11725^3\) | \(0.32411^8\) | \(0.32359^6\) | \(0.09143^2\) | \(0.27826^5\) | \(0.23658^4\) | |
\(\beta \) | \(0.85382^2\) | \(0.03382^1\) | \(1.18852^6\) | \(0.96771^3\) | \(1.10747^4\) | \(1.13214^5\) | \(2.21890^8\) | \(1.26914^7\) | ||
\(\lambda \) | \(0.37880^2\) | \(0.07119^1\) | \(0.41376^4\) | \(0.41116^3\) | \(0.44449^6\) | \(0.41429^5\) | \(0.74748^8\) | \(0.47520^7\) | ||
\(\theta \) | \(0.04824^2\) | \(0.04583^1\) | \(0.06675^7\) | \(0.05544^3\) | \(0.05609^4\) | \(0.06734^7\) | \(0.26678^8\) | \(0.06646^5\) | ||
\(\sum Ranks\) | \(\alpha \) | \(0.56896^7\) | \(0.01635^1\) | \(0.34241^3\) | \(0.56931^8\) | \(0.56885^6\) | \(0.30237^2\) | \(0.52750^5\) | \(0.48639^4\) | |
| Â | \(40^2\) | \(29^1\) | \(61^6\) | \(52^3\) | \(58^4\) | \(60^5\) | \(66^{7.5}\) | \(66^{7.5}\) | ||
500 | AEs | \(\beta \) | \(2.47798^4\) | \( 2.71355^8 \) | \(2.53867^5\) | \(2.40238^3\) | \(2.24405^2\) | \(2.63327^7\) | \(0.96478^1\) | \(2.55025^6\) |
\(\lambda \) | \(0.91458^3\) | \(0.80055^2\) | \(0.93710^4\) | \(0.94581^6\) | \(1.03398^8\) | \(0.93930^5\) | \(0.59929^1 \) | \(0.96909^7\) | ||
\(\theta \) | \(0.99810^6\) | \(0.99689^4\) | \( 0.99793^5 \) | \(0.99466^2\) | \(0.99476^3\) | \(1.00013^8\) | \(0.94612^1\) | \(0.99906^7\) | ||
MSEs | \(\alpha \) | \(1.32791^1\) | \(1.57009^8\) | \(1.50813^7 \) | \(1.39771^5\) | \(1.38220^4\) | \(1.47807^6\) | \(1.35952^2\) | \(1.36438^3\) | |
\(\beta \) | \(0.42910^2\) | \(0.00043^1\) | \(0.93844^5\) | \(0.56404^3\) | \( 0.79650^4 \) | \(0.94482^6\) | \(4.71787^8\) | \(1.13902^7\) | ||
\(\lambda \) | \(0.09749^2\) | \(0.00186^1\) | \(0.13919^4\) | \(0.13849^3 \) | \(0.15757^6\) | \(0.14148^5\) | \(0.55901^8\) | \(0.17418^7\) | ||
\(\theta \) | \(0.00155^2\) | \(0.00098^1\) | \(0.00294^7\) | \(0.00162^3 \) | \(0.00189^4 \) | \(0.00283^6\) | \(0.05501^8\) | \(0.00262^5\) | ||
ABs | \(\alpha \) | \(0.32308^6\) | \(0.00011^1\) | \(0.11861^3\) | \(0.32412^8\) | \(0.32383^7\) | \(0.09910^2 \) | \(0.23761^4\) | \(0.24336^5\) | |
\(\beta \) | \(0.65506^2\) | \(0.02066^1\) | \(0.96873^5 \) | \(0.75102^3\) | \(0.89247^4\) | \(0.97202^6\) | \(2.17207^8\) | \(1.06725^7\) | ||
\(\lambda \) | \(0.31223^2\) | \(0.04314^1\) | \(0.37308^4\) | \(0.37214^3\) | \(0.39695^6\) | \(0.37613^5\) | \(0.74767^8\) | \(0.41734^7\) | ||
\(\theta \) | \(0.03941^2 \) | \( 0.03136^1\) | \(0.05426^7\) | \(0.04019^3\) | \(0.04345^4\) | \(0.05315^6\) | \(0.23453^8\) | \(0.05118^5\) | ||
\(\sum Ranks\) | \(\alpha \) | \(0.56840^6\) | \(0.01037^1 \) | \(0.34440^3\) | \(0.56931^8 \) | \(0.56906^7\) | \(0.31481^2\) | \(0.48745^4\) | \(0.49332^5\) | |
| Â | \(38^2\) | \(30^1\) | \(59^{4.5}\) | \(50^3\) | \(59^{4.5}\) | \(64^7\) | \(61^6\) | \(71^8\) | ||
