Table 7 Simulation results of eight different estimators for \(\beta =1.8, \lambda =0.8, \theta =1, \alpha =1.57\).
From: Properties, estimation, and applications of the extended log-logistic distribution
n | Measures | Parameters | MLEs | MPSEs | OLSEs | WLSEs | ADEs | CVMEs | PCEs | RADEs |
|---|---|---|---|---|---|---|---|---|---|---|
20 | AEs | \(\beta \) | \(2.40605^8\) | \(1.70906^6\) | \(1.70647^4\) | \(1.27940^2\) | \(1.44168^3\) | \(1.83402^7\) | \(0.72092^1\) | \(1.86964^5\) |
\(\lambda \) | \(1.44759^8 \) | \(0.84548^3\) | \(0.85629^4\) | \(1.15860^7\) | \(1.15423^6\) | \(0.83037^2\) | \(0.43884^1\) | \(0.96393^5\) | ||
\(\theta \) | \(1.03600^6\) | \(0.92683^1\) | \(0.97252^3 \) | \(0.95373^2\) | \(1.01515^5\) | \(1.03711^7\) | \(0.98104^4\) | \(1.04259^8\) | ||
MSEs | \(\alpha \) | \(1.00024^1\) | \(1.58825^6 \) | \(1.62559^8\) | \(1.13345^2\) | \(1.18353^3\) | \(1.59529^7\) | \(1.35438^4\) | \(1.36723^5\) | |
\(\beta \) | \(0.63984^1\) | \(0.64000^4\) | \( 0.64000^4\) | \(0.64000^4\) | \(0.64000^4\) | \(0.64000^4 \) | \(2.28449^7\) | \(2.38166^8 \) | ||
\(\lambda \) | \(0.45304^6 \) | \(0.26100^1\) | \(0.29216^2 \) | \(0.33851^5\) | \( 0.33119^4\) | \(0.29968^3\) | \(0.56008^8\) | \(0.51945^7\) | ||
\(\theta \) | \(0.02163^1\) | \(0.02877^3\) | \(0.02999^4\) | \(0.03359^6 \) | \(0.02407^2\) | \(0.03085^5\) | \(0.19684^8\) | \(0.05564^7\) | ||
ABs | \(\alpha \) | \(0.32490^{5.5}\) | \(0.02040^1\) | \( 0.05269^3\) | \(0.32489^4\) | \(0.32490^{5.5}\) | \(0.05029^2\) | \(1.11602^8\) | \(0.32695^7\) | |
\(\beta \) | \(0.79990^1\) | \(0.80000^4\) | \(0.80000^4\) | \(0.80000^4 \) | \(0.80000^4\) | \(0.80000^4 \) | \(1.51145^7\) | \(1.54326^8\) | ||
\(\lambda \) | \(0.67308^6\) | \(0.51088^1 \) | \(0.54052^2\) | \(0.58182^5 \) | \(0.57549^4\) | \(0.54743^3\) | \(0.74839^8\) | \(0.72073^7\) | ||
\(\theta \) | \(0.14708^1\) | \(0.16961^3\) | \(0.17319^4\) | \(0.18327^6\) | \(0.15514^2\) | \(0.17565^5\) | \(0.44367^8\) | \(0.23589^7\) | ||
\(\sum Ranks\) | \(\alpha \) | \(0.57000^{5.5}\) | \(0.14283^1\) | \(0.22955^3\) | \(0.56999^4 \) | \(0.57000^{5.5}\) | \(0.22425^2\) | \(1.05642^8\) | \(0.57180^7\) | |
| Â | \(50^4\) | \(34^1\) | \(45^2\) | \(51^{5.5}\) | \(48^3\) | \(51^{5.5}\) | \(72^7\) | \(81^8\) | ||
50 | AEs | \(\beta \) | \(1.78446^6\) | \(1.76392^5\) | \(1.69096^4\) | \(1.37499^3\) | \(1.33285^2\) | \(1.85071^8\) | \(0.58666^1\) | \(1.78552^7\) |
\(\lambda \) | \(1.15608^7\) | \(0.79969^2\) | \(0.90721^3 \) | \(1.14449^6\) | \(1.17978^8\) | \(0.91861^4\) | \(0.39058^1\) | \(0.95613^5\) | ||
\(\theta \) | \(0.99597^5\) | \(0.96009^1\) | \(0.99074^4\) | \(0.98077^2\) | \(1.00163^6\) | \(1.01869^8\) | \(0.98618^3\) | \(1.00883^7\) | ||
MSEs | \(\alpha \) | \(1.00131^1\) | \(1.58021^8 \) | \(1.52486^7\) | \(1.17194^3\) | \(1.22175^4\) | \(1.45174^6\) | \(1.32363^5\) | \(1.07363^2 \) | |
\(\beta \) | \(0.63984^2\) | \(0.57173^1\) | \(0.64000^{4.5}\) | \(0.64000^{4.5}\) | \(0.64000^{4.5}\) | \(0.64000^{4.5}\) | \(2.27661^8\) | \(1.65709^7\) | ||
\(\lambda \) | \(0.29360^5\) | \(0.17676^1\) | \(0.24173^2\) | \(0.28535^4\) | \(0.29770^6\) | \(0.24842^3 \) | \(0.54472^8\) | \(0.38550^7\) | ||
\(\theta \) | \(0.01366^2\) | \(0.01553^3\) | \(0.01754^6\) | \(0.01577^4\) | \(0.01350^1\) | \(0.01738^5\) | \(0.12597^8\) | \(0.02504^7\) | ||
ABs | \(\alpha \) | \(0.32488^6\) | \(0.01195^1\) | \(0.12787^2\) | \(0.32486^5\) | \(0.32490^7\) | \(0.14647^3 \) | \(0.47601^8\) | \(0.32473^4\) | |
\(\beta \) | \(0.79990^2\) | \(0.75613^1\) | \(0.80000^{4.5}\) | \(0.80000^{4.5}\) | \(0.80000^{4.5}\) | \(0.80000^{4.5}\) | \(1.50884^8\) | \(1.28728^7\) | ||
\(\lambda \) | \(0.54185^5\) | \(0.42043^1\) | \(0.49166^2\) | \(0.53418^4\) | \(0.54562^6\) | \(0.49842^3\) | \(0.73805^8\) | \( 0.62089^7\) | ||
\(\theta \) | \(0.11688^2\) | \(0.12463^3\) | \(0.13243^6\) | \(0.12558^4\) | \(0.11618^1\) | \(0.13184^5 \) | \(0.35492^8\) | \(0.15823^7\) | ||
\(\sum Ranks\) | \(\alpha \) | \(0.56998^6\) | \(0.10930^1 \) | \(0.35759^2\) | \(0.56996^5\) | \(0.57000^7\) | \(0.38271^3\) | \(0.68994^8\) | \(0.56985^4\) | |
| Â | \(49^{3.5}\) | \(28^1\) | \(47^2\) | \(49^{3.5}\) | \(57^{5.5}\) | \(57^{5.5}\) | \(74^8\) | \(71^7\) | ||
100 | AEs | \(\beta \) | \(1.61190^5\) | \(1.74976^8\) | \(1.72152^6 \) | \(1.45504^3\) | \(1.35007^2\) | \(1.72737^7\) | \(0.61898^1\) | \(1.57376^4\) |
\(\lambda \) | \(1.06663^7\) | \(0.77534^2\) | \(1.00824^5\) | \(1.07983^8\) | \(1.04610^6\) | \(0.93019^3\) | \(0.44217^1\) | \(1.00676^4\) | ||
\(\theta \) | \(0.99473^5\) | \(0.98635^2\) | \(0.99042^4\) | \(0.98866^3\) | \(1.00311^6\) | \(1.01694^8\) | \(0.95510^1\) | \(1.01322^7\) | ||
MSEs | \(\alpha \) | \(1.00480^1\) | \(1.57883^8\) | \(1.45900^7\) | \(1.19737^3\) | \(1.26341^4\) | \(1.44237^6\) | \(1.28958^5\) | \(1.15004^2\) | |
\(\beta \) | \(0.49533^2\) | \(0.29973^1\) | \(0.64000^{4.5}\) | \(0.64000^{4.5}\) | \(0.64000^{4.5}\) | \(0.64000^{4.5}\) | \(2.27225^8\) | \(1.02345^7 \) | ||
\(\lambda \) | \(0.23216^5\) | \(0.10940^1\) | \(0.22407^3\) | \(0.23063^4\) | \(0.23671^6\) | \(0.21868^2 \) | \(0.54863^8\) | \(0.30929^7\) | ||
\(\theta \) | \(0.00775^2\) | \(0.00817^3\) | \( 0.01057^5\) | \(0.00883^4\) | \(0.00765^1\) | \(0.01082^6\) | \(0.09409^8\) | \(0.01310^7\) | ||
ABs | \(\alpha \) | \(0.32484^7\) | \(0.00648^1\) | \(0.19743^3\) | \(0.32458^5\) | \(0.32466^6 \) | \(0.15337^2 \) | \(0.37547^8\) | \(0.32390^4\) | |
\(\beta \) | \(0.70380^2\) | \( 0.54748^1\) | \(0.80000^{4.5}\) | \(0.80000^{4.5}\) | \(0.80000^{4.5}\) | \(0.80000^{4.5}\) | \(1.50740^8\) | \(1.01166^7\) | ||
\(\lambda \) | \(0.48183^5\) | \(0.33076^1\) | \(0.47337^3\) | \(0.48024^4\) | \(0.48653^6\) | \(0.46763^2\) | \(0.74070^8\) | \(0.55614^7\) | ||
\(\theta \) | \(0.08806^2\) | \(0.09039^3\) | \(0.10281^5\) | \(0.09399^4\) | \(0.08749^1\) | \(0.10404^6\) | \(0.30673^8\) | \(0.11447^7\) | ||
\(\sum Ranks\) | \(\alpha \) | \(0.56994^7\) | \(0.08052^1\) | \(0.44432^3\) | \(0.56972^5\) | \(0.56979^6\) | \(0.39163^2\) | \(0.61275^8\) | \(0.56912^4\) | |
| Â | \(50^2\) | \(32^1\) | \(53^{4.5}\) | \(52^3\) | \(53^{4.5}\) | \(57^6\) | \(72^8\) | \(67^7\) | ||
300 | AEs | \(\beta \) | \(1.64013^7\) | \(1.78639^8 \) | \(1.55607^5\) | \(1.55258^4\) | \( 1.46469^2\) | \(1.56842^6\) | \(0.67874^1\) | \(1.53963^3\) |
\(\lambda \) | \(0.99022^6\) | \( 0.79249^2\) | \(1.01253^8\) | \(0.93167^4\) | \(0.99474^7\) | \(0.91569^3\) | \(0.61077^1\) | \(0.95918^5\) | ||
\(\theta \) | \(0.98989^3\) | \(0.99362^5\) | \(0.99304^4\) | \(0.98951^2\) | \(0.99383^6\) | \(1.00593^8\) | \(0.93409^1\) | \(1.00158^7\) | ||
MSEs | \(\alpha \) | \(1.16031^1\) | \(1.57242^8\) | \(1.48064^6\) | \(1.42548^4\) | \(1.52672^7\) | \(1.48053^5\) | \(1.34723^3\) | \(1.29471^2\) | |
\(\beta \) | \(0.20980^2\) | \(0.10166^1 \) | \(0.45191^6\) | \(0.23408^3\) | \( 0.32930^4\) | \(0.39015^5\) | \(2.16190^8\) | \(0.47519^7\) | ||
\(\lambda \) | \(0.13312^3\) | \(0.03817^1\) | \(0.16962^6\) | \(0.12743^2\) | \(0.15141^5\) | \(0.14292^4\) | \(0.53932^8\) | \(0.20437^7\) | ||
\(\theta \) | \(0.00296^3 \) | \(0.00238^1\) | \(0.00450^6\) | \(0.00335^4 \) | \(0.00295^2\) | \(0.00446^5\) | \(0.07166^8\) | \(0.00477^7\) | ||
ABs | \(\alpha \) | \(0.32460^7\) | \(0.00177^1\) | \(0.31807^2 \) | \(0.32426^8\) | \(0.32453^6\) | \(0.32059^3\) | \(0.32104^4\) | \(0.32324^5\) | |
\(\beta \) | \(0.45804^2\) | \(0.31884^1 \) | \(0.67224^6\) | \(0.48381^3\) | \(0.57385^4\) | \(0.62462^5\) | \(1.47034^8\) | \(0.68934^7\) | ||
\(\lambda \) | \(0.36485^3\) | \(0.19537^1\) | \(0.41185^6\) | \(0.35697^2\) | \(0.38912^5\) | \(0.37805^4\) | \(0.73439^8\) | \(0.45207^7\) | ||
\(\theta \) | \(0.05437^3\) | \(0.04878^1\) | \(0.06712^6\) | \(0.05788^4\) | \(0.05434^2\) | \(0.06677^5\) | \(0.26769^8\) | \(0.06906^7\) | ||
\(\sum Ranks\) | \(\alpha \) | \(0.56973^7\) | \(0.04209^1\) | \(0.56398^2\) | \(0.56944^8\) | \(0.56968^6\) | \(0.56621^3\) | \(0.56660^4\) | \(0.56854^5 \) | |
| Â | \(47^2\) | \(31^1\) | \(63^6\) | \(48^3\) | \(56^{4.5}\) | \(56^{4.5}\) | \(64^7\) | \(69^8\) | ||
500 | AEs | \(\beta \) | \(1.66480^7\) | \(1.79223^8 \) | \(1.57804^3\) | \(1.62616^6\) | \( 1.52002^2 \) | \(1.61942^5\) | \(0.72885^1\) | \(1.58496^4\) |
\(\lambda \) | \(0.93083^4\) | \(0.79564^2\) | \(1.00028^7\) | \(0.87846^3 \) | \(0.96266^6\) | \(0.95835^5\) | \(0.68702^1\) | \(1.01412^8\) | ||
\(\theta \) | \(0.98978^2\) | \( 0.99670^8 \) | \( 0.99317^7\) | \(0.99096^4 \) | \(0.99069^3 \) | \(0.99254^5 \) | \(0.92539^1\) | \(0.99265^6\) | ||
MSEs | \(\alpha \) | \(1.19724^1\) | \(1.57099^8\) | \(1.47376^5\) | \(1.55625^7\) | \(1.55564^6\) | \(1.45404^4\) | \(1.35852^3\) | \(1.32584^2\) | |
\(\beta \) | \(0.11866^3\) | \(0.00270^1\) | \(0.30485^6 \) | \(0.11563^2\) | \(0.18873^4\) | \(0.28790^5 \) | \(2.02183^8\) | \(0.35532^7\) | ||
\(\lambda \) | \(0.09756^3\) | \(0.00461^1 \) | \(0.13278^6\) | \(0.08035^2\) | \(0.11769^4\) | \( 0.13078^5\) | \(0.54828^8\) | \(0.16448^7\) | ||
\(\theta \) | \(0.00195^3\) | \(0.00146^1\) | \(0.00260^6\) | \(0.00179^2\) | \(0.00208^4\) | \(0.00272^7\) | \(0.06404^8\) | \(0.00250^5\) | ||
ABs | \(\alpha \) | \(0.32446^7\) | \( 0.00028^1 \) | \(0.32211^3\) | \(0.32416^6\) | \(0.32481^8\) | \(0.32221^4\) | \(0.28438^2\) | \(0.32322^5\) | |
\(\beta \) | \(0.34448^3\) | \(0.05192^1\) | \(0.55214^6\) | \(0.34004^2\) | \(0.43443^4\) | \(0.53656^5\) | \(1.42191^8\) | \(0.59608^7\) | ||
\(\lambda \) | \(0.31235^3\) | \(0.06789^1\) | \(0.36439^6\) | \(0.28346^2\) | \(0.34305^4\) | \(0.36164^5\) | \(0.74046^8\) | \(0.40556^7\) | ||
\(\theta \) | \(0.04420^3\) | \(0.03818^1 \) | \(0.05095^6\) | \(0.04227^2\) | \( 0.04561^4\) | \(0.05211^7\) | \(0.25306^8\) | \(0.04999^5\) | ||
\(\sum Ranks\) | \(\alpha \) | \(0.56961^7\) | \(0.01674^1\) | \(0.56755^3\) | \(0.56935^6\) | \(0.56992^8\) | \(0.56763^4\) | \(0.53327^2\) | \(0.56852^5\) | |
| Â | \(46^3\) | \(34^1\) | \(64^7\) | \(44^2\) | \(57^4\) | \(61^6\) | \(58^5\) | \(68^8\) |
