Table 4 Simulation results of eight different estimators for \(\beta =1.8, \lambda =0.75, \theta =0.6, \alpha =2.75\).
From: Properties, estimation, and applications of the extended log-logistic distribution
n | Measures | Parameters | MLEs | MPSEs | OLSEs | WLSEs | ADEs | CVMEs | PCEs | RADEs |
|---|---|---|---|---|---|---|---|---|---|---|
20 | AEs | \(\beta \) | \(2.30918^7\) | \(1.73038^5\) | \(1.62072^4\) | \(1.10976^1\) | \(1.27347^3\) | \(1.98980^6\) | \(1.14088^2\) | \(2.02430^8\) |
\(\lambda \) | \(1.34738^8\) | \(0.86261^3\) | \( 0.68810^2\) | \(1.16629^7\) | \(0.97032^6\) | \(0.90218^4\) | \(0.43964^1\) | \(0.96514^5\) | ||
\(\theta \) | \(0.62140^6\) | \(0.56018^1\) | \(0.58548^4\) | \(0.58113^2\) | \(0.61694^5\) | \(0.62548^7\) | \(0.58156^3\) | \(0.63153^8\) | ||
MSEs | \(\alpha \) | \(1.00020^1\) | \(2.72547^8\) | \(2.30898^6\) | \(1.12781^3\) | \(1.21752^4\) | \(1.83258^5\) | \(2.71709^7\) | \(1.04026^2\) | |
\(\beta \) | \(0.63984^2\) | \(0.64000^5\) | \(0.64000^5\) | \(0.64000^5\) | \( 0.64000^5\) | \(0.64000^5\) | \(0.33850^1\) | \(2.02862^8\) | ||
\(\lambda \) | \(0.51213^6\) | \(0.40127^1\) | \( 0.45512^2\) | \(0.48816^5\) | \(0.47457^3\) | \(0.48114^4\) | \(0.86239^7\) | \(0.97720^8\) | ||
\(\theta \) | \(0.00890^1\) | \(0.01096^5\) | \(0.01077^4\) | \(0.01061^3\) | \(0.01009^2\) | \(0.01138^6\) | \(0.06070^8\) | \(0.02007^7\) | ||
ABs | \(\alpha \) | \(3.06241^6\) | \(0.02106^1\) | \(1.09984^3\) | \(3.06246^7 \) | \(3.06250^8\) | \(1.49623^4\) | \(2.94840^5\) | \(0.96502^2\) | |
\(\beta \) | \(0.79990^2\) | \(0.80000^5\) | \(0.80000^5\) | \(0.80000^5\) | \(0.80000^5\) | \(0.80000^5\) | \(0.58180^1\) | \(1.42430^8\) | ||
\(\lambda \) | \(0.71563^6\) | \(0.63346^1\) | \(0.67463^2 \) | \(0.69868^5\) | \(0.68889^3\) | \(0.69364^4 \) | \(0.92865^7\) | \(0.98853^8\) | ||
\(\theta \) | \(0.09436^1\) | \(0.10471^5\) | \(0.10378^4\) | \(0.10302^3\) | \(0.10045^2\) | \(0.10668^6 \) | \(0.24637^8\) | \(0.14167^7\) | ||
\(\sum Ranks\) | \(\alpha \) | \(1.74998^6 \) | \(0.14514^1\) | \(1.04873^3\) | \(1.74999^7\) | \(1.75000^8\) | \(1.22320^4\) | \(1.71709^5\) | \(0.98236^2\) | |
| Â | \(52^3\) | \(41^1\) | \(44^2\) | \(53^4\) | \(54^5\) | \(60^7\) | \(55^6\) | \(73^8\) | ||
50 | AEs | \(\beta \) | \(1.90847^8\) | \(1.71174^4\) | \(1.79350^5\) | \(1.37898^3\) | \(1.34919^2\) | \(1.81052^6\) | \(1.08159^1\) | \(1.86446^7\) |
\(\lambda \) | \(1.11951^7 \) | \(0.75245^2\) | \( 0.89347^4\) | \(1.05591^6\) | \(1.37838^8\) | \(0.78640^3\) | \(0.48067^1\) | \(0.95828^5\) | ||
\(\theta \) | \(0.59671^4\) | \( 0.58277^2\) | \(0.59264^3\) | \(0.59807^5\) | \(0.59828^6 \) | \(0.61489^8\) | \(0.56804^1\) | \(0.60509^7\) | ||
MSEs | \(\alpha \) | \(1.00222^1\) | \(2.75406^8\) | \(2.30314^6\) | \(1.30344^2\) | \(1.59166^4\) | \( 2.17652^5\) | \(2.66161^7\) | \(1.41061^3\) | |
\(\beta \) | \(0.63984^3\) | \(0.55288^2\) | \(0.64000^{5.5}\) | \(0.64000^{5.5}\) | \(0.64000^{5.5}\) | \(0.64000^{5.5}\) | \(0.27110^1 \) | \( 1.59887^8 \) | ||
\(\lambda \) | \(0.45329^5\) | \(0.30358^1\) | \(0.42676^2\) | \(0.44349^4 \) | \(0.49364^6 \) | \(0.43305^3\) | \(0.86533^7\) | \(0.90302^8\) | ||
\(\theta \) | \(0.00516^2\) | \(0.00512^1 \) | \(0.00604^5\) | \(0.00533^4\) | \(0.00517^3 \) | \(0.00627^6 \) | \(0.05104^8\) | \(0.00832^7\) | ||
ABs | \(\alpha \) | \(3.06232^6 \) | \(0.01200^1\) | \(1.16161^3\) | \(3.06245^7\) | \(3.06250^8\) | \(1.17075^4\) | \(2.76096^5\) | \(0.9353^2\) | |
\(\beta \) | \(0.79990^3\) | \(0.74356^2\) | \(0.80000^{5.5}\) | \(0.80000^{5.5}\) | \( 0.80000^{5.5}\) | \(0.80000^{5.5}\) | \(0.52068^1\) | \(1.26446^8\) | ||
\(\lambda \) | \(0.67327^5\) | \(0.55098^1\) | \(0.65327^2\) | \(0.66595^4\) | \(0.70259^6 \) | \(0.65806^3\) | \(0.93023^7\) | \(0.95028^8\) | ||
\(\theta \) | \(0.07183^2\) | \(0.07153^1\) | \(0.07774^5\) | \(0.07300^4\) | \(0.07191^3\) | \(0.07916^6\) | \(0.22592^8\) | \(0.09123^7\) | ||
\(\sum Ranks\) | \(\alpha \) | \(1.74995^6\) | \(0.10955^1\) | \(1.07778^3 \) | \(1.74999^7 \) | \(1.75000^8\) | \(1.08201^4\) | \(1.66161^5\) | \( 0.96711^2\) | |
| Â | \(52^{3.5}\) | \(24^1\) | \(49^2\) | \(57^5\) | \(65^7\) | \(59^6\) | \(52^{3.5}\) | \(72^8\) | ||
100 | AEs | \(\beta \) | \(1.61324^5 \) | \(1.74387^7 \) | \(1.58740^4 \) | \(1.42594^3 \) | \(1.34350^2\) | \(1.78835^8\) | \(1.12875^1\) | \(1.7283^6\) |
\(\lambda \) | \(0.93339^5 \) | \(0.75448^2\) | \(0.85191^3\) | \(0.93943^6\) | \(1.03594^7\) | \(0.89111^4\) | \(0.52064^1\) | \(1.04084^8\) | ||
\(\theta \) | \(0.59964^5\) | \(0.58852^1\) | \(0.60018^6\) | \(0.59711^3 \) | \(0.59950^4\) | \(0.60627^8\) | \(0.59277^2\) | \(0.60209^7\) | ||
MSEs | \(\alpha \) | \(1.01823^1 \) | \(2.74961^8\) | \(2.40158^6\) | \(1.59791^2 \) | \(1.80575^4 \) | \(2.30641^5\) | \(2.64374^7\) | \(1.70493^3\) | |
\(\beta \) | \(0.54732^3\) | \(0.30312^1\) | \(0.64000^{5.5}\) | \(0.64000^{5.5}\) | \(0.64000^{5.5}\) | \(0.64000^{5.5}\) | \(0.30996^2\) | \(1.27306^8\) | ||
\(\lambda \) | \(0.39023^2\) | \(0.23891^1\) | \(0.40670^4\) | \(0.40243^3 \) | \(0.43476^6\) | \(0.42111^5\) | \(0.86207^8\) | \(0.81844^7\) | ||
\(\theta \) | \(0.00302^3\) | \(0.00292^1\) | \(0.00416^6\) | \(0.00295^2\) | \(0.00305^4\) | \( 0.00404^5\) | \(0.03880^8\) | \(0.00466^7\) | ||
ABs | \(\alpha \) | \(3.06210^7\) | \(0.00728^1\) | \(1.52178^4\) | \(3.06164^6\) | \(3.06250^8 \) | \(1.33813^3\) | \(2.70188^5\) | \(0.91841^2\) | |
\(\beta \) | \(0.73981^3\) | \(0.55057^1\) | \( 0.80000^{5.5}\) | \(0.80000^{5.5}\) | \(0.80000^{5.5}\) | \(0.80000^{5.5}\) | \(0.55674^2\) | \(1.1283^8\) | ||
\(\lambda \) | \(0.62468^2 \) | \(0.48879^1\) | \( 0.63773^4\) | \( 0.63437^3\) | \(0.65936^6\) | \(0.64893^5\) | \(0.92848^8\) | \(0.90468^7\) | ||
\(\theta \) | \(0.05493^3 \) | \(0.05404^1\) | \(0.06453^6\) | \(0.05430^2\) | \(0.05525^4\) | \( 0.06354^5\) | \(0.19698^8\) | \(0.06828^7\) | ||
\(\sum Ranks\) | \(\alpha \) | \(1.74989^7\) | \(0.08534^1\) | \(1.23360^4\) | \( 1.74975^6\) | \(1.75000^8\) | \(1.15677^3\) | \(1.64374^5\) | \(0.95834^2\) | |
| Â | \(46^2\) | \(26^1\) | \(58^5\) | \(47^3\) | \(64^7\) | \(62^6\) | \(57^4\) | \(72^8\) | ||
300 | AEs | \(\beta \) | \(1.64047^6\) | \(1.77558^8\) | \( 1.59409^4\) | \(1.57214^3 \) | \(1.42582^2\) | \(1.68963^7\) | \(0.95532^1\) | \(1.61968^5\) |
\(\lambda \) | \(0.78274^3\) | \(0.74914^2\) | \(0.94333^7\) | \(0.82878^4 \) | \(0.98881^8\) | \(0.89013^5\) | \(0.52280^1\) | \(0.90073^6\) | ||
\(\theta \) | \(0.59411^2\) | \(0.59710^5\) | \(0.60051^8\) | \(0.59618^3\) | \(0.59688^4\) | \(0.59925^6\) | \(0.59252^1\) | \(0.60041^7\) | ||
MSEs | \(\alpha \) | \(1.59383^1\) | \(2.75038^8\) | \(2.55426^6\) | \(2.71605^7 \) | \(2.34249^3\) | \( 2.41842^4\) | \(2.47185^5\) | \(1.88476^2\) | |
\(\beta \) | \(0.19496^2\) | \(0.08322^1\) | \(0.42438^7 \) | \(0.22547^3 \) | \(0.33092^5\) | \(0.38857^6\) | \(0.25367^4\) | \(1.03975^8\) | ||
\(\lambda \) | \(0.27608^3\) | \(0.07199^1\) | \(0.34230^5\) | \(0.26977^2\) | \(0.35612^6\) | \(0.33378^4\) | \(0.88833^8\) | \(0.63047^7\) | ||
\(\theta \) | \(0.00101^3\) | \(0.00089^1\) | \(0.00171^7 \) | \(0.00097^2\) | \(0.00128^4\) | \(0.00155^5\) | \(0.04369^8\) | \(0.00158^6\) | ||
ABs | \(\alpha \) | \(3.06111^7\) | \(0.00098^1\) | \(1.95127^3\) | \(3.06073^6\) | \(3.06250^8\) | \( 2.48202^5\) | \(2.16636^4\) | \(0.88243^2\) | |
\(\beta \) | \(0.44154^2\) | \(0.28848^1 \) | \(0.65144^7\) | \( 0.47484^3\) | \(0.57525^5\) | \(0.62335^6\) | \(0.50365^4\) | \(1.01968^8\) | ||
\(\lambda \) | \(0.52543^3\) | \(0.26831^1\) | \(0.58506^5\) | \(0.51939^2\) | \(0.59675^6\) | \(0.57774^4\) | \(0.94251^8 \) | \(0.79402^7\) | ||
\(\theta \) | \(0.03185^3 \) | \(0.02980^1\) | \(0.04140^7 \) | \(0.03117^2\) | \(0.03578^4\) | \(0.03937^5\) | \(0.20901^8\) | \(0.03973^6\) | ||
\(\sum Ranks\) | \(\alpha \) | \(1.74960^7 \) | \(0.03123^1\) | \( 1.39688^3 \) | \( 1.74950^6 \) | \(1.75000^8\) | \(1.57544^5\) | \(1.47185^4\) | \(0.93938^2\) | |
| Â | \(42^2\) | \(31^1\) | \(69^8\) | \(43^3\) | \(63^6\) | \(62^5\) | \(56^4\) | \(66^7\) | ||
500 | AEs | \(\beta \) | \(1.65881^6\) | \(1.78382^8\) | \(1.58341^3\) | \(1.64070^5 \) | \(1.48414^2\) | \(1.66351^7\) | \(0.93181^1\) | \(1.62609^4\) |
\(\lambda \) | \(0.73330^3\) | \(0.75017^4\) | \(0.93770^5\) | \(0.72597^2\) | \(1.00969^8 \) | \(0.95051^6\) | \(0.50345^1\) | \(0.96262^7\) | ||
\(\theta \) | \(0.59718^3\) | \( 0.59715^2\) | \(0.60078^8 \) | \(0.59746^4\) | \(0.59814^6 \) | \(0.59765^5\) | \(0.58425^1\) | \(0.59991^7\) | ||
MSEs | \(\alpha \) | \(2.02497^1 \) | \(2.75017^8\) | \(2.56480^5\) | \(2.73855^7 \) | \(2.60460^6\) | \(2.51114^4\) | \(2.43156^3\) | \(2.02909^2\) | |
\(\beta \) | \(0.12583^3 \) | \(0.00391^1\) | \(0.30285^7\) | \( 0.10440^2 \) | \(0.23034^4\) | \(0.28490^6\) | \(0.26919^5\) | \(1.05286^8\) | ||
\(\lambda \) | \(0.23375^3\) | \(0.00305^1\) | \(0.30349^5 \) | \(0.19211^2\) | \(0.31794^6\) | \(0.29551^4\) | \(0.87795^8\) | \(0.54197^7\) | ||
\(\theta \) | \(0.00067^3 \) | \(0.00046^1\) | \(0.00098^6 \) | \(0.00062^2\) | \(0.00070^4\) | \(0.00097^5\) | \(0.03896^8\) | \(0.00099^7\) | ||
ABs | \(\alpha \) | \(3.06142^7\) | \(0.00015^1\) | \(3.02884^5\) | \(3.05993^6\) | \(3.06250^8\) | \(2.89958^4\) | \(2.04936^3\) | \(1.05902^2\) | |
\(\beta \) | \(0.35472^3\) | \(0.06250^1\) | \(0.55032^7\) | \(0.32311^2\) | \(0.47994^4\) | \(0.53376^6\) | \(0.51884^5\) | \(1.02609^8\) | ||
\(\lambda \) | \(0.48348^3\) | \(0.05525^1 \) | \(0.55090^5\) | \(0.43830^2\) | \(0.56386^6\) | \(0.54361^4\) | \(0.93699^8\) | \(0.73618^7\) | ||
\(\theta \) | \(0.02595^3\) | \(0.02135^1\) | \(0.03131^6\) | \(0.02487^2 \) | \(0.02639^4\) | \(0.03122^5\) | \(0.19739^8\) | \(0.03149^7\) | ||
\(\sum Ranks\) | \(\alpha \) | \(1.74969^7\) | \(0.01210^1\) | \(1.74036^5 \) | \(1.74926^6 \) | \(1.75000^8 \) | \(1.70281^4\) | \(1.43156^3\) | \(1.02909^2\) | |
| Â | \(45^3\) | \(30^1\) | \(67^7\) | \(42^2\) | \(66^6\) | \(60^5\) | \(54^4\) | \(68^8\) |
