Table 3 Simulation results of eight different estimators for \(\beta =1.8, \lambda =0.75, \theta =0.6, \alpha =1.57\).
From: Properties, estimation, and applications of the extended log-logistic distribution
n | Measures | Parameters | MLEs | MPSEs | OLSEs | WLSEs | ADEs | CVMEs | PCEs | RADEs |
|---|---|---|---|---|---|---|---|---|---|---|
20 | AEs | \(\beta \) | \(2.09746^7\) | \(1.78359^6\) | \(1.71928^4\) | \(1.36221^2\) | \(1.42514^3\) | \(1.77951^5\) | \(1.21904^1\) | \(2.20069^8 \) |
\(\lambda \) | \(1.30780^8\) | \(0.83407^4\) | \(0.58000^2\) | \(1.27839^7\) | \(1.24909^5 \) | \(0.79788^3\) | \(0.44958^1\) | \(1.25369^6\) | ||
\(\theta \) | \( 0.64299^7\) | \(0.56072^1\) | \(1.58000^8\) | \(0.57783^2\) | \(0.61660^5\) | \(0.63314^6\) | \(0.59385^3\) | \(0.61482^4\) | ||
MSEs | \(\alpha \) | \(1.00004^1\) | \(1.56748^8\) | \(1.00017^2\) | \(1.01227^5\) | \(1.00537^4\) | \(1.53722^7\) | \(1.49703^6\) | \(1.00022^3\) | |
\(\beta \) | \(0.63984^2\) | \(0.64000^5\) | \(0.64000^5\) | \(0.64000^5\) | \(0.64000^5\) | \(0.64000^5\) | \(0.38321^1\) | \(2.56222^8\) | ||
\(\lambda \) | \(0.45529^4\) | \(0.38809^1\) | \(0.40314^3\) | \(0.45824^6\) | \(0.45615^5\) | \(0.40266^2\) | \(0.87720^7\) | \(0.96407^8\) | ||
\(\theta \) | \(0.00835^2\) | \(0.01008^3\) | \(0.01043^4\) | \(0.01100^5\) | \(0.00806^1\) | \(0.01193^6\) | \(0.06674^8\) | \(0.02247^7\) | ||
ABs | \(\alpha \) | \(0.32490^{5.5}\) | \(0.07911^1\) | \(0.15864^3\) | \(0.32489^4\) | \(0.32490^{5.5} \) | \(0.15409^2\) | \(0.51727^7\) | \(0.80459^8\) | |
\(\beta \) | \(0.79990^2 \) | \(0.80000^5\) | \(0.80000^5\) | \(0.80000^5\) | \(0.80000^5\) | \(0.80000^5\) | \(0.61904^1\) | \(1.60069^8\) | ||
\(\lambda \) | \(0.67475^4\) | \(0.62297^1\) | \(0.63494^3 \) | \(0.67694^6\) | \(0.67539^5\) | \(0.63455^2\) | \(0.93659^7\) | \(0.98187^8\) | ||
\(\theta \) | \(0.09136^2\) | \(0.10038^3\) | \(0.10215^4\) | \(0.10490^5\) | \(0.08976^1\) | \(0.10924^6\) | \(0.25834^8\) | \(0.14991^7\) | ||
\(\sum Ranks\) | \(\alpha \) | \(0.57000^{5.5}\) | \(0.28127^1\) | \(0.39830^3\) | \(0.56999^4\) | \(0.57000^{5.5}\) | \(0.39254^2\) | \(0.71921^7\) | \(0.89699^8\) | |
| Â | \(50^{3.5}\) | \(39^1\) | \(46^2\) | \(56^6\) | \(50^{3.5}\) | \(51^5\) | \(61^7\) | \(83^8\) | ||
50 | AEs | \(\beta \) | \(1.80585^7\) | \(1.73494^4\) | \(1.80285^6\) | \(1.49615^3\) | \(1.44570^2\) | \(1.78379^5\) | \(1.14991^1\) | \(1.87590^8\) |
\(\lambda \) | \(1.31902^8\) | \(0.74899^2 \) | \(0.95478^4 \) | \(1.10741^5\) | \(1.26827^7 \) | \(0.88721^3\) | \(0.40490^1\) | \(1.09852^6\) | ||
\(\theta \) | \(0.61093^6\) | \(0.58263^1\) | \(0.59294^3 \) | \(0.59671^4\) | \(0.61245^8\) | \(0.61239^7\) | \(0.58768^2\) | \(0.60750^5\) | ||
MSEs | \(\alpha \) | \(1.00066^1\) | \(1.57887^8\) | \(1.36249^6\) | \(1.02458^3\) | \(1.04555^4\) | \(1.29198^5\) | \( 1.41052^7\) | \(1.00092^2\) | |
\(\beta \) | \(0.63984^3\) | \(0.55165^2\) | \(0.64000^{5.5}\) | \(0.64000^{5.5}\) | \(0.64000^{5.5}\) | \(0.64000^{5.5}\) | \(0.34101^1\) | \(1.62809^8\) | ||
\(\lambda \) | \(0.41582^5\) | \(0.28695^1\) | \(0.37292^2\) | \( 0.39218^4\) | \(0.41843^6\) | \(0.37641^3\) | \(0.88101^8\) | \(0.86812^7\) | ||
\(\theta \) | \(0.00383^1 \) | \(0.00466^3\) | \(0.00596^5\) | \(0.00570^4\) | \( 0.00430^2 \) | \(0.00612^6\) | \(0.05837^8\) | \(0.00919^7\) | ||
ABs | \(\alpha \) | \(0.32490^{5.5}\) | \(0.04873^1\) | \(0.30435^2\) | \( 0.32489^4 \) | \(0.32490^{5.5}\) | \(0.31433^3\) | \(0.41104^7\) | \(0.44785^8\) | |
\(\beta \) | \(0.79990^3\) | \(0.74273^2\) | \(0.80000^{5.5}\) | \(0.80000^{5.5}\) | \(0.80000^{5.5} \) | \(0.80000^{5.5}\) | \(0.58396^1\) | \(1.2759^8\) | ||
\(\lambda \) | \(0.64484^5\) | \(0.53568^1\) | \(0.61067^2\) | \(0.62624^4\) | \(0.64686^6\) | \(0.61352^3\) | \(0.93862^8\) | \(0.93173^7\) | ||
\(\theta \) | \(0.06188^1\) | \(0.06828^3\) | \(0.07723^5\) | \(0.07553^4\) | \(0.06557^2\) | \(0.07821^6\) | \(0.24161^8\) | \(0.09585^7\) | ||
\(\sum Ranks\) | \(\alpha \) | \(0.57000^{5.5}\) | \(0.22074^1\) | \(0.55168^2\) | \(0.56999^4\) | \(0.57000^{5.5}\) | \(0.56065^3\) | \(0.64112^7\) | \(0.66922^8\) | |
| Â | \(51^4\) | \(29^1 \) | \(48^2\) | \(50^3\) | \(59^{6.5}\) | \(55^5 \) | \(59^{6.5}\) | \(81^8 \) | ||
100 | AEs | \(\beta \) | \(1.77425^8\) | \(1.75071^6\) | \( 1.65464^4\) | \(1.55109^3\) | \(1.42874^2\) | \(1.76222^7\) | \(1.11177^1\) | \(1.69885^5\) |
\(\lambda \) | \(1.25980^7\) | \(0.73063^2\) | \(0.98402^3\) | \(1.17632^6\) | \(1.32867^8\) | \(0.99105^4\) | \(0.50348^1\) | \(1.06644^5\) | ||
\(\theta \) | \(0.60018^5\) | \(0.59366^3\) | \(0.59909^4\) | \(0.59292^2\) | \(0.60205^8\) | \(0.60143^6\) | \(0.58040^1\) | \(0.60156^7\) | ||
MSEs | \(\alpha \) | \(1.00199^1\) | \(1.57961^8\) | \(1.28893^6\) | \(1.07731^3\) | \(1.12086^4\) | \(1.21141^5\) | \(1.36716^7\) | \(1.00556^2\) | |
\(\beta \) | \(0.50490^3\) | \(0.28701^1\) | \(0.64000^{5.5}\) | \(0.64000^{5.5}\) | \(0.64000^{5.5}\) | \(0.64000^{5.5}\) | \(0.32576^2\) | \(1.20747^8\) | ||
\(\lambda \) | \(0.38041^5\) | \(0.19940^1\) | \(0.34738^2\) | \(0.37289^4\) | \(0.40140^6\) | \(0.35386^3 \) | \(0.89047^8 \) | \(0.72337^7\) | ||
\(\theta \) | \(0.00222^1\) | \(0.00250^2\) | \(0.00382^5\) | \(0.00315^4 \) | \( 0.00258^3\) | \(0.00420^6\) | \(0.04769^8\) | \(0.00450^7\) | ||
ABs | \(\alpha \) | \(0.32489^6\) | \(0.02916^1\) | \(0.32076^4\) | \(0.32481^5\) | \(0.32490^7\) | \(0.32064^3\) | \(0.38096^8\) | \(0.25413^2\) | |
\(\beta \) | \(0.71057^3 \) | \(0.53574^1\) | \(0.80000^{5.5}\) | \(0.80000^{5.5}\) | \(0.80000^{5.5}\) | \(0.80000^{5.5}\) | \( 0.57075^2\) | \(1.09885^8\) | ||
\(\lambda \) | \(0.61677^5\) | \(0.44654^1\) | \(0.58939^2 \) | \(0.61065^4\) | \(0.63356^6\) | \(0.59486^3\) | \(0.94364^8\) | \(0.85051^7\) | ||
\(\theta \) | \(0.04708^1\) | \(0.05001^2\) | \(0.06180^5\) | \(0.05615^4\) | \(0.05084^3\) | \( 0.06481^6\) | \(0.21837^8 \) | \(0.06709^7\) | ||
\(\sum Ranks\) | \(\alpha \) | \(0.56999^6\) | \(0.17078^1\) | \(0.56636^4\) | \(0.56992^5\) | \(0.57000^7\) | \(0.56625^3\) | \(0.61722^8 \) | \(0.50411^2\) | |
| Â | \(51^{3.5}\) | \(29^1\) | \(50^2\) | \(51^{3.5}\) | \(65^7\) | \(57^5\) | \(62^6\) | \(67^8\) | ||
| Â | AEs | \(\beta \) | \(1.65761^5\) | \(1.77057^8\) | \(1.66057^6\) | \(1.61653^3\) | \(1.50661^2\) | \(1.67375^7\) | \(1.00234^1\) | \(1.61819^4\) |
| Â | \(\lambda \) | \(1.11724^6\) | \(0.74471^2\) | \(1.02341^5 \) | \(0.93275^3\) | \(1.13634^7 \) | \(1.02224^4\) | \(0.56169^1\) | \(1.17334^8\) | |
| Â | \(\theta \) | \(0.59272^2\) | \(0.59777^5\) | \(0.59516^3\) | \(0.59626^4\) | \(0.59825^6\) | \(0.60010^8\) | \(0.59249^1 \) | \(0.59969^7\) | |
| Â | MSEs | \(\alpha \) | \(1.06505^1\) | \(1.57213^8\) | \(1.31336^7 \) | \(1.29559^6\) | \(1.20721^3 \) | \(1.27060^5\) | \(1.26796^4\) | \(1.10575^2\) |
| Â | \(\beta \) | \(0.20521^2\) | \(0.08808^1\) | \(0.42990^7\) | \(0.22794^3\) | \(0.35183^5 \) | \(0.41926^6\) | \(0.30386^4\) | \(0.52084^8\) | |
| Â | \(\lambda \) | \(0.26541^3\) | \(0.06368^1\) | \(0.27543^5\) | \(0.22326^2\) | \(0.29660^6\) | \(0.26658^4\) | \(0.91692^7 \) | \(1.03671^8\) | |
300 | \(\theta \) | \(0.00104^3\) | \(0.00092^1\) | \(0.00167^7\) | \(0.00101^2\) | \(0.00109^4\) | \(0.00166^6\) | \( 0.04669^8 \) | \(0.00154^5\) | |
ABs | \(\alpha \) | \(0.32476^6\) | \(0.00855^1\) | \(0.32249^4\) | \(0.32445^5 \) | \(0.32490^7\) | \(0.32012^3\) | \(0.39760^8\) | \(0.08507^2\) | |
\(\beta \) | \(0.45300^2\) | \(0.29678^1\) | \(0.65567^7\) | \(0.47743^3\) | \(0.59316^5\) | \(0.64750^6\) | \( 0.55124^4\) | \(1.01819^8\) | ||
\(\lambda \) | \(0.51518^3\) | \(0.25235^1\) | \(0.52481^5\) | \(0.47250^2\) | \(0.54461^6\) | \(0.51632^4\) | \( 0.65756^7 \) | \(0.72169^8\) | ||
\(\theta \) | \(0.03227^3\) | \(0.03038^1\) | \(0.04083^7\) | \(0.03183^2\) | \(0.03299^4\) | \(0.04075^6\) | \(0.21607^8\) | \(0.03927^5\) | ||
\(\sum Ranks\) | \(\alpha \) | \(0.56988^6\) | \(0.09248^1\) | \(0.56788^4\) | \(0.56961^5 \) | \(0.57000^7\) | \(0.56579^3 \) | \(0.63055^8 \) | \(0.29167^2\) | |
| Â | \(42^3\) | \(31^1\) | \(67^{7.5}\) | \(40^2\) | \(62^{5.5}\) | \(62^{5.5}\) | \(61^4\) | \(67^{7.5}\) | ||
500 | AEs | \(\beta \) | \(1.67086^7\) | \(1.79009^8\) | \(1.62632^4\) | \(1.64936^6\) | \(1.55506^2\) | \(1.61795^3\) | \(0.84703^1\) | \(1.63722^5\) |
\(\lambda \) | \(0.90824^4\) | \(0.74903^2\) | \(1.04692^7\) | \(0.85683^3\) | \(1.15103^8 \) | \(0.98194^5\) | \(0.49168^1\) | \(1.03080^6\) | ||
\(\theta \) | \(0.59329^2\) | \(0.59837^7\) | \(0.59707^5\) | \(0.59428^4\) | \(0.59376^3\) | \(0.60036^8\) | \(0.58603^1\) | \(0.59827^6\) | ||
MSEs | \(\alpha \) | \(1.12894^2\) | \(1.57046^8\) | \(1.35327^6\) | \(1.53839^7\) | \(1.20849^3\) | \(1.31855^5\) | \(1.06943^1\) | \(1.16244^4\) | |
\(\beta \) | \(0.11403^2\) | \(0.00366^1\) | \(0.31584^7\) | \(0.12041^3\) | \(0.23957^4\) | \(0.29410^5\) | \(0.31004^6\) | \(1.07583^8 \) | ||
\(\lambda \) | \(0.17897^3\) | \(0.00308^1\) | \(0.23841^5\) | \(0.15116^2\) | \(0.26813^6\) | \(0.22464^4\) | \(0.92226^8\) | \(0.40847^7\) | ||
\(\theta \) | \(0.00067^3\) | \(0.00049^1\) | \(0.00094^6\) | \(0.00061^2\) | \(0.00074^4\) | \(0.00099^7\) | \(0.04308^8\) | \(0.00092^5\) | ||
ABs | \(\alpha \) | \(0.32441^5\) | \(0.00048^1\) | \(0.32260^4\) | \(0.32447^6 \) | \(0.32490^7\) | \(0.32182^3\) | \(0.38011^8\) | \(0.07641^2\) | |
\(\beta \) | \(0.33768^2\) | \(0.06047^1\) | \(0.56200^7\) | \(0.34700^3\) | \(0.48946^4\) | \(0.54231^5\) | \( 0.55681^6\) | \(1.03722^8\) | ||
\(\lambda \) | \(0.42305^3\) | \(0.05547^1\) | \(0.48828^5\) | \(0.38879^2\) | \(0.51781^6\) | \(0.47396^4\) | \(0.96034^8 \) | \(0.63912^7\) | ||
\(\theta \) | \(0.02589^3\) | \(0.02207^1\) | \(0.03068^6\) | \(0.02478^2\) | \(0.02712^4\) | \(0.03148^7\) | \(0.20755^8 \) | \(0.03036^5\) | ||
\(\sum Ranks\) | \(\alpha \) | \(0.56957^5 \) | \(0.02198^1\) | \(0.56798^4\) | \(0.56962^6\) | \( 0.57000^7\) | \(0.56729^3\) | \(0.61653^8\) | \(0.27642^2\) | |
| Â | \(41^2\) | \(33^1\) | \(66^8\) | \(46^3\) | \(58^4\) | \(59^5\) | \(64^6\) | \(65^7\) |
