Table 6 Simulation results of eight different estimators for \(\beta =2.72, \lambda =0.75, \theta =0.6, \alpha =1.57\).
From: Properties, estimation, and applications of the extended log-logistic distribution
n | Measures | Parameters | MLEs | MPSEs | OLSEs | WLSEs | ADEs | CVMEs | PCEs | RADEs |
|---|---|---|---|---|---|---|---|---|---|---|
20 | AEs | \(\beta \) | \(3.45758^8\) | \( 2.64609^5\) | \(2.58345^4\) | \(2.18844^2\) | \(2.31879^3\) | \(2.77140^6\) | \(1.87986^1\) | \(3.20587^7\) |
\(\lambda \) | \(1.30289^8\) | \(0.82749^4\) | \(0.80068^3\) | \(1.11445^7 \) | \( 0.93248^5\) | \(0.67959^2\) | \(0.39421^1\) | \(1.07693^6\) | ||
\(\theta \) | \(0.65188^7\) | \(0.58108^1\) | \(0.60997^4\) | \(0.60183^3\) | \(0.64205^6\) | \(0.66307^8\) | \(0.60036^2\) | \(0.63728^5\) | ||
MSEs | \(\alpha \) | \(1.00002^1\) | \(1.54537^8\) | \(1.51483^7\) | \(1.00441^3\) | \(1.00534^4\) | \(1.51385^6\) | \(1.44357^5\) | \(1.00017^2 \) | |
\(\beta \) | \(2.95806^2\) | \(2.95840^5\) | \(2.95840^5\) | \(2.95840^5\) | \(2.95840^5\) | \(2.95840^5\) | \(2.45478^1\) | \(6.08272^8\) | ||
\(\lambda \) | \(0.50049^6\) | \(0.45470^1\) | \(0.49217^3 \) | \(0.50092^7\) | \(0.47753^2\) | \(0.49832^5\) | \(0.49328^4 \) | \(0.55464^8\) | ||
\(\theta \) | \(0.00844^1\) | \(0.00863^2\) | \(0.01087^4\) | \(0.01304^5\) | \(0.01051^3\) | \(0.01402^6\) | \(0.06779^8\) | \(0.01780^7\) | ||
ABs | \(\alpha \) | \(0.32490^{5.5} \) | \(0.11566^1 \) | \(0.32254^3\) | \(0.32489^4 \) | \(0.32490^{5.5}\) | \(0.31812^2\) | \(0.49758^7\) | \(1.21913^8\) | |
\(\beta \) | \(1.71990^2\) | \( 1.72000^5\) | \(1.72000^5\) | \(1.72000^5\) | \(1.72000^5\) | \(1.72000^5\) | \(1.56677^1\) | \(2.46632^8\) | ||
\(\lambda \) | \(0.70745^6\) | \(0.67432^1\) | \(0.70155^3\) | \( 0.70776^7\) | \(0.69103^2\) | \(0.70591^5\) | \(0.70234^4 \) | \( 0.74474^8\) | ||
\(\theta \) | \(0.09186^1\) | \(0.09291^2\) | \(0.10427^4\) | \(0.11420^5\) | \(0.10253^3\) | \(0.11841^6\) | \(0.26036^8\) | \(0.13343^7\) | ||
\(\sum Ranks\) | \(\alpha \) | \(0.57000^{5.5}\) | \( 0.34009^1 \) | \(0.56793^3\) | \(0.56999^4\) | \(0.57000^{5.5}\) | \(0.56402^2\) | \(0.70539^7\) | \(1.10414^8\) | |
| Â | \(53^5\) | \(36^1\) | \(48^2\) | \(57^6\) | \(49^{3.5}\) | \(58^7\) | \(49^{3.5}\) | \(82^8\) | ||
50 | AEs | \(\beta \) | \(3.00939^8\) | \(2.65220^6\) | \(2.70223^7 \) | \(2.05584^2\) | \(2.20039^3\) | \(2.63174^4\) | \(1.93128^1\) | \(2.62357^4\) |
\(\lambda \) | \(1.27821^8\) | \(0.76490^4\) | \(0.85529^5\) | \(1.11997^6\) | \(1.14012^7\) | \(0.72029^2\) | \(0.44413^1\) | \(0.74069^3\) | ||
\(\theta \) | \(0.61677^6\) | \(0.58547^1\) | \(0.59983^3\) | \(0.60354^4\) | \(0.61520^5\) | \( 0.62753^8\) | \(0.59467^2\) | \(0.61953^7\) | ||
MSEs | \(\alpha \) | \(1.00028^1\) | \(1.57005^8\) | \(1.37710^6\) | \(1.01228^3\) | \(1.00829^3\) | \(1.25882^5\) | \(1.38215^7\) | \(1.00123^2\) | |
\(\beta \) | \(2.24055^2\) | \(1.83301^1\) | \( 2.95840^6\) | \(2.95840^6\) | \(2.77128^4\) | \(2.95840^6\) | \(2.3166^3\) | \(4.22151^8\) | ||
\(\lambda \) | \(0.46800^4\) | \(0.36632^1\) | \(0.45465^2\) | \(0.47170^6 \) | \(0.47042^5\) | \(0.46476^3\) | \(0.49500^7\) | \(0.51350^8 \) | ||
\(\theta \) | \(0.00395^1 \) | \(0.00505^3\) | \(0.00567^4 \) | \(0.00604^5\) | \(0.00479^2\) | \(0.00631^6\) | \(0.05234^8 \) | \(0.00741^7 \) | ||
ABs | \(\alpha \) | \(0.32489^{4.5}\) | \(0.04264^1 \) | \(0.32319^2\) | \(0.32489^{4.5} \) | \(0.32490^6\) | \(0.32391^3\) | \(0.44995^8\) | \(0.34286^7\) | |
\(\beta \) | \(1.49685^2\) | \(1.35389^1\) | \( 1.72000^6\) | \(1.72000^6\) | \(1.66472^4\) | \(1.72000^6\) | \(1.52204^3\) | \(2.05463^8\) | ||
\(\lambda \) | \(0.68411^4\) | \(0.60524^1\) | \( 0.67427^2 \) | \(0.68681^6\) | \(0.68587^5\) | \(0.68173^3\) | \(0.70356^7\) | \(0.71659^8\) | ||
\(\theta \) | \(0.06286^1\) | \(0.07107^3 \) | \(0.07527^4 \) | \(0.07771^5\) | \(0.06920^2\) | \(0.07947^6\) | \(0.22878^8\) | \(0.08608^7\) | ||
\(\sum Ranks\) | \(\alpha \) | \(0.56999^{4.5}\) | \(0.20649^1\) | \(0.56850^2\) | \(0.56999^{4.5}\) | \(0.57000^6 \) | \(0.56913^3\) | \(0.67078 ^8\) | \(0.58554^7\) | |
| Â | \(46^2\) | \(31^1\) | \(49^3\) | \(58^6\) | \(52^4\) | \(55^5\) | \(63^7\) | \( 76^8\) | ||
100 | AEs | \(\beta \) | \(2.77687^7\) | \(2.65992^6\) | \(2.63080^5\) | \(2.25057^3\) | \(2.22733^2\) | \( 2.61156^4\) | \(1.86254^1\) | \( 3.02124^8\) |
\(\lambda \) | \(1.18418^7\) | \( 0.74730^2\) | \(0.82262^4\) | \(1.18670^8\) | \(1.11692^6\) | \(0.79190^3\) | \(0.51700^1\) | \( 1.07479^5\) | ||
\(\theta \) | \(0.60337^5\) | \(0.59248^1\) | \(0.60527^6\) | \(0.59837^2\) | \(0.60713^7\) | \(0.61335^8\) | \(0.59291^3\) | \(0.59852^4\) | ||
MSEs | \(\alpha \) | \(1.00141^1\) | \(1.57365^8\) | \(1.42037^7\) | \(1.01434^3\) | \(1.03549^4\) | \(1.33082^6\) | \(1.31455^5\) | \(1.00253^2 \) | |
\(\beta \) | \(1.55235^2\) | \(0.91658^1\) | \(2.38591^5 \) | \(2.54240^7\) | \(2.13491^4\) | \(2.49961^6\) | \(1.98836^3\) | \(3.24822^8\) | ||
\(\lambda \) | \(0.41405^3 \) | \(0.20532^1\) | \(0.40796^2 \) | \(0.44704^6 \) | \(0.42599^5\) | \(0.41499^4\) | \(0.50163^8\) | \(0.48085^7\) | ||
\(\theta \) | \(0.00211^1\) | \(0.00257^3\) | \(0.00374^6\) | \(0.00343^4\) | \(0.00256^2\) | \(0.00353^5\) | \(0.04306^8\) | \(0.00422^7\) | ||
300 | ABs | \(\alpha \) | \(0.32487^4\) | \(0.01927^1\) | \(0.32186^2\) | \(0.32489^5\) | \(0.32490^6\) | \(0.32317^3\) | \(0.42601^8\) | \(0.33143^7\) |
\(\beta \) | \(1.24594^2\) | \(0.95738^1\) | \(1.54464^5\) | \(1.59449^7\) | \(1.46113^4\) | \(1.58102^6\) | \(1.41009^3\) | \(1.80228^8\) | ||
\(\lambda \) | \(0.64347^3\) | \( 0.45312^1\) | \(0.63872^2\) | \(0.66861^6\) | \(0.65268^5 \) | \(0.64420^4\) | \(0.70826^8\) | \(0.69343^7\) | ||
\(\theta \) | \(0.04594^1\) | \( 0.05072^3\) | \( 0.06114^6\) | \(0.05853^4\) | \(0.05063^2 \) | \(0.05940^5\) | \(0.20750^8\) | \(0.06497^7\) | ||
\(\sum Ranks\) | \(\alpha \) | \(0.56997^4\) | \(0.13880^1\) | \(0.56733^2\) | \(0.56999^5\) | \(0.57000^6\) | \( 0.56848^3\) | \(0.65270^8\) | \( 0.57570^7\) | |
| Â | \(40^2\) | \(29^1\) | \(52^3\) | \(60^6\) | \(53^4\) | \(57^5\) | \(64^7\) | \(77^8\) | ||
AEs | \(\beta \) | \(2.59788^5\) | \(2.70275^8\) | \( 2.62434^6\) | \(2.34032^2\) | \(2.43272^3\) | \(2.57995^4\) | \(1.59694^1\) | \(2.63740^7\) | |
\(\lambda \) | \(1.06740^6\) | \(0.74888^2\) | \(0.94853^4\) | \(1.12280^7\) | \(1.15166^8\) | \(0.92458^3\) | \(0.46136^1\) | \(0.98446^5\) | ||
\(\theta \) | \(0.59757^2\) | \(0.59665^1\) | \( 0.59937^5\) | \(0.59878^4\) | \(0.59865^3\) | \(0.60363^8\) | \(0.59979^7\) | \(0.59963^6\) | ||
MSEs | \(\alpha \) | \(1.05055^2 \) | \(1.57063^8\) | \(1.40350^7 \) | \(1.03966^1\) | \(1.15125^4\) | \(1.35775^6\) | \(1.15493^5\) | \(1.09217^3\) | |
\(\beta \) | \(0.79194^2\) | \(0.00288^1\) | \(1.22647^5 \) | \(1.08088^3 \) | \(1.20488^4\) | \(1.24769^6\) | \(2.88604^8\) | \(1.53777^7\) | ||
\(\lambda \) | \(0.26824^2\) | \(0.00611^1\) | \(0.28852^4\) | \(0.32398^5\) | \(0.32582^6 \) | \(0.28216^3\) | \(0.52870^8\) | \(0.34003^7\) | ||
\(\theta \) | \(0.00075^2\) | \(0.00067^1\) | \(0.00140^5\) | \(0.00099^4\) | \(0.00095^3\) | \(0.00154^6\) | \(0.04713^8\) | \(0.00163^7 \) | ||
ABs | \(\alpha \) | \(0.32450^5\) | \(0.00099^1\) | \(0.28330^2\) | \(0.32484^6\) | \(0.32490^7\) | \(0.30812^3\) | \(0.64507^8 \) | \(0.32444^4\) | |
\(\beta \) | \(0.88991^2\) | \(0.05367^1\) | \(1.10746^5\) | \(1.03965^3\) | \( 1.09767^4\) | \(1.11700^6\) | \(1.69883^8\) | \(1.24006^7\) | ||
\(\lambda \) | \(0.51791^2\) | \( 0.07813^1\) | \( 0.53714^4\) | \(0.56920^5\) | \(0.57081^6\) | \(0.53119^3\) | \(0.72712^8\) | \(0.58312^7\) | ||
\(\theta \) | \(0.02740^2\) | \(0.02591^1\) | \(0.03742^5\) | \(0.03146^4\) | \(0.03085^3\) | \(0.03923^6\) | \(0.21711^8\) | \(0.04034^7\) | ||
\(\sum Ranks\) | \(\alpha \) | \(0.56965^5\) | \(0.03148^1\) | \( 0.53226^2\) | \(0.56995^6 \) | \(0.57000^7\) | \(0.55509^3\) | \(0.80316^8 \) | \(0.56960^4\) | |
| Â | \(37^2\) | \(27^1\) | \(54^5\) | \(50^4\) | \(58^7\) | \(57^6\) | \(46^3\) | \(71^8\) | ||
500 | AEs | \(\beta \) | \(2.57388^5\) | \( 2.70815^8\) | \(2.59432^7\) | \(2.38078^3\) | \(2.31559^2 \) | \(2.54036^4\) | \(1.40882^1\) | \(2.58689^6\) |
\(\lambda \) | \(0.99487^6 \) | \(0.75231^2\) | \(0.92833^5\) | \(1.08109^7 \) | \(1.09365^8\) | \(0.86885^3\) | \(0.52734^1\) | \(0.91153^4\) | ||
\(\theta \) | \(0.59640^2\) | \(0.59679^4\) | \( 0.59854^5\) | \(0.59678^3\) | \(0.59874^6\) | \(0.60350^8\) | \(0.56823^1 \) | \(0.60041^7\) | ||
MSEs | \(\alpha \) | \(1.14124^4\) | \(1.56922^8\) | \(1.40925^7\) | \(1.08962^1\) | \(1.23440^5\) | \(1.36721^6\) | \(1.09240^2\) | \(1.11867^3\) | |
\(\beta \) | \(0.50451^2\) | \(0.00131^1\) | \(0.91128^6\) | \(0.70638^3\) | \( 0.86603^5\) | \(0.83181^4\) | \(3.19292^8\) | \(1.07978^7\) | ||
\(\lambda \) | \(0.20959^2 \) | \(0.00269^1\) | \(0.23147^3 \) | \(0.27549^5 \) | \( 0.27975^6\) | \(0.23430^4\) | \(0.54323^8\) | \(0.28593^7\) | ||
\(\theta \) | \(0.00058^{2.5}\) | \(0.00033^1\) | \( 0.00098^6\) | \(0.00061^4\) | \(0.00058^{2.5}\) | \(0.00095^5\) | \(0.04585^8\) | \(0.00100^7\) | ||
ABs | \(\alpha \) | \(0.32370^5\) | \(0.00043^1\) | \(0.26808^3\) | \(0.32471^7\) | \(0.32470^6\) | \(0.26576^2\) | \(0.73211^8\) | \(0.32361^4\) | |
\(\beta \) | \(0.71029^2\) | \(0.03622^1\) | \(0.95461^6\) | \(0.84046^3\) | \(0.93061^5\) | \(0.91204^4\) | \(1.78687^8\) | \(1.03912^7\) | ||
\(\lambda \) | \(0.45781^2\) | \(0.05187^1\) | \(0.48111^3\) | \(0.52487^5\) | \(0.52891^6\) | \(0.48405^4\) | \(0.73704^8 \) | \(0.53473^7\) | ||
\(\theta \) | \(0.02407^{2.5}\) | \(0.01827^1\) | \(0.03131^6\) | \(0.02480^4 \) | \(0.02414^{2.5}\) | \(0.03087^5\) | \(0.21414^8\) | \(0.03159^7\) | ||
\(\sum Ranks\) | \(\alpha \) | \(0.56895^5\) | \(0.02075^1\) | \(0.51776^3 \) | \(0.56983^7\) | \(0.56982^6\) | \(0.51552^2\) | \(0.85563^8 \) | \(0.56887^4\) | |
| Â | \(40^2\) | \(30^1\) | \(60^{5.5}\) | \(52^4\) | \(60^{5.5}\) | \(51^3\) | \(69^7\) | \(70^8\) |
