A New Extended Model with Bathtub-Shaped Failure Rate: Properties, Inference, Simulation, and Applications
Abstract
:1. Introduction
2. Materials and Methods
3. Mathematical Properties
3.1. Shape
3.2. Linear Representation
3.3. Reliability Characteristics of the E-NPF Model
3.4. Limiting Behavior
f_(E-NPF) (x)~θζ(1+η),
S_(E-NPF) (x)~1,
h_(E-NPF) (x)~0.
f_(E-NPF) (x)~0,
S_(E-NPF) (x)~0,
h_(E-NPF) (x)~Indeterminate.
3.5. Quantile Function, Skewness, and Kurtosis
3.6. Moments and Associated Measures
3.7. Incomplete Moments
3.8. Entropy
3.9. Stress–Strength Reliability
3.10. Stochastic Ordering
- (i)
- stochastic order (st) (denoted by X1 X2) if Q1(X) Q2(X) x;
- (ii)
- reversed hazard rate (rhr) order (denoted by X1 X2) if Q1(X)/Q2(X) is decreasing for x 0;
- (iii)
- likelihood ratio (lkr) order (denoted by X1 X2) if Q1(x)/Q2(x) is decreasing for x 0;
- (iv)
- hazard rate (hr) order (denoted by X1 X2) if Q1(X)/Q2(X) is decreasing for x 0.
3.11. Order Statistics
4. Estimation of the E-NPF Distribution
4.1. Estimation under Complete Samples
4.1.1. Maximum Likelihood Estimation under Complete Samples
4.1.2. Ordinary and Weighted Least-Squares Estimators
4.1.3. Maximum Product of Spacing
4.1.4. The Cramér–von Mises Estimators
4.1.5. The Anderson–Darling and Right-Tail Anderson–Darling Estimators
4.1.6. Method of Percentile Estimation
4.2. Estimation under Type II Censored Samples
5. Simulation Study
5.1. Simulation Results under Complete Samples
- Step 1:
- A random sample of sizes , and are generated from the QF in Equation (14).
- Step 2:
- The required results are obtained based on 36 combinations of the parameters and
- Step 3:
- Each sample is replicated times.
- Step 4:
5.2. Simulation Results under Type II Censored Samples
6. Results and Discussion
7. Conclusions
Supplementary Materials
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
- Lehmann, E.L. The power of rank tests. Ann. Math. Stat. 1953, 24, 23–43. [Google Scholar] [CrossRef]
- Gupta, R.C.; Gupta, P.; Gupta, R.D. Modeling failure time data by Lehmann alternatives. Commun. Stat.-Theory Methods 1998, 27, 887–904. [Google Scholar] [CrossRef]
- Cordeiro, G.M.; De Castro, M. A new family of generalized distributions. J. Stat. Comput. Simul. 2011, 81, 883–893. [Google Scholar] [CrossRef]
- Dallas, A.C. Characterization of Pareto and power function distribution. Ann. Math. Stat. 1976, 28, 491–497. [Google Scholar] [CrossRef]
- Meniconi, M.; Barry, D. The power function distribution: A useful and simple distribution to assess electrical component reliability. Microelectron Reliab. 1996, 36, 1207–1212. [Google Scholar] [CrossRef]
- Chang, S.K. Characterizations of the power function distribution by the independence of record values. J. Chungcheong Math. Soc. 2007, 20, 139–146. [Google Scholar]
- Tavangar, M. Power function distribution characterized by dual generalized order statistics. J. Iran. Stat. Soc. 2011, 10, 13–27. [Google Scholar]
- Ahsanullah, M.; Shakil, M.; Golam-Kibria, B.M.G. A characterization of the power function distribution based on lower records. ProbStat Forum. 2013, 6, 68–72. [Google Scholar]
- Zaka, A.; Akhter, A.S.; Farooqi, N. Methods for estimating the parameters of the power function distribution. J. Stat. 2014, 21, 90–102. [Google Scholar] [CrossRef]
- Shahzad, M.N.; Asghar, Z.; Shehzad, F.; Shahzadi, M. Parameter estimation of power function distribution with TL-moments. Rev. Colomb. Estad. 2015, 38, 321–334. [Google Scholar] [CrossRef]
- Cordeiro, G.M.; Brito, R.D.S. The beta power distribution. Braz. J. Probab. 2012, 26, 88–112. [Google Scholar]
- Tahir, M.H.; Alizadeh, M.; Mansoor, M.; Cordeiro, G.M.; Zubair, M. The Weibull power function distribution with applications. Hacettepe J. Math. Stat. 2014, 45, 245–265. [Google Scholar] [CrossRef]
- Tahir, M.H.; Cordeiro, G.M.; Alizadeh, M.; Mansoor, M.; Zubair, M.; Hamedani, G.G. The odd generalized exponential family of distributions with applications. J. Stat. Distrib. Appl. 2015, 2, 1–28. [Google Scholar] [CrossRef] [Green Version]
- Ahsan-ul-Haq, M.; Butt, N.S.; Usman, R.M.; Fattah, A.A. Transmuted power function distribution. Gazi Univ. J. Sci. 2016, 29, 177–185. [Google Scholar]
- Al-Mutairi, A.O. Transmuted weighted power function distribution. Pak. J. Statist. 2017, 33, 491–498. [Google Scholar]
- Okorie, I.E.; Akpanta, A.C.; Ohakwe, J. Marshall-Olkin extended power function distribution. Eur. J. Stat. Probab. 2017, 5, 16–29. [Google Scholar]
- Abdul-Moniem, I.B.; Diab, L.S. Generalized transmuted power function distribution. J. Stat. Appl. Probab. 2018, 7, 401–411. [Google Scholar] [CrossRef]
- Usman, R.M.; Ahsan-ul-Haq, M.; Bursa, N.; Özel, G. Exponentiated transmuted power function distribution: Theory & applications. Gazi Univ. J. Sci. 2018, 31, 660–675. [Google Scholar]
- Hassan, A.; Elshrpieny, E.; Mohamed, R. Odd generalized exponential power function distribution. Gazi Univ. J. Sci. 2019, 32, 351–370. [Google Scholar]
- Arshad, M.Z.; Iqbal, M.Z.; Ahmad, M. Exponentiated power function distribution: Properties and applications. JSTA 2020, 19, 297–313. [Google Scholar] [CrossRef]
- Arshad, M.Z.; Iqbal, M.Z.; Anees, A.; Ahmad, Z.; Balogun, O.S. A new bathtub shaped failure rate model: Properties, and applications to engineering sector. Pak. J. Statist. 2021, 37, 57–80. [Google Scholar]
- Sen, S.; Afify, A.Z.; Al-Mofleh, H.; Ahsanullah, M. The quasi xgamma-geometric distribution with application in medicine. Filomat 2019, 33, 5291–5330. [Google Scholar] [CrossRef] [Green Version]
- Ramos, P.L.; Louzada, F.; Ramos, E.; Dey, S. The Fréchet distribution: Estimation and application-An overview. J. Stat. Manag. Syst. 2020, 23, 549–578. [Google Scholar] [CrossRef] [Green Version]
- Afify, A.Z.; Mohamed, O.A. A new three-parameter exponential distribution with variable shapes for the hazard rate: Estimation and applications. Mathematics 2020, 8, 135. [Google Scholar] [CrossRef] [Green Version]
- Alsubie, A.; Akhter, Z.; Athar, H.; Alam, M.; Ahmad, A.E.B.A.; Cordeiro, G.M.; Afify, A.Z. On the omega distribution: Some properties and estimation. Mathematics 2021, 9, 656. [Google Scholar] [CrossRef]
- Iqbal, M.Z.; Arshad, M.Z.; Özel, G.; Balogun, O.S. A better approach to discuss medical science and engineering data with a modified Lehmann Type–II model. F1000Research 2021, 10, 823. [Google Scholar] [CrossRef]
- Pu, S.; Oluyede, B.O.; Qiu, Y.; Linder, D. A generalized class of exponentiated modified Weibull distribution with applications. Data Sci. J. 2016, 14, 585–614. [Google Scholar] [CrossRef]
- Oluyede, B.O.; Pu, S.; Makubate, B.; Qiu, Y. The gamma-Weibull-G family of distributions with applications. Austrian J. Stat. 2018, 47, 45–76. [Google Scholar] [CrossRef] [Green Version]
- Hyndman, R.J.; Fan, Y. Sample quantile in statistical packages. Am. Stat. 1996, 50, 361–365. [Google Scholar]
- Bowley, A.L. Elements of Statistics, 4th ed.; Charles Scribner: New York, NY, USA, 1920. [Google Scholar]
- Moors, J.J.A. A quantile alternative for kurtosis. J. R. Stat. Soc. Ser. D Stat. 1998, 37, 25–32. [Google Scholar] [CrossRef] [Green Version]
- Rényi, A. On measures of entropy and information. In Proceedings of the 4th Fourth Berkeley Symposium on Mathematical Statistics and Probability, Berkeley, CA, USA, 20–30 July 1960; University of California Press: Berkeley, CA, USA, 1961; pp. 547–561. [Google Scholar]
- Shaked, M.; Shanthikumar, J.G. Stochastic Orders and Their Applications; Academic Press: New York, NY, USA, 1994. [Google Scholar]
- R Core Team. R: A Language and Environment for Statistical Computing; R Foundation for Statistical Computing: Vienna, Austria, 2021; Available online: https://www.R-project.org/ (accessed on 18 May 2021).
- Frery, A.C.; Muller, H.J.; Yanasse, C.C.F.; Sant’Anna, S.J.S. A model for extremely heterogeneous clutter. IEEE Trans. Geosci. Remote Sens. 1997, 35, 648–659. [Google Scholar] [CrossRef]
- Murthy, D.N.P.; Xie, M.; Jiang, R. Weibull Models; John Wiley & Sons: Hoboken, NJ, USA, 2004. [Google Scholar]
- Rafael, P.; Bourguignon, M.; Rafael, C. Adequacy Model: Adequacy of Probabilistic Models and General Purpose Optimization. R Package Version 2.0.0. 2016. Available online: https://CRAN.R-project.org/package=AdequacyModel (accessed on 20 June 2021).
- Mood, A.M.; Graybill, F.A.; Boes, D.C. Introduction to the Theory of Statistics; McGraw-Hill: New York, NY, USA, 1974. [Google Scholar]
- Topp, C.W.; Leone, F.C. A family of J-shaped frequency functions. J. Am. Stat. 1955, 50, 209–219. [Google Scholar] [CrossRef]
- Kumaraswamy, P. Generalized probability density-function for double-bounded random-processes. J. Hydrol. 1980, 46, 79–88. [Google Scholar] [CrossRef]
- Saran, J.; Pandey, A. Estimation of parameters of power function distribution and its characterization by the kth record values. Statistica 2004, 14, 523–536. [Google Scholar]
- Abdul-Moniem, I.B. The Kumaraswamy power function distribution. J. Stat. Appl. Probab. 2017, 6, 81–90. [Google Scholar] [CrossRef]
- Muhammad, M. A new lifetime model with a bounded support. Asian J. Math. 2017, 7, 1–11. [Google Scholar] [CrossRef]
S-I | S-II | S-III | S-IV | S-V | |
0.2874 | 0.3528 | 0.0599 | 0.4500 | 0.3581 | |
0.1582 | 0.1794 | 0.0107 | 0.2543 | 0.1714 | |
0.1064 | 0.1101 | 0.0031 | 0.1645 | 0.0990 | |
0.0790 | 0.0753 | 0.0011 | 0.1162 | 0.0647 | |
0.0701 | 0.0342 | 0.0013 | 0.0010 | 0.0094 | |
0.0203 | 0.0621 | 8.3037 | 0.0694 | 0.3347 | |
0.5630 | 0.2697 | 2.7600 | −0.1506 | −0.1713 |
50 | 0.22949{2} | 0.25491{5} | 0.19893{1} | 0.25592{6} | 0.59344{8} | 0.23115{3} | 0.24367{4} | 0.31888{7} | ||
0.04667{3} | 0.05170{5} | 0.18196{7} | 0.06960{6} | 0.31282{8} | 0.04117{1} | 0.04246{2} | 0.04977{4} | |||
0.02636{3} | 0.02929{4} | 0.05176{7} | 0.03909{6} | 0.10345{8} | 0.02486{2} | 0.02304{1} | 0.03038{5} | |||
0.13184{3} | 0.16396{6} | 0.06754{1} | 0.15261{5} | 0.91318{8} | 0.12109{2} | 0.13419{4} | 0.35193{7} | |||
0.00677{3} | 0.00845{4} | 0.10140{7} | 0.01070{6} | 0.42939{8} | 0.00573{1} | 0.00587{2} | 0.00995{5} | |||
0.00223{3} | 0.00275{4} | 0.00450{7} | 0.00355{5.5} | 0.02340{8} | 0.00208{1} | 0.00213{2} | 0.00355{5.5} | |||
−0.45897{7} | −0.50983{4} | −0.39786{8} | −0.51184{3} | −1.18688{1} | −0.46229{6} | −0.48734{5} | −0.63776{2} | |||
0.09333{3} | 0.10340{5} | 0.36391{7} | 0.13921{6} | 0.62564{8} | 0.08234{1} | 0.08492{2} | 0.09953{4} | |||
0.06590{3} | 0.07323{4} | 0.12941{7} | 0.09772{6} | 0.25861{8} | 0.06214{2} | 0.05759{1} | 0.07596{5} | |||
30{3} | 41{4} | 52{7} | 49.5{6} | 65{8} | 19{1} | 23{2} | 44.5{5} | |||
80 | 0.17731{3} | 0.18881{5} | 0.15427{1} | 0.19846{6} | 0.43540{8} | 0.17521{2} | 0.18167{4} | 0.23895{7} | ||
0.03774{3} | 0.04176{4} | 0.12121{7} | 0.05495{6} | 0.18780{8} | 0.03414{2} | 0.03376{1} | 0.04447{5} | |||
0.02112{2} | 0.02396{4} | 0.04053{7} | 0.03054{6} | 0.07637{8} | 0.02144{3} | 0.01970{1} | 0.02658{5} | |||
0.06778{4} | 0.07756{5} | 0.03938{1} | 0.08202{6} | 0.51767{8} | 0.06301{2} | 0.06548{3} | 0.15423{7} | |||
0.00463{3} | 0.00564{4} | 0.03587{7} | 0.00699{5} | 0.12950{8} | 0.00403{2} | 0.00375{1} | 0.00770{6} | |||
0.00149{1} | 0.00185{4} | 0.00267{7} | 0.00229{5} | 0.01157{8} | 0.00151{2} | 0.00160{3} | 0.00260{6} | |||
−0.35462{6} | −0.37762{4} | −0.30854{8} | −0.39692{3} | −0.8708{1} | −0.35042{7} | −0.36334{5} | −0.4779{2} | |||
0.07548{3} | 0.08352{4} | 0.24243{7} | 0.10990{6} | 0.37560{8} | 0.06828{2} | 0.06752{1} | 0.08895{5} | |||
0.05281{2} | 0.05990{4} | 0.10134{7} | 0.07635{6} | 0.19092{8} | 0.05359{3} | 0.04925{1} | 0.06644{5} | |||
27{3} | 38{4} | 52{7} | 49{6} | 65{8} | 25{2} | 20{1} | 48{5} | |||
120 | 0.14157{3} | 0.15071{5} | 0.12488{1} | 0.15701{6} | 0.33864{8} | 0.13949{2} | 0.14476{4} | 0.18448{7} | ||
0.03095{3} | 0.03374{4} | 0.09034{7} | 0.04572{6} | 0.13792{8} | 0.02903{2} | 0.02884{1} | 0.03617{5} | |||
0.01823{3} | 0.01927{4} | 0.03334{7} | 0.02489{6} | 0.06217{8} | 0.01704{1} | 0.01706{2} | 0.02058{5} | |||
0.03753{3} | 0.04486{5} | 0.02492{1} | 0.04625{6} | 0.29530{8} | 0.03533{2} | 0.03942{4} | 0.07910{7} | |||
0.00322{3} | 0.00387{4} | 0.01580{7} | 0.00489{5} | 0.05979{8} | 0.00296{2} | 0.00280{1} | 0.00551{6} | |||
0.00109{2} | 0.00124{4} | 0.00178{7} | 0.00158{5} | 0.00718{8} | 0.00102{1} | 0.00117{3} | 0.00168{6} | |||
−0.28313{6} | −0.30143{4} | −0.24976{8} | −0.31402{3} | −0.67728{1} | −0.27899{7} | −0.28951{5} | −0.36896{2} | |||
0.06189{3} | 0.06748{4} | 0.18069{7} | 0.09144{6} | 0.27583{8} | 0.05807{2} | 0.05768{1} | 0.07234{5} | |||
0.04558{3} | 0.04819{4} | 0.08336{7} | 0.06223{6} | 0.15542{8} | 0.04261{1} | 0.04266{2} | 0.05144{5} | |||
29{3} | 38{4} | 52{7} | 49{6} | 65{8} | 20{1} | 23{2} | 48{5} | |||
200 | 0.10664{3} | 0.11928{5} | 0.09892{1} | 0.12024{6} | 0.23829{8} | 0.10411{2} | 0.11143{4} | 0.13639{7} | ||
0.02358{2} | 0.02825{4} | 0.06684{7} | 0.03499{6} | 0.09182{8} | 0.02231{1} | 0.02393{3} | 0.03000{5} | |||
0.01428{3} | 0.01553{4} | 0.02558{7} | 0.01909{6} | 0.04646{8} | 0.01396{2} | 0.01361{1} | 0.01745{5} | |||
0.02023{3} | 0.02580{6} | 0.01563{1} | 0.02485{5} | 0.12311{8} | 0.01853{2} | 0.02153{4} | 0.03900{7} | |||
0.00193{3} | 0.00270{4} | 0.00792{7} | 0.00300{5} | 0.01711{8} | 0.00179{1} | 0.00192{2} | 0.00374{6} | |||
0.00069{2} | 0.00081{4} | 0.00104{6} | 0.00095{5} | 0.00374{8} | 0.00067{1} | 0.00075{3} | 0.00117{7} | |||
−0.21328{6} | −0.23857{4} | −0.19785{8} | −0.24047{3} | −0.47659{1} | −0.20822{7} | −0.22286{5} | −0.27279{2} | |||
0.04715{2} | 0.05650{4} | 0.13367{7} | 0.06998{6} | 0.18363{8} | 0.04463{1} | 0.04785{3} | 0.05999{5} | |||
0.03569{3} | 0.03882{4} | 0.06396{7} | 0.04771{6} | 0.11615{8} | 0.03490{2} | 0.03403{1} | 0.04363{5} | |||
27{3} | 39{4} | 51{7} | 48{5} | 65{8} | 19{1} | 26{2} | 49{6} | |||
300 | 0.08835{3} | 0.09392{5} | 0.07939{1} | 0.09739{6} | 0.18183{8} | 0.08621{2} | 0.09360{4} | 0.11021{7} | ||
0.01974{2} | 0.02301{4} | 0.05371{7} | 0.02904{6} | 0.07244{8} | 0.01876{1} | 0.01985{3} | 0.02533{5} | |||
0.01189{3} | 0.01239{4} | 0.02043{7} | 0.01573{6} | 0.03646{8} | 0.01102{1} | 0.01173{2} | 0.01472{5} | |||
0.01318{3} | 0.01549{5} | 0.01010{1} | 0.01596{6} | 0.06384{8} | 0.01270{2} | 0.01515{4} | 0.02281{7} | |||
0.00138{3} | 0.00186{4} | 0.00510{7} | 0.00213{5} | 0.00970{8} | 0.00128{1} | 0.00135{2} | 0.00263{6} | |||
0.00048{2} | 0.00054{3} | 0.00072{6} | 0.00067{5} | 0.00221{8} | 0.00044{1} | 0.00056{4} | 0.00084{7} | |||
−0.1767{6} | −0.18785{4} | −0.15877{8} | −0.19478{3} | −0.36365{1} | −0.17241{7} | −0.18719{5} | −0.22042{2} | |||
0.03948{2} | 0.04601{4} | 0.10742{7} | 0.05807{6} | 0.14487{8} | 0.03752{1} | 0.03971{3} | 0.05066{5} | |||
0.02973{3} | 0.03096{4} | 0.05108{7} | 0.03932{6} | 0.09115{8} | 0.02755{1} | 0.02934{2} | 0.03680{5} | |||
27{2} | 37{4} | 51{7} | 49{5.5} | 65{8} | 17{1} | 29{3} | 49{5.5} |
50 | ||||||||||
80 | ||||||||||
120 | ||||||||||
200 | ||||||||||
300 | ||||||||||
50 | ||||||||||
. | ||||||||||
80 | ||||||||||
. | ||||||||||
120 | ||||||||||
200 | ||||||||||
80 | ||||||||||
120 | ||||||||||
200 | ||||||||||
300 | ||||||||||
Θ | Est. | Type II Censored Sample | Complete Sample | ||||||
---|---|---|---|---|---|---|---|---|---|
n = 20, r = 16 | |||||||||
−0.50 | 0.50 | 0.40 | Mean of Estimates | 0.02536 | 1.38871 | 0.35431 | −0.46446 | 0.93292 | 0.43112 |
|BIAS| | 0.90594 | 1.13302 | 0.14491 | 0.32995 | 0.53123 | 0.08899 | |||
MSE | 1.83855 | 2.69930 | 0.04002 | 0.20946 | 0.99588 | 0.01408 | |||
MRE | −1.81189 | 2.26604 | 0.36228 | −0.65990 | 1.06247 | 0.22248 | |||
1.60 | Mean of Estimates | 1.13152 | 0.41430 | 2.47183 | −0.49234 | 0.59221 | 1.69942 | ||
|BIAS| | 1.76837 | 0.30984 | 1.18411 | 0.21307 | 0.15888 | 0.31490 | |||
MSE | 4.49482 | 0.35483 | 2.18777 | 0.08266 | 0.08176 | 0.17602 | |||
MRE | −3.53675 | 0.61968 | 0.74007 | −0.42614 | 0.31775 | 0.19682 | |||
3.00 | 0.40 | Mean of Estimates | −0.89077 | 2.95476 | 0.41281 | −0.43525 | 4.56513 | 0.43834 | |
|BIAS| | 0.43124 | 0.57669 | 0.09044 | 0.36158 | 2.27523 | 0.08909 | |||
MSE | 0.20129 | 1.31193 | 0.01649 | 0.30529 | 6.13979 | 0.01470 | |||
MRE | −0.86247 | 0.19223 | 0.22611 | −0.72316 | 0.75841 | 0.22272 | |||
1.60 | Mean of Estimates | 0.60246 | 2.89592 | 2.25226 | −0.50873 | 4.11418 | 1.73030 | ||
|BIAS| | 1.36299 | 1.99720 | 0.94201 | 0.25631 | 1.75142 | 0.33951 | |||
MSE | 3.25105 | 4.73545 | 1.68761 | 0.10308 | 4.29353 | 0.20842 | |||
MRE | −2.72598 | 0.66573 | 0.58875 | −0.51261 | 0.58381 | 0.21220 | |||
0.75 | 0.50 | 0.40 | Mean of Estimates | 1.43620 | 0.96711 | 0.43501 | 0.79006 | 0.95802 | 0.42758 |
|BIAS| | 1.94245 | 0.94918 | 0.11519 | 1.07585 | 0.55713 | 0.08699 | |||
MSE | 4.74513 | 2.08058 | 0.02723 | 1.74201 | 1.08031 | 0.01321 | |||
MRE | 2.58994 | 1.89835 | 0.28797 | 1.43447 | 1.11426 | 0.21746 | |||
1.60 | Mean of Estimates | 1.96219 | 1.07707 | 2.26591 | 0.76298 | 0.58404 | 1.69855 | ||
|BIAS| | 1.91551 | 0.65689 | 1.06406 | 0.72764 | 0.14955 | 0.30952 | |||
MSE | 4.91114 | 1.55156 | 1.86972 | 0.87650 | 0.06153 | 0.17425 | |||
MRE | 2.55401 | 1.31379 | 0.66504 | 0.97019 | 0.29910 | 0.19345 | |||
3.00 | 0.40 | Mean of Estimates | −0.89087 | 2.12209 | 0.39709 | 0.82649 | 4.50031 | 0.43758 | |
|BIAS| | 1.64438 | 1.21784 | 0.08103 | 1.08455 | 2.20812 | 0.08709 | |||
MSE | 2.73656 | 1.89040 | 0.01168 | 1.87625 | 5.88662 | 0.01387 | |||
MRE | 2.19251 | 0.40595 | 0.20258 | 1.44607 | 0.73604 | 0.21772 | |||
1.60 | Mean of Estimates | 1.27091 | 2.07891 | 1.53803 | 0.74257 | 4.03618 | 1.73340 | ||
|BIAS| | 1.3406 | 2.07753 | 0.50886 | 0.88465 | 1.70161 | 0.33898 | |||
MSE | 2.64421 | 4.62694 | 0.41624 | 1.17173 | 4.11937 | 0.20624 | |||
MRE | 1.78746 | 0.69251 | 0.31804 | 1.17953 | 0.56720 | 0.21186 | |||
4.00 | 0.50 | 0.40 | Mean of Estimates | 4.74374 | 0.07115 | 0.42604 | 3.92379 | 0.63644 | 0.43506 |
|BIAS| | 2.22384 | 0.43326 | 0.09351 | 1.84629 | 0.21978 | 0.08486 | |||
MSE | 5.85760 | 0.19703 | 0.01829 | 4.12508 | 0.10116 | 0.01312 | |||
MRE | 0.55596 | 0.86653 | 0.23379 | 0.46157 | 0.43956 | 0.21216 | |||
1.60 | Mean of Estimates | 4.56242 | 2.03698 | 1.72026 | 3.97352 | 0.54780 | 1.72696 | ||
|BIAS| | 0.81567 | 1.55182 | 0.41545 | 1.60168 | 0.11034 | 0.32785 | |||
MSE | 2.19999 | 3.93102 | 0.39893 | 3.33003 | 0.01878 | 0.19543 | |||
MRE | 0.20392 | 3.10365 | 0.25965 | 0.40042 | 0.22069 | 0.20491 | |||
3.00 | 0.40 | Mean of Estimates | 2.00166 | 1.00066 | 0.82099 | 3.63991 | 4.09687 | 0.43824 | |
|BIAS| | 1.99834 | 1.99934 | 0.42099 | 1.90492 | 1.62451 | 0.08687 | |||
MSE | 3.99524 | 3.99788 | 0.18097 | 4.27413 | 3.65230 | 0.01375 | |||
MRE | 0.49959 | 0.66645 | 1.05248 | 0.47623 | 0.54150 | 0.21718 | |||
1.60 | Mean of Estimates | 2.64871 | 1.04407 | 1.34941 | 3.71724 | 3.76750 | 1.74039 | ||
|BIAS| | 1.77045 | 1.95664 | 0.37753 | 1.85213 | 1.31979 | 0.34592 | |||
MSE | 3.47854 | 3.85526 | 0.18168 | 4.06150 | 2.38659 | 0.21290 | |||
MRE | 0.44261 | 0.65221 | 0.23595 | 0.46303 | 0.43993 | 0.21620 | |||
Est. | Type II Censored Sample | Complete Sample | |||||||
−0.50 | 0.50 | 0.40 | Mean of Estimates | −0.03462 | 0.80194 | 0.35457 | −0.48778 | 0.66403 | 0.41525 |
|BIAS| | 0.73585 | 0.62085 | 0.11616 | 0.22967 | 0.24637 | 0.05897 | |||
MSE | 1.39700 | 1.12454 | 0.02863 | 0.09242 | 0.24034 | 0.00577 | |||
MRE | −1.47170 | 1.24171 | 0.29039 | −0.45935 | 0.49273 | 0.14743 | |||
1.60 | Mean of Estimates | 1.09431 | 0.24986 | 2.15271 | −0.49145 | 0.53442 | 1.64629 | ||
|BIAS| | 1.6690 | 0.26218 | 0.80954 | 0.15192 | 0.09129 | 0.21139 | |||
MSE | 4.14544 | 0.08500 | 1.03798 | 0.03826 | 0.01669 | 0.07485 | |||
MRE | −3.33799 | 0.52436 | 0.50596 | −0.30385 | 0.18258 | 0.13212 | |||
3.00 | 0.40 | Mean of Estimates | −0.96139 | 2.87124 | 0.37446 | −0.46550 | 4.23377 | 0.41951 | |
|BIAS| | 0.46299 | 0.18748 | 0.06008 | 0.31142 | 1.94842 | 0.05990 | |||
MSE | 0.21791 | 0.28813 | 0.00560 | 0.18681 | 4.97484 | 0.00614 | |||
MRE | −0.92598 | 0.06249 | 0.15021 | −0.62285 | 0.64947 | 0.14975 | |||
1.60 | Mean of Estimates | 0.39332 | 2.34107 | 1.94614 | −0.50736 | 3.77870 | 1.65730 | ||
|BIAS| | 1.14635 | 1.31139 | 0.71288 | 0.20899 | 1.38150 | 0.22197 | |||
MSE | 2.43720 | 2.74940 | 0.92182 | 0.06604 | 3.03827 | 0.08322 | |||
MRE | −2.29270 | 0.43713 | 0.44555 | −0.41799 | 0.46050 | 0.13873 | |||
0.75 | 0.50 | 0.40 | Mean of Estimates | 1.84211 | 0.34312 | 0.41428 | 0.77368 | 0.66332 | 0.41334 |
|BIAS| | 1.83931 | 0.54291 | 0.07473 | 0.76897 | 0.24331 | 0.05776 | |||
MSE | 4.62772 | 0.65856 | 0.00994 | 0.96286 | 0.24666 | 0.00563 | |||
MRE | 2.45242 | 1.08582 | 0.18682 | 1.02529 | 0.48662 | 0.14440 | |||
1.60 | Mean of Estimates | 0.90352 | 0.61288 | 1.76750 | 0.75899 | 0.53770 | 1.64111 | ||
|BIAS| | 0.50055 | 0.18753 | 0.36987 | 0.52382 | 0.09199 | 0.21079 | |||
MSE | 1.11173 | 0.15566 | 0.41445 | 0.45658 | 0.01689 | 0.07433 | |||
MRE | 0.66740 | 0.37507 | 0.23117 | 0.69843 | 0.18399 | 0.13174 | |||
3.00 | 0.40 | Mean of Estimates | −0.96774 | 1.95206 | 0.37015 | 0.83267 | 4.18991 | 0.41664 | |
|BIAS| | 1.71774 | 1.10911 | 0.05674 | 1.02422 | 1.92245 | 0.05781 | |||
MSE | 2.95404 | 1.41682 | 0.00471 | 1.63273 | 4.86474 | 0.00558 | |||
MRE | 2.29031 | 0.36970 | 0.14184 | 1.36562 | 0.64082 | 0.14452 | |||
1.60 | Mean of Estimates | 0.75810 | 1.36259 | 1.28724 | 0.68700 | 3.84043 | 1.65042 | ||
|BIAS| | 0.63656 | 1.97456 | 0.41903 | 0.72026 | 1.40726 | 0.22249 | |||
MSE | 0.72589 | 4.06606 | 0.23244 | 0.76949 | 3.13434 | 0.08148 | |||
MRE | 0.84874 | 0.65819 | 0.26189 | 0.96035 | 0.46909 | 0.13906 | |||
4.00 | 0.50 | 0.40 | Mean of Estimates | 5.30331 | 0.02956 | 0.39698 | 4.03736 | 0.56895 | 0.41627 |
|BIAS| | 2.23675 | 0.47044 | 0.05865 | 1.62497 | 0.14409 | 0.05713 | |||
MSE | 6.13685 | 0.22233 | 0.00568 | 3.43612 | 0.0388 | 0.00558 | |||
MRE | 0.55919 | 0.94089 | 0.14661 | 0.40624 | 0.28818 | 0.14283 | |||
1.60 | Mean of Estimates | 4.02169 | 1.54639 | 1.50348 | 3.97160 | 0.52998 | 1.64968 | ||
|BIAS| | 0.03829 | 1.04788 | 0.20024 | 1.31378 | 0.08119 | 0.21689 | |||
MSE | 0.01843 | 1.55876 | 0.06842 | 2.42673 | 0.01042 | 0.07843 | |||
MRE | 0.00957 | 2.09577 | 0.12515 | 0.32844 | 0.16239 | 0.13556 | |||
3.00 | 0.40 | Mean of Estimates | 2 | 1 | 0.96089 | 3.73488 | 3.84223 | 0.41781 | |
|BIAS| | 2 | 2 | 0.56089 | 1.90041 | 1.39078 | 0.05820 | |||
MSE | 4 | 4 | 0.31671 | 4.26720 | 2.78207 | 0.00576 | |||
MRE | 0.5 | 0.66667 | 1.40222 | 0.47510 | 0.46359 | 0.14551 | |||
1.60 | Mean of Estimates | 2.05978 | 1.00152 | 1.36241 | 3.83447 | 3.53228 | 1.66829 | ||
|BIAS| | 1.95311 | 1.99848 | 0.24716 | 1.65001 | 1.06500 | 0.22882 | |||
MSE | 3.86420 | 3.99459 | 0.06605 | 3.43137 | 1.62522 | 0.08826 | |||
MRE | 0.48828 | 0.66616 | 0.15448 | 0.41250 | 0.35500 | 0.14301 | |||
Est. | Type II Censored Sample | Complete Sample | |||||||
−0.50 | 0.50 | 0.40 | Mean of Estimates | −0.25434 | 0.50998 | 0.36723 | −0.48972 | 0.58125 | 0.40933 |
|BIAS| | 0.40992 | 0.33777 | 0.07209 | 0.18482 | 0.15656 | 0.04655 | |||
MSE | 0.72077 | 0.40866 | 0.01465 | 0.05690 | 0.07344 | 0.00356 | |||
MRE | −0.81984 | 0.67554 | 0.18022 | −0.36963 | 0.31312 | 0.11638 | |||
1.60 | Mean of Estimates | 0.98115 | 0.22748 | 2.02669 | −0.49669 | 0.52312 | 1.63179 | ||
|BIAS| | 1.5235 | 0.27278 | 0.65365 | 0.12229 | 0.07096 | 0.17481 | |||
MSE | 3.60873 | 0.07824 | 0.68068 | 0.02411 | 0.00894 | 0.05034 | |||
MRE | −3.04699 | 0.54556 | 0.40853 | −0.24458 | 0.14192 | 0.10926 | |||
3 | 0.4 | Mean of Estimates | −0.96960 | 2.91108 | 0.36606 | −0.49170 | 4.09860 | 0.41066 | |
|BIAS| | 0.46964 | 0.12398 | 0.05196 | 0.26870 | 1.76791 | 0.04734 | |||
MSE | 0.22495 | 0.13286 | 0.00393 | 0.12248 | 4.34794 | 0.00372 | |||
MRE | −0.93928 | 0.04133 | 0.12989 | −0.53741 | 0.58930 | 0.11834 | |||
1.60 | Mean of Estimates | −0.40550 | 2.88608 | 1.51892 | −0.51462 | 3.63807 | 1.63794 | ||
|BIAS| | 0.49871 | 0.43340 | 0.46798 | 0.17540 | 1.15813 | 0.18618 | |||
MSE | 0.79117 | 0.63580 | 0.43635 | 0.04604 | 2.32818 | 0.05707 | |||
MRE | −0.99743 | 0.14447 | 0.29249 | −0.35079 | 0.38604 | 0.11636 | |||
0.75 | 0.50 | 0.40 | Mean of Estimates | 2.00017 | 0.16424 | 0.40759 | 0.77780 | 0.58915 | 0.41013 |
|BIAS| | 1.69157 | 0.44490 | 0.06004 | 0.64728 | 0.16030 | 0.04704 | |||
MSE | 4.35212 | 0.30073 | 0.00603 | 0.68097 | 0.08309 | 0.00360 | |||
MRE | 2.25542 | 0.88981 | 0.15011 | 0.86305 | 0.32059 | 0.11761 | |||
1.60 | Mean of Estimates | 0.77059 | 0.53864 | 1.63200 | 0.75155 | 0.52383 | 1.63163 | ||
|BIAS| | 0.05656 | 0.09876 | 0.07775 | 0.41808 | 0.07017 | 0.17269 | |||
MSE | 0.07157 | 0.02972 | 0.03575 | 0.28432 | 0.00896 | 0.04845 | |||
MRE | 0.07541 | 0.19752 | 0.04859 | 0.55744 | 0.14035 | 0.10793 | |||
3.00 | 0.40 | Mean of Estimates | −0.95074 | 1.94419 | 0.36349 | 0.76503 | 4.08439 | 0.40995 | |
|BIAS| | 1.70074 | 1.09492 | 0.05090 | 0.91762 | 1.76293 | 0.04726 | |||
MSE | 2.94726 | 1.37834 | 0.00371 | 1.27842 | 4.31348 | 0.00371 | |||
MRE | 2.26765 | 0.36497 | 0.12726 | 1.22349 | 0.58764 | 0.11814 | |||
1.60 | Mean of Estimates | 0.58256 | 1.19342 | 1.22947 | 0.69887 | 3.64777 | 1.63428 | ||
|BIAS| | 0.43963 | 1.90741 | 0.40425 | 0.62402 | 1.17837 | 0.18385 | |||
MSE | 0.33276 | 3.83657 | 0.20708 | 0.57606 | 2.38031 | 0.05497 | |||
MRE | 0.58617 | 0.63580 | 0.25266 | 0.83202 | 0.39279 | 0.11490 | |||
4.00 | 0.50 | 0.40 | Mean of Estimates | 5.41344 | 0.02784 | 0.38913 | 4.05867 | 0.54748 | 0.41029 |
|BIAS| | 2.03075 | 0.47216 | 0.04804 | 1.46592 | 0.11703 | 0.04615 | |||
MSE | 5.65874 | 0.22703 | 0.00365 | 2.92575 | 0.02396 | 0.00347 | |||
MRE | 0.50769 | 0.94432 | 0.12010 | 0.36648 | 0.23406 | 0.11537 | |||
1.60 | Mean of Estimates | 4.02210 | 1.39921 | 1.46971 | 4.00365 | 0.52099 | 1.63338 | ||
|BIAS| | 0.02225 | 0.89923 | 0.13136 | 1.15002 | 0.06745 | 0.17134 | |||
MSE | 0.00080 | 1.27251 | 0.02781 | 1.94327 | 0.00729 | 0.04804 | |||
MRE | 0.00556 | 1.79846 | 0.08210 | 0.28750 | 0.13490 | 0.10709 | |||
3.00 | 0.40 | Mean of Estimates | 2.03040 | 1.03040 | 1.02112 | 3.79309 | 3.71697 | 0.41065 | |
|BIAS| | 1.96960 | 1.96960 | 0.62112 | 1.85369 | 1.27107 | 0.04617 | |||
MSE | 3.93920 | 3.93920 | 0.39308 | 4.09412 | 2.34912 | 0.00351 | |||
MRE | 0.49240 | 0.65653 | 1.55279 | 0.46342 | 0.42369 | 0.11542 | |||
1.60 | Mean of Estimates | 2.04411 | 1.03966 | 1.43178 | 3.86118 | 3.44297 | 1.64051 | ||
|BIAS| | 1.95589 | 1.96034 | 0.16870 | 1.50608 | 0.93256 | 0.18524 | |||
MSE | 3.90615 | 3.92058 | 0.03111 | 2.97689 | 1.28946 | 0.05644 | |||
MRE | 0.48897 | 0.65345 | 0.10544 | 0.37652 | 0.31085 | 0.11578 | |||
Est. | Type II Censored Sample | Complete Sample | |||||||
n = 80, r = 64 | |||||||||
−0.50 | 0.50 | 0.40 | Mean of Estimates | −0.44447 | 0.44278 | 0.38860 | −0.49365 | 0.55789 | 0.40759 |
|BIAS| | 0.13195 | 0.12553 | 0.03060 | 0.15565 | 0.12343 | 0.04077 | |||
MSE | 0.18512 | 0.08844 | 0.00442 | 0.04022 | 0.03894 | 0.00274 | |||
MRE | −0.26391 | 0.25106 | 0.07650 | −0.31130 | 0.24686 | 0.10193 | |||
1.60 | Mean of Estimates | 0.56941 | 0.27545 | 1.88606 | −0.49756 | 0.51655 | 1.61984 | ||
|BIAS| | 1.09074 | 0.22457 | 0.46346 | 0.10416 | 0.05899 | 0.14572 | |||
MSE | 2.42954 | 0.06320 | 0.42680 | 0.01779 | 0.00602 | 0.03456 | |||
MRE | −2.18148 | 0.44914 | 0.28966 | −0.20832 | 0.11798 | 0.09108 | |||
3.00 | 0.40 | Mean of Estimates | −0.83580 | 2.93908 | 0.37527 | −0.49316 | 3.94824 | 0.40958 | |
|BIAS| | 0.3358 | 0.08854 | 0.03370 | 0.24751 | 1.60699 | 0.04060 | |||
MSE | 0.16221 | 0.08440 | 0.00231 | 0.10069 | 3.79387 | 0.00273 | |||
MRE | −0.67160 | 0.02951 | 0.08425 | −0.49502 | 0.53566 | 0.10150 | |||
1.60 | Mean of Estimates | −0.59949 | 3.03916 | 1.48909 | −0.51071 | 3.51706 | 1.62467 | ||
|BIAS| | 0.17699 | 0.14837 | 0.18026 | 0.15836 | 1.02430 | 0.15556 | |||
MSE | 0.11099 | 0.11114 | 0.09585 | 0.03680 | 1.90466 | 0.03925 | |||
MRE | −0.35399 | 0.04946 | 0.11266 | −0.31673 | 0.34143 | 0.09722 | |||
0.75 | 0.50 | 0.40 | Mean of Estimates | 1.63645 | 0.21192 | 0.40235 | 0.77679 | 0.55292 | 0.40701 |
|BIAS| | 1.11403 | 0.31081 | 0.03627 | 0.54404 | 0.12009 | 0.04023 | |||
MSE | 2.89971 | 0.15255 | 0.00281 | 0.48384 | 0.03497 | 0.00262 | |||
MRE | 1.48537 | 0.62162 | 0.09068 | 0.72539 | 0.24017 | 0.10057 | |||
1.60 | Mean of Estimates | 0.75288 | 0.53288 | 1.60023 | 0.77706 | 0.51422 | 1.62043 | ||
|BIAS| | 0.00755 | 0.05057 | 0.01446 | 0.37689 | 0.06024 | 0.14981 | |||
MSE | 0.00199 | 0.0153 | 0.00315 | 0.23075 | 0.00628 | 0.03596 | |||
MRE | 0.01006 | 0.10114 | 0.00904 | 0.50252 | 0.12047 | 0.09363 | |||
3.00 | 0.40 | Mean of Estimates | −0.45435 | 2.24969 | 0.37493 | 0.77530 | 3.94513 | 0.40923 | |
|BIAS| | 1.20435 | 0.77242 | 0.03185 | 0.86458 | 1.61783 | 0.04075 | |||
MSE | 2.09319 | 0.95827 | 0.00204 | 1.13223 | 3.82698 | 0.00274 | |||
MRE | 1.60580 | 0.25747 | 0.07963 | 1.15278 | 0.53928 | 0.10187 | |||
1.60 | Mean of Estimates | 0.60585 | 1.79682 | 1.35269 | 0.70014 | 3.53740 | 1.62552 | ||
|BIAS| | 0.23935 | 1.22169 | 0.25546 | 0.55078 | 1.02209 | 0.15488 | |||
MSE | 0.14717 | 2.43875 | 0.12453 | 0.46335 | 1.90801 | 0.03946 | |||
MRE | 0.31913 | 0.40723 | 0.15966 | 0.73438 | 0.34070 | 0.09680 | |||
4.00 | 0.50 | 0.40 | Mean of Estimates | 5.06826 | 0.15632 | 0.39130 | 4.04705 | 0.53891 | 0.40756 |
|BIAS| | 1.3662 | 0.34368 | 0.03030 | 1.37861 | 0.10507 | 0.04020 | |||
MSE | 3.87684 | 0.16678 | 0.00200 | 2.64085 | 0.01922 | 0.00260 | |||
MRE | 0.34155 | 0.68737 | 0.07574 | 0.34465 | 0.21013 | 0.10050 | |||
1.60 | Mean of Estimates | 4.00996 | 0.89035 | 1.54137 | 4.02660 | 0.51513 | 1.62403 | ||
|BIAS| | 0.00996 | 0.39035 | 0.05863 | 1.03334 | 0.05856 | 0.14807 | |||
MSE | 0.00037 | 0.56146 | 0.01292 | 1.60182 | 0.00550 | 0.03570 | |||
MRE | 0.00249 | 0.78071 | 0.03665 | 0.25833 | 0.11711 | 0.09254 | |||
3.00 | 0.40 | Mean of Estimates | 2.59320 | 1.59320 | 0.86636 | 3.76797 | 3.69231 | 0.40789 | |
|BIAS| | 1.40680 | 1.40680 | 0.46636 | 1.80526 | 1.22321 | 0.03948 | |||
MSE | 2.81360 | 2.81360 | 0.30972 | 3.93873 | 2.16536 | 0.00253 | |||
MRE | 0.35170 | 0.46893 | 1.16591 | 0.45132 | 0.40774 | 0.09870 | |||
1.60 | Mean of Estimates | 2.71107 | 1.71080 | 1.51325 | 3.93342 | 3.35525 | 1.63242 | ||
|BIAS| | 1.28893 | 1.28920 | 0.08676 | 1.41664 | 0.84134 | 0.15582 | |||
MSE | 2.57745 | 2.57840 | 0.01248 | 2.71057 | 1.07953 | 0.04032 | |||
MRE | 0.32223 | 0.42973 | 0.05422 | 0.35416 | 0.28045 | 0.09739 | |||
Est. | Type II Censored Sample | Complete Sample | |||||||
−0.50 | 0.50 | 0.40 | Mean of Estimates | −0.49708 | 0.48406 | 0.40029 | −0.49826 | 0.54489 | 0.40508 |
|BIAS| | 0.01916 | 0.02311 | 0.00779 | 0.13648 | 0.10342 | 0.03573 | |||
MSE | 0.01727 | 0.01200 | 0.00093 | 0.03020 | 0.02506 | 0.00204 | |||
MRE | −0.03833 | 0.04623 | 0.01948 | −0.27297 | 0.20685 | 0.08933 | |||
1.60 | Mean of Estimates | −0.24955 | 0.43244 | 1.67070 | −0.49755 | 0.51330 | 1.62400 | ||
|BIAS| | 0.25627 | 0.06756 | 0.12570 | 0.09537 | 0.05357 | 0.13341 | |||
MSE | 0.53812 | 0.01797 | 0.13017 | 0.01452 | 0.00484 | 0.02855 | |||
MRE | −0.51253 | 0.13513 | 0.07856 | −0.19074 | 0.10714 | 0.08338 | |||
3.00 | 0.40 | Mean of Estimates | −0.60199 | 2.98716 | 0.39250 | −0.49661 | 3.87087 | 0.40767 | |
|BIAS| | 0.10199 | 0.02155 | 0.00939 | 0.23318 | 1.51153 | 0.03686 | |||
MSE | 0.04961 | 0.01607 | 0.00059 | 0.08509 | 3.47402 | 0.00218 | |||
MRE | −0.20398 | 0.00718 | 0.02347 | −0.46636 | 0.50384 | 0.09215 | |||
1.60 | Mean of Estimates | −0.52137 | 3.00325 | 1.58892 | −0.51634 | 3.47978 | 1.61988 | ||
|BIAS| | 0.03305 | 0.02503 | 0.02414 | 0.14563 | 0.93673 | 0.14105 | |||
MSE | 0.01518 | 0.01798 | 0.00944 | 0.03146 | 1.62734 | 0.03162 | |||
MRE | −0.06609 | 0.00834 | 0.01509 | −0.29127 | 0.31224 | 0.08815 | |||
0.75 | 0.50 | 0.40 | Mean of Estimates | 1.00439 | 0.4078 | 0.39972 | 0.77100 | 0.53941 | 0.40587 |
|BIAS| | 0.31646 | 0.09521 | 0.00922 | 0.47349 | 0.10015 | 0.03551 | |||
MSE | 0.81499 | 0.04442 | 0.00062 | 0.36932 | 0.02051 | 0.00205 | |||
MRE | 0.42195 | 0.19041 | 0.02305 | 0.63132 | 0.20031 | 0.08876 | |||
1.60 | Mean of Estimates | 0.75135 | 0.51655 | 1.59731 | 0.75581 | 0.51376 | 1.62092 | ||
|BIAS| | 0.00147 | 0.01777 | 0.00305 | 0.33022 | 0.05326 | 0.13322 | |||
MSE | 0.00004 | 0.0054 | 0.00021 | 0.17552 | 0.00487 | 0.02886 | |||
MRE | 0.00196 | 0.03554 | 0.00191 | 0.4403 | 0.10653 | 0.08326 | |||
3.00 | 0.40 | Mean of Estimates | 0.42807 | 2.79703 | 0.39401 | 0.79667 | 3.78996 | 0.40624 | |
|BIAS| | 0.32193 | 0.20565 | 0.00793 | 0.81234 | 1.46896 | 0.03567 | |||
MSE | 0.56025 | 0.25105 | 0.00048 | 1.02104 | 3.31672 | 0.00208 | |||
MRE | 0.42925 | 0.06855 | 0.01982 | 1.08312 | 0.48965 | 0.08917 | |||
1.60 | Mean of Estimates | 0.71442 | 2.71982 | 1.54349 | 0.69564 | 3.46989 | 1.61355 | ||
|BIAS| | 0.05528 | 0.28507 | 0.05794 | 0.50327 | 0.92182 | 0.13587 | |||
MSE | 0.03426 | 0.56972 | 0.02723 | 0.37901 | 1.59651 | 0.02958 | |||
MRE | 0.07371 | 0.09502 | 0.03621 | 0.67102 | 0.30727 | 0.08492 | |||
4.00 | 0.50 | 0.40 | Mean of Estimates | 4.29423 | 0.3942 | 0.3967 | 4.04726 | 0.53223 | 0.40624 |
|BIAS| | 0.35635 | 0.1058 | 0.00846 | 1.25444 | 0.09237 | 0.03544 | |||
MSE | 1.01896 | 0.05175 | 0.00051 | 2.26044 | 0.01472 | 0.00208 | |||
MRE | 0.08909 | 0.21160 | 0.02115 | 0.31361 | 0.18473 | 0.08860 | |||
1.60 | Mean of Estimates | 4.00321 | 0.62236 | 1.58113 | 4.01883 | 0.51211 | 1.62139 | ||
|BIAS| | 0.00321 | 0.12236 | 0.01887 | 0.91969 | 0.05175 | 0.13376 | |||
MSE | 0.00012 | 0.17977 | 0.00429 | 1.30638 | 0.00437 | 0.02817 | |||
MRE | 0.0008 | 0.24473 | 0.01179 | 0.22992 | 0.1035 | 0.08360 | |||
3.00 | 0.40 | Mean of Estimates | 3.65760 | 2.65760 | 0.51563 | 3.87503 | 3.57281 | 0.40620 | |
|BIAS| | 0.34240 | 0.34240 | 0.11563 | 1.74919 | 1.13575 | 0.03612 | |||
MSE | 0.68480 | 0.68480 | 0.07821 | 3.79583 | 1.88352 | 0.00210 | |||
MRE | 0.08560 | 0.11413 | 0.28908 | 0.43730 | 0.37858 | 0.09030 | |||
1.60 | Mean of Estimates | 3.69760 | 2.69760 | 1.58209 | 3.91556 | 3.32605 | 1.62096 | ||
|BIAS| | 0.30240 | 0.30240 | 0.01791 | 1.31845 | 0.77827 | 0.13790 | |||
MSE | 0.60480 | 0.60480 | 0.00228 | 2.40502 | 0.94921 | 0.03066 | |||
MRE | 0.07560 | 0.10080 | 0.01120 | 0.32961 | 0.25942 | 0.08619 | |||
Est. | Type II Censored Sample | Complete Sample | |||||||
n = 150, r = 120 | |||||||||
−0.50 | 0.50 | 0.40 | Mean of Estimates | −0.50012 | 0.49977 | 0.40011 | −0.49446 | 0.52279 | 0.40404 |
|BIAS| | 0.00012 | 0.00023 | 0.00016 | 0.11024 | 0.07664 | 0.02925 | |||
MSE | 0.00001 | 0.00005 | 0.00002 | 0.01961 | 0.01052 | 0.00136 | |||
MRE | −0.00024 | 0.00046 | 0.00039 | −0.22047 | 0.15328 | 0.07312 | |||
1.60 | Mean of Estimates | −0.49963 | 0.49977 | 1.60009 | −0.49991 | 0.50911 | 1.61178 | ||
|BIAS| | 0.00041 | 0.00023 | 0.00015 | 0.07715 | 0.04269 | 0.10744 | |||
MSE | 0.00028 | 0.00006 | 0.00007 | 0.00939 | 0.00301 | 0.01863 | |||
MRE | −0.00082 | 0.00046 | 0.00010 | −0.15430 | 0.08537 | 0.06715 | |||
3.00 | 0.40 | Mean of Estimates | −0.50137 | 2.99984 | 0.39989 | −0.50689 | 3.72250 | 0.40475 | |
|BIAS| | 0.00137 | 0.00019 | 0.00012 | 0.19754 | 1.28906 | 0.02957 | |||
MSE | 0.00067 | 0.00002 | 0.00001 | 0.05975 | 2.74650 | 0.00141 | |||
MRE | −0.00274 | 0.00006 | 0.00031 | −0.39509 | 0.42969 | 0.07392 | |||
1.60 | Mean of Estimates | −0.50011 | 2.99998 | 1.60002 | −0.51088 | 3.32551 | 1.61378 | ||
|BIAS| | 0.00011 | 0.00002 | 0.00002 | 0.11945 | 0.73009 | 0.11463 | |||
MSE | 0.00001 | 0 | 0 | 0.02183 | 1.04586 | 0.02102 | |||
MRE | −0.00021 | 0.00001 | 0.00001 | −0.23890 | 0.24336 | 0.07164 | |||
0.75 | 0.50 | 0.40 | Mean of Estimates | 0.75383 | 0.49857 | 0.40000 | 0.77153 | 0.52511 | 0.40455 |
|BIAS| | 0.00393 | 0.00143 | 0.00009 | 0.39829 | 0.08041 | 0.02941 | |||
MSE | 0.01081 | 0.00064 | 0.00000 | 0.25491 | 0.01215 | 0.00136 | |||
MRE | 0.00524 | 0.00287 | 0.00022 | 0.53106 | 0.16083 | 0.07353 | |||
1.60 | Mean of Estimates | 0.75 | 0.50002 | 1.60000 | 0.76007 | 0.50777 | 1.61389 | ||
|BIAS| | 0 | 0.00002 | 0.00000 | 0.26679 | 0.04185 | 0.10970 | |||
MSE | 0.00000 | 0.00000 | 0.00000 | 0.11418 | 0.00290 | 0.01908 | |||
MRE | 0.00000 | 0.00003 | 0.00000 | 0.35573 | 0.08371 | 0.06856 | |||
3.00 | 0.40 | Mean of Estimates | 0.74791 | 2.99870 | 0.39997 | 0.73353 | 3.70019 | 0.40502 | |
|BIAS| | 0.00209 | 0.00130 | 0.00004 | 0.68921 | 1.28229 | 0.02945 | |||
MSE | 0.00365 | 0.00146 | 0.00000 | 0.72113 | 2.70530 | 0.00141 | |||
MRE | 0.00279 | 0.00043 | 0.00010 | 0.91895 | 0.42743 | 0.07362 | |||
1.60 | Mean of Estimates | 0.74991 | 2.99960 | 1.59990 | 0.70689 | 3.34933 | 1.61295 | ||
|BIAS| | 0.00009 | 0.00040 | 0.00010 | 0.42766 | 0.75586 | 0.11124 | |||
MSE | 0.00004 | 0.00080 | 0.00005 | 0.27871 | 1.11243 | 0.02000 | |||
MRE | 0.00011 | 0.00013 | 0.00006 | 0.57021 | 0.25195 | 0.06952 | |||
4.00 | 0.50 | 0.40 | Mean of Estimates | 4.00380 | 0.49853 | 0.39998 | 4.02845 | 0.52406 | 0.40459 |
|BIAS| | 0.00461 | 0.00147 | 0.00008 | 1.06817 | 0.07642 | 0.02931 | |||
MSE | 0.01338 | 0.00072 | 0.00000 | 1.73134 | 0.00982 | 0.00138 | |||
MRE | 0.00115 | 0.00294 | 0.00019 | 0.26704 | 0.15284 | 0.07328 | |||
1.60 | Mean of Estimates | 4.00001 | 0.50085 | 1.59986 | 4.00796 | 0.50865 | 1.61070 | ||
|BIAS| | 0.00001 | 0.00085 | 0.00014 | 0.77649 | 0.04265 | 0.10730 | |||
MSE | 0.00000 | 0.00090 | 0.00002 | 0.94515 | 0.00298 | 0.01835 | |||
MRE | 0.00000 | 0.00170 | 0.00009 | 0.19412 | 0.08531 | 0.06706 | |||
3.00 | 0.40 | Mean of Estimates | 3.99600 | 2.99600 | 0.40144 | 3.93716 | 3.46171 | 0.40455 | |
|BIAS| | 0.00400 | 0.00400 | 0.00144 | 1.60899 | 1.00107 | 0.02900 | |||
MSE | 0.00800 | 0.00800 | 0.00104 | 3.34629 | 1.51007 | 0.00135 | |||
MRE | 0.00100 | 0.00133 | 0.00361 | 0.40225 | 0.33369 | 0.07251 | |||
1.60 | Mean of Estimates | 4.00000 | 3.00000 | 1.60000 | 3.86922 | 3.29227 | 1.61184 | ||
|BIAS| | 0.00000 | 0.00000 | 0.00000 | 1.14505 | 0.67471 | 0.11224 | |||
MSE | 0.00000 | 0.00000 | 0.00000 | 1.89665 | 0.75127 | 0.02024 | |||
MRE | 0.00000 | 0.00000 | 0.00000 | 0.28626 | 0.22490 | 0.07015 |
Competitive Models | The CDF of Competitive Models | Author(s) |
---|---|---|
Lehmann Type I (L-I) | , 0 < x < 1 | Lehmann [1] |
Lehmann Type II (L-II) | 0 < x < 1 | |
Beta | 0 < x < 1 | Mood et al. [38] |
Topp–Leone (TL) | , 0 < x < 1 | Topp & Leone [39] |
Kumaraswamy (Kum) | 0 < x < 1 | Kumaraswamy [40] |
Generalized-PF (GPF) | Saran & Pandey [41] | |
Weibull-PF (WPF) | Tahir et al. [11] | |
Kumaraswamy-PF (Kum-PF) | Abdul-Moniem [42] | |
Mustapha Type II (MT-II) | Muhammad [43] | |
New PF (NPF) | 0 < x < 1 | Iqbal et al. [26] |
Exponentiated-PF (EPF) | Arshad et al. [20] |
Dataset | Minimum | 1st Quartile | Mean | 3rd Quartile | 95% CI | Maximum |
---|---|---|---|---|---|---|
1 | 0.0115 | 0.0285 | 0.0458 | 0.0559 | (0.0418,0.0497) | 0.1258 |
2 | 0.0670 | 0.0848 | 0.1215 | 0.1220 | (0.0797,0.1634) | 0.4850 |
3 | 0.0903 | 0.1623 | 0.2181 | 0.2627 | (0.1938,0.2423) | 0.4641 |
Model | Parameters (S.E.) | Goodness-of-Fit | ||||||
---|---|---|---|---|---|---|---|---|
AIC | CM | AD | KS | KS p-Value | ||||
E-NPF | 5.3053 (7.9348) | 10.3963 (10.0501) | 7.8870 (3.1445) | −649.6312 | 0.0323 | 0.1968 | 0.0502 | 0.8961 |
Beta | 4.2288 (0.5033) | 87.9827 (11.0854) | - | −647.5600 | 0.0726 | 0.5288 | 0.0577 | 0.7757 |
EPF | 3.5459 (0.3552) | 2.1359 (0.2966) | - | −637.5190 | 0.1914 | 1.3168 | 0.0765 | 0.4259 |
Kum | 2.13 (0.13) | 555.64 (214.56) | - | −635.6234 | 0.1973 | 1.3760 | 0.0709 | 0.5263 |
WPF | 60.5265 (48.8081) | 0.8117 (0.2644) | 2.1967 (0.5041) | −627.8117 | 0.2526 | 1.7310 | 0.0882 | 0.2593 |
KPF | 1.2907 (39.5678) | 1.3841 (42.4311) | 4.2046 (0.5992) | −615.4130 | 0.4094 | 2.6764 | 0.1093 | 0.0872 |
GPF | 2.2542 (0.1969) | - | - | −611.4720 | 0.2454 | 1.6535 | 0.1734 | 0.0008 |
NPF | −0.6094 (0.2327) | 52.6932 (31.0382) | - | −549.0419 | 0.0764 | 0.5568 | 0.2887 | 0.0000 |
L-II | 21.1874 (1.8511) | - | - | −548.3578 | 0.0715 | 0.5212 | 0.2949 | 0.0000 |
TL | 0.3953 (0.0345) | - | - | −325.8514 | 0.0464 | 0.3354 | 0.4501 | 0.0000 |
L-I | 0.3125 (0.0273) | - | - | −269.5753 | 0.0417 | 0.2990 | 0.4841 | 0.0000 |
MT-II | 0.2364 (0.0273) | - | - | −250.0246 | 0.0452 | 0.3264 | 0.4769 | 0.0000 |
Model | Parameters (S.E.) | Goodness-of-Fit | ||||||
---|---|---|---|---|---|---|---|---|
AIC | CM | AD | KS | KS p-Value | ||||
E-NPF | 448.2682 (562.8587) | 3.3050 (1.0733) | 322803.8 (25116.79) | −69.8565 | 0.0599 | 0.4713 | 0.1232 | 0.9217 |
EPF | 1.7986 (0.6898) | 0.5082 (0.1424) | - | −54.8662 | 0.4301 | 2.5719 | 0.3438 | 0.0177 |
Beta | 3.1111 (0.9366) | 21.8184 (7.0402) | - | −51.7626 | 0.3700 | 2.3154 | 0.2538 | 0.1520 |
WPF | 25.3217 (10.9815) | 8.6983 (30.6164) | 0.1887 (0.6641) | −46.8444 | 0.3971 | 2.4524 | 0.2641 | 0.1226 |
GPF | 3.1354 (0.7011) | - | - | −26.2083 | 0.4155 | 2.5011 | 0.4262 | 0.0014 |
Kum | 1.5877 (0.2444) | 21.8682 (10.2103) | −50.4166 | 0.4369 | 2.6507 | 0.2626 | 0.1267 | |
NPF | −0.5085 (1.1201) | 14.0472 (30.0514) | - | −41.3505 | 0.4083 | 2.5089 | 0.3856 | 0.0052 |
L-II | 7.3407 (1.6414) | - | - | −43.1863 | 0.3697 | 2.3141 | 0.3989 | 0.0034 |
KPF | 1.0534 (87.4390) | 0.9593 (79.6359) | 2.2244 (0.6820) | −32.2740 | 0.7622 | 4.1593 | 0.3704 | 0.0083 |
TL | 0.6248 (1.1397) | - | - | −25.4857 | 0.3390 | 2.1564 | 0.4841 | 0.0002 |
L-I | 0.4485 (0.1003) | - | - | −15.1163 | 0.3211 | 2.0626 | 0.5104 | 0.0001 |
MT-II | 0.3403 (0.1003) | - | - | −12.1936 | 0.3385 | 2.1537 | 0.5001 | 0.0001 |
Model | Parameters (S.E.) | Goodness-of-Fit | ||||||
---|---|---|---|---|---|---|---|---|
AIC | CM | AD | KS | KS p-value | ||||
E-NPF | 6.6638 (10.9778) | 3.9381 (2.0050) | 46.9570 (76.5344) | −110.8073 | 0.0289 | 0.1892 | 0.0831 | 0.8948 |
GPF | 1.7877 (0.2580) | - | - | −103.4055 | 0.2315 | 1.4418 | 0.1557 | 0.1944 |
Beta | 5.9417 (1.1813) | 21.2057 (4.3469) | - | −107.2004 | 0.1280 | 0.7782 | 0.1428 | 0.2820 |
EPF | 2.2058 (0.4036) | 1.3835 (0.2923) | - | −103.6836 | 0.2183 | 1.3640 | 0.1570 | 0.1874 |
WPF | 42.9950 (15.7909) | 8.7742 (28.6244) | 0.3131 (1.0209) | −99.4828 | 0.2000 | 1.2259 | 0.1499 | 0.2308 |
Kum | 2.7187 (0.2936) | 44.6671 (17.5871) | - | −100.9830 | 0.2083 | 1.2803 | 0.1533 | 0.2095 |
KPF | 1.4411 (90.5464) | 1.4050 (88.2744) | 2.6326 (0.5554) | −86.0842 | 0.4173 | 2.5450 | 0.1863 | 0.0716 |
NPF | −0.9953 (0.0020) | 724.2227 (33.3339) | - | −65.2928 | 0.1704 | 1.0482 | 0.3311 | 0.0001 |
L-II | 3.9644 (0.5722) | - | - | −58.4411 | 0.1279 | 0.7777 | 0.3599 | 0.0000 |
TL | 0.9894 (0.1428) | - | - | −40.3319 | 0.1190 | 0.7218 | 0.3680 | 0.0000 |
L-I | 0.6300 (0.0909) | - | - | −10.0237 | 0.1139 | 0.6904 | 0.4295 | 0.0000 |
MT-II | 0.4786 (0.0909) | - | - | −3.1089 | 0.1224 | 0.7434 | 0.4241 | 0.0000 |
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Al Mutairi, A.; Iqbal, M.Z.; Arshad, M.Z.; Alnssyan, B.; Al-Mofleh, H.; Afify, A.Z. A New Extended Model with Bathtub-Shaped Failure Rate: Properties, Inference, Simulation, and Applications. Mathematics 2021, 9, 2024. https://doi.org/10.3390/math9172024
Al Mutairi A, Iqbal MZ, Arshad MZ, Alnssyan B, Al-Mofleh H, Afify AZ. A New Extended Model with Bathtub-Shaped Failure Rate: Properties, Inference, Simulation, and Applications. Mathematics. 2021; 9(17):2024. https://doi.org/10.3390/math9172024
Chicago/Turabian StyleAl Mutairi, Alya, Muhammad Z. Iqbal, Muhammad Z. Arshad, Badr Alnssyan, Hazem Al-Mofleh, and Ahmed Z. Afify. 2021. "A New Extended Model with Bathtub-Shaped Failure Rate: Properties, Inference, Simulation, and Applications" Mathematics 9, no. 17: 2024. https://doi.org/10.3390/math9172024
APA StyleAl Mutairi, A., Iqbal, M. Z., Arshad, M. Z., Alnssyan, B., Al-Mofleh, H., & Afify, A. Z. (2021). A New Extended Model with Bathtub-Shaped Failure Rate: Properties, Inference, Simulation, and Applications. Mathematics, 9(17), 2024. https://doi.org/10.3390/math9172024