Inference procedures based on the likelihood function are considered for the

one logistic distribution with time censored data. The finite sample performances

of the maximum likelihood estimator as well as the large sample likelihood

inferential procedures based on the Wald, the Rao, and the likelihood ratio

statistics are investigated. It is found that the obtained from the asymptotic

normal distribution of the maximum likelihood estimator are found no accurate.

It is found also that interval estimation based on the Wald and Rao statistics

need much more sample size than interval estimation based on the likelihood

ratio statistics to attain reasonable accuracy

Baklizi, Ayman

Universiti Putra Malaysia Institutional Repository

Pertanika J. Sci. & Techno!. 10(1): 55 - 62 (2002)ISSN: 0128-7680© Universiti Putra Malaysia PressLikelihood Based Estimation in the Logistic Modelwith Time Censored DataAyman S. BakliziDepartment of StatisticsYarmouk UniversityIrbid-jordanReceived: 23 October 2000ABSTRAKProsedur-prosedur kesimpulan berdasarkan fungsi kemungkinandipertimbangkan untuk satu pengagihan logistik dengan data tapisan masa.Pelaksanaan sampel penganggar kemungkinan maksimum yang terhadsepertimana prosedur kesimpulan kemungkinan sampel besar berdasarkansampel Wald, Rao dan statistik nisbah kemungkinan disiasat. Hasilnya, apa yangdiperolehi daripada pengagihan normal asimptot penganggar kemungkinanmaksimum didapati tidak tepat. Hasil siasatanjuga menunjukkan penganggaranjeda yang berdasarkan statistik Wald dan Rao memerlukan lebih banyak saizsampel berbanding penganggaran jeda berdasarkan statistik nisbah kemungkinanuntuk memperolehi ketepatan yang munasabah.ABSTRACTInference procedures based on the likelihood function are considered for theone logistic distribution with time censored data. The finite sample performancesof the maximum likelihood estimator as well as the large sample likelihoodinferential procedures based on the Wald, the Rao, and the likelihood ratiostatistics are investigated. It is found that the obtained from the asymptoticnormal distribution of the maximum likelihood estimator are found no accurate.It is found also that interval estimation based on the Wald and Rao statisticsneed much more sample size than interval estimation based on the likelihoodratio statistics to attain reasonable accuracy.Keywords: Confidence intervals, logistic distribution, maximum likelihoodestimator, time censored dataINTRODUCTIONThe logistic distribution arises in a variety of fields, for example it can beapplied as a growth model in human populations and certain biologicalorganisms (Pearl et al. 1940). Oliver (1964) used this distribution to modelagricultural production data. It also arises in the analysis of survival data(Placket 1959) as well as the analysis of income distributions (Fisk 1961). Manyother applications and motivations for this distribution are discussed inBalakrishnan (1992) and Johnson et al. (1994).Inference procedures for the parameters of this distribution have beendiscussed in the literature. Harter and Moore (1967) and Balakrishnan (1992)discussed and investigated the properties the maximum likelihood estimator insample of size 10 and 20, for various choices of type 2 censoring. ConfidenceAyrnan S. BakJiziintervals for the parameters of this models are constructed by Antle et al.(1970) by simulating the percentage points of some pivotal quantities based onthe maximum likelihood estimator. Schaffer and Sheffield (1973) have furtherdiscussion on this problem while Bain et al. (1992) considered this problemwhen the data is type 2 censored.In this paper I shall consider maximum likelihood estimation and asymptoticinterval estimators based on inverting the likelihood based statistics when thedata is time censored. These statistics are the Wald, the Rao, and the likelihoodratio (Barndorff- Nielsen and Cox 1994). These statistics are often used forinterval estimation with censored data ( elson 1990), and are known to havean asymptotic chi-squared distribution (Rao 1973). However their performancesin finite samples are different and change from one model to another (Lawless1982). Hence it is desirable to use the statistics which has a faster covergencerate to its limiting distribution and therefore is applicable for small sample sizesthat may be in practice (Cox 1988).METHODThe Model and the Likelihood StatisticsThe probability density function and the cummulative distribution function ofthe logistic distribution are given respectively by (Johnson et al. 1994)( c(x-l-t)c exp crf(x, l-t, cr) = ( )2cr c(x-l-t)1+exp (- --'-----'--'-crx > 0, cr> 0, - 00 < l-t < 001twhere c = .J3' Consider a random sample of size N from this distribution, ofwhich n are less than or equal to t (some predetermined censoring time) andthe remaining no observations are censored and are only known to exceed t.Thus N = n + no" Here t is fixed but n and no are both random the likelihoodfunction is given by(c)n J ~ c(x -l-t)nn c(x -l-t)-2L(x,l-t,cr)= -; ex~- -'2 ~ ;_1 1+ exp (- 'cr xnn" [( c(t -l-t»)-I);=1 1- l+exp(---cr-The maximum likelihood estimator can be found by solving the system of firstpartial derivatives of the log-likelihood function, describe in the appendix. To56 PertanikaJ. Sci. & Technol. Vol. 10 No. 1,2002Likelihood Based Estimation in the Logistic Model with Time Censored Datapresent the Wald statistics, the Rao and the likelihood ratio statistics for thismodel we need the following quantities.The Wald statistics for l.l and 0' are given respectively byml (l.l) =()l- 1.l)2 (II' (l.l, 0-))-15, (0') = (0- - 0')2 (1 22 ()l, 0-) r 1The Rao statistics for l.l and 0' are given respectively by- 2 II -m2 (l.l) =(VI (l.l, cr)) I (l.l, cr),52(cr) = (U2(jl,cr))2e2(jl,cr)The likelihood ratio statistics for l.l and cr are given respectively bym3(1.l) = 2(l()l, 0-) -l(l.l, 6),53(cr) = 2(l()l,0-) -1(jl,cr))Where l(l.l,cr) = In(L(l.l,cr)) is the log-likelihood function, )l and 0- are themaximum likelihood estimator of l.l and cr. )l is the maximum likelihoodestimator of l.l for a given value of cr. lij is the ij-th element of 11th element of1-1, the inverse of the information matrix.Finite Sample Performance of the Likelihood StatisticsIn this section we shall describe a simulation study conducted to investigate thefinite sample behaviour of the maximum likelihood estimator, and confidenceintervals based on the Wald, the Rao, and the likelihood ratio statistics.The criteria used for the evaluation of the performance of the maximumlikelihood estimator are the bias, the finite sample variance, and the adequacyof the asymptotic variance estimates (Elperin and Gertsbakh 1987). For theconfidence intercals, we use the attainment of the nominal error probabilitiesand the symmetry of lower and upper error probabilities (Jennings 1987).Attainment of nominal error probabilities and our conclusions therefore areimprecise and can be misleading. The symmetry of lower and upper errorprobabilities means that of the interval fails to contain the true value of theparameter, it is equally expect this symmetry because they are using symmetericpercentiles of the approximating distributions to form their confidence intervals.PertanikaJ. Sci. & Techno!. Vo!. 10 No.1, 2002 57Ayman S. BakliziTABLE 1Values of the bias, finite sample variance, mean squared error and asymptoticvariance for the maximum likelihood estimatorLocation Parameter Scale ParameterCP Bias FSV MSE ASV Bias FSV MSE ASV0.0 2.0 -59 436 437 449 -308 345 354 33940 -40 219 219 225 176 163 166 17160 -27 152 152 151 -113 115 116 11580 6 110 110 113 -80 82 82 87100 8 88 88 91 -77 68 68 69120 11 76 76 76 -69 58 59 58140 13 64 64 65 -60 49 50 500.1 60 -26 152 152 152 -94 126 127 12880 6 110 110 114 -71 89 90 96100 9 87 87 91 -69 74 75 77120 11 76 76 76 -65 63 63 64140 13 64 64 65 -58 53 54 550.3 60 -13 157 157 160 -69 170 171 17580 23 114 114 120 -35 129 129 132100 21 90 90 96 -42 103 103 104120 23 77 78 79 -38 86 86 87140 24 66 66 68 -33 73 74 740.5 60 22 203 203 208 -41 269 269 27480 53 143 144 154 -5 197 197 205100 56 116 117 122 -2 162 162 163120 57 103 104 101 0 133 133 135140 56 87 88 86 4 113 113 115However, symmetry of error probabilities may not occur due to the skewness ofthe actual sampling distribution (Jennings 1987).In the simulations we use values of the sample size N as 60, 80, 100, 120 and140. In each case we examined the censoring proportions 0.1, 0.3, and 0.5. Forevery combination of the sample size and censoring proportion we generated2000 samples which are used to determine, the bias, the finite sample variance,the asymptotic variance of the maximum likelihood estimator, the meansquared errors of the estimators (MSE), and the error probabilities of theconfidence intervals obtained from the Wald the Rao, and the likelihood ratiostatistics. The levels of significance used are cx= 0.05, and 0.1. The results aregiven in Tables 1, 2 and 3. All values in the tables are multiplied by 10000.FINDINGS AND CONCLUSIONSConcerning the behaviour of the maximum likelihood estimator for thelocation and scale parameters, it appears that it is almost unbiased. It appears58 PertanikaJ. Sci. & Techno!. Vo!. 10 No. 1,2002Likelihood Based Estimation in the Logistic Model with Time Censored DataTABLE 2Lower and upper error probabilities of confidence intenrals based on the Wald,Rao and likelihood ratio statistics. a = 0.1Location Parameter Scale ParameterWald Rao LR Wald Rao LRCP N L U L U L U L U L U L U0.0 20 300 235 100 95 255 195 995 20 0 525 435 11540 350 225 195 130 315 190 730 30 0 445 340 11560 310 290 225 200 285 265 590 50 0 430 345 14080 225 210 180 185 210 210 490 70 20 340 315 175100 225 225 190 195 220 220 500 65 55 375 335 155120 260 250 210 220 250 240 515 85 80 380 365 155140 275 275 240 235 265 260 480 110 55 395 340 1800.1 60 310 275 215 190 290 255 585 40 0 490 310 15580 235 210 175 185 215 210 475 55 15 385 300 140100 220 225 175 200 215 220 540 65 40 390 325 140120 250 250 210 210 240 250 505 55 65 375 37 165140 290 275 240 240 270 265 450 90 55 380 340 1750.3 60 295 210 185 255 260 225 570 25 0 570 330 18580 260 195 140 220 235 210 470 65 10 550 290 230100 260 220 165 255 235 235 510 85 20 445 310 225120 280 220 210 260 255 230 485 75 35 495 330 185140 295 220 220 270 275 245 475 120 30 485 265 2300.5 60 365 85 45 520 295 275 610 40 0 615 295 19580 300 115 60 435 220 240 545 50 0 610 250 190100 295 110 65 425 225 215 595 90 0 560 275 235120 360 165 105 470 255 250 510 105 0 500 300 210140 340 165 105 460 235 265 460 105 0 530 270 210also that the bias decreases when increasing the sample size, as anticipatedfrom the asymptotic unbiasedness of maximum likelihood estimators. The biasand variance increase when increasing the censoring proportion. The varianceof the maximum likelihood estimator decreases as the sample size increases,this is because of its consistency (Rao 1973). The asymptotic approximation tothe variance of the maximum likelihood estimator provided by the observedinformation matrix seems to hold very well in this model, even for smallsamples with high censoring level. This also shows the high efficiency of themaximum likelihood estimator in this model.For the interval estimation of the location parameter, it is clear that intervalsbased on the Wald statistics perform well. However, as the censoring proportionincreases they tend to be asymmetric, especially when the sample size is small.Intervals based in the Rao statistics are generally conservative, that is, having anactual coverage probability that is less than the nominal. Moreover they tendPertanikaJ. Sci. & Techno\. Vol. 10 No.1, 2002 59Ayman S. BakliziTABLE 3Lower and upper error probabilities of confidence intervals based on the Wald,Rao and likelihood ratio statistics. a = 0.1Location Parameter Scale ParameterWald Rao LR Wald Rao LRCP L U L U L U L U L U L U0.0 20 555 555 340 285 505 515 1340 100 0 735 865 31040 615 535 470 385 590 500 1055 115 25 635 780 27060 590 505 520 450 570 495 980 180 135 700 655 33080 480 445 445 425 475 435 800 230 180 565 580 315100 470 505 430 460 460 495 795 230 240 655 640 365120 535 490 490 460 525 475 810 275 325 640 675 380140 505 500 470 450 500 495 785 280 270 625 625 4250.1 60 570 510 495 460 570 505 930 165 110 705 660 38580 495 440 430 430 455 440 770 190 150 605 575 340100 475 520 415 445 470 490 820 200 215 690 66- 375120 525 490 475 455 520 480 860 235 290 650 650 360140 490 495 460 480 485 495 775 235 270 665 635 3900.3 60 605 440 375 515 515 470 930 170 40 820 605 40080 505 370 375 465 475 420 830 255 105 745 560 425100 465 455 405 520 445 485 820 245 160 725 590 410120 515 435 430 500 500 455 815 240 165 7-5 630 440140 535 490 475 540 525 510 785 315 155 735 580 4500.5 60 705 300 290 740 570 460 935 145 0 955 565 51080 520 285 235 715 435 415 820 165 10 860 555 465100 545 280 255 740 470 455 885 250 40 840 63 410120 615 370 335 730 520 500 820 255 80 700 585 380140 615 385 320 745 495 530 785 285 90 790 540 440to be asymmetric as the censoring proportion increases. On the other hand,interval based on likelihood ratio statistics tend to attain their nominal coverageprobability and are symmetric in almost all situations considered.For the scale parameter, it is clear that intervals based on the Wald and theRao statistics are highly asymmetric, even for samples as large as 140, and hencecannot be recommended for use in practice. While intervals based on thelikelihood ratio statistics still attain their nominal coverage probability and aresymmetric in almost all situations considered.The high asymmetry of Wald and Rao intervals indicates that the actualsampling distributions of the Wald and the Rao statistics are highly skewed(Jennings 1986), and they require a large sample size for the asymptotic chi-squared approximation to hold well. While the likelihood ratio statistics appearsto need much lesser sample size to justify its use.60 PertanikaJ. Sci. & Techno!. Vo!. 10 No. 1,2002Likelihood Based Estimation in the Logistic Model with Time Censored DataThis shows that the likelihood ratio statistics converges of its limitingdistribution much faster than the Wald and the Rao statistics.The likelihood ratio statistics are applicable for one sided interval estimationand for one sided hypotheses testing because of their symmetric lower andupper error probabilities. This is, however, not the case for Wald and Raostatistics unless the sample size is high.For all kinds of intervals considered it appears that as the sample sizeincreases, all intervals tend to have error probabilities that are more symmetricand closer to the nominal ones. Also, larger nominal error probabilities areattained faster than smaller error probabillities.ACKNOWLEDGEMENTThe author is grateful to the referee for his comments which have improvedthe paper.REFERENCESANTLE, c., KLIMKO and W. HARKNESS. 1970. Confidence intervals for the parameters of thelogistic distribution. Biometrica 57:397-402.BAlN, L. J. EASTMAN, M. ENGELHARDT, N. BALAKRISHNAN and C. ANTLE. 1922. ReliabilityEstimation Based on MLEs for Complete and Censored Samples. In Handbook of theLogistic Distribution, ed. N. Balakrishznan. New York: Marcel Dekker.BALAKRISHNAN, N. 1922~andbook of the Logistic Distribution. In Inference and Asymptotics,ed. Barndorff-Nielsen, O. and D.R. Cox. 1994. New York: Marcel Dekker. Chapmanand Hall.Cox, D. R. 1988. Some aspects of conditional and asymptotic inference: a review.Sankhya. A. 3: 314-337.ELPERlN, T. and I. GERTSBAKH. 1987. Maximum likelihood estimation in a Weilbullregression model with type 1 censored data: a Monte Carlo study. Comm. Stat. Simul.Comput. 16(2): 349-371.FISK, P. 1961. The graduation of income distributions. Econometrika 29: 171-185.JENNINGS, D. 1986. Judging inference adequacy in logistic regression.JASA 81(394): 471-476.JENNINGS, D. 1987. How do we judge confidence intervals adequacy? The AmericanStatistician 41(40): 335-337.JOHNSON, N., S. KOTZ and N. BALAKRISHNAN. 1994. Continuous Univariate Distributions.Volume 1. New York: Wiley.HARTER, H. and A. MOORE. 1967. Maximum likelihood estimation from censored samplesof the parameters of the logistic distributions. J Amer. Stat. Assoc. 62:657-684.LAwLESS, J. 1982. StatiModels and Methods for Ligetime Data. New York: Wiley.OLIVER, F. 1964. Methods of estimating the logistic growth function. Appl. Statist. 13: 57-66.PertanikaJ. Sci. & Techno!. Vol. 10 No.1, 2002 61Ayrnan S. BakliziPEARL, R., L. REED and J. KISH. 1940. The logistic curve and the census count of 1940.Science 92: 486-488.PLACKETT, R. 1959. The analysis of Life Test Data. Technometrics I: 9-19.RAo, C. 1973. Linear Statistical Inference and Its Applications. New York: Wiley.Schafer, R. and T. Sheffield. 1973. Inferences on the parameters of the logisticdistribution. Biometrika 29: 449-455.62 PertanikaJ. Sci. & Technol. Vol. 10 o. 1,2002

Likelihood Based Estimation in the Logistic Model with Time Censored Data

http://psasir.upm.edu.my/3763/1/Likelihood_Based_Estimation_in_the_Logistic_Model.pdf

Likelihood Based Estimation in the Logistic Model with Time Censored Data

Abstract

Similar works

Full text

Available Versions

Universiti Putra Malaysia Institutional Repository