90 IEEE TRANSACTIONS ON RELIABILITY, VOL. 52, NO. 1, MARCH 2003 Point and Interval Estimation for Gaussian Distribution, Based on Progressively Type-II Censored Samples N. Balakrishnan, N. Kannan, C. T. Lin, and H. K. T. Ng Abstract The likelihood equations based on a progressively Type-II censored sample from a Gaussian distribution do not provide explicit solutions in any situation except the complete sample case. This paper examines numerically the bias and mean square error of the MLE, and demonstrates that the probability coverages of the pivotal quantities (for location and scale parameters) based on asymptotic -normality are unsatisfactory, and particularly so when the effective sample size is small. Therefore, this paper suggests using unconditional simulated percentage points of these pivotal quantities for constructing -confidence intervals. An approximation of the Gaussian hazard function is used to develop approximate estimators which are explicit and are almost as efficient as the MLE in terms of bias and mean square error; however, the probability coverages of the corresponding pivotal quantities based on asymptotic -normality are also unsatisfactory. A wide range of sample sizes and progressive censoring schemes are used in this study. Index Terms Gaussian distribution, hazard function, maximum likelihood estimator, Monte Carlo simulation, optimal censoring scheme, pivotal quantity, progressive type-ii censoring, statistical-confidence interval. BLUE Cdf hf IMSL MLE MSE pdf Sf ACRONYMS 1 best linear unbiased estimate cumulative distribution function hazard function International Mathematical and Statistical Library maximum likelihood estimate (estimation) mean square error probability density function survivor function. NOTATION location parameter of Gaussian distribution scale parameter of Gaussian distribution Manuscript received October 14, 2000; revised May 28, and June 7, 2001. Responsible Editor: G.-A. Klutke. N. Balakrishnan is with McMaster University (e-mail: Bala@mcmail.cis. mcmaster.ca). N. Kannan is with the University of Texas at San Antonio, San Antonio, TX (e-mail: NKannan@utsa.edu). C. T. Lin is with the Department of Mathematics, Tamkang University, Taiwan (e-mail: Chen@math.tku.edu.tw). H. K. T. Ng is with the Department of Statistical Scince, Southern Methodist University, Dallas, TX (e-mail: Ngh@post.cis.smu.edu). Digital Object Identifier 10.1109/TR.2002.805786 1 The singular and plural of an acronym are always spelled the same. MLE of MLE of approximate MLE of approximate MLE of sample size number of complete failures in a progressivelycensored sample of size order-statistic # from a progressively-censored sample of size number of surviving items withdrawn from the life test at the time of failure # Gaussian pdf with mean and std. dev. standard Gaussian pdf Gaussian Cdf with mean and std. dev. standard Gaussian Cdf : standard Gaussian Sf standard Gaussian hf likelihood function based on a progressively censored sample. I. INTRODUCTION In many life-testing and reliability studies, the experimenter might not always obtain complete information on failure times for all experimental units. For example, individuals in a clinical trial may drop out of the study, or the study may have to be terminated for lack of funds. In an industrial experiment, units may break accidentally. There are, however, many situations in which the removal of units prior to failure is pre-planned. One of the main reasons for this is to save time and cost associated with testing. Data obtained from such experiments are called censored data. The most common censoring schemes are Type-I and Type-II censoring. Consider units placed on a life-test. In conventional Type-I censoring, the experiment continues up to a prespecified time,. Any failures that occur after are not observed. The termination point of the experiment is assumed to be -independent of the failure times. In conventional Type-II censoring, the experimenter decides to terminate the experiment after a prespecified number of units fail. In this scenario, only the smallest lifetimes are observed. In Type-I censoring, the number of failures observed is random and the endpoint of the experiment is fixed, whereas in Type-II censoring the endpoint is random, while the number of 0018-9529/03$17.00 2003 IEEE
BALAKRISHNAN et al.: POINT AND INTERVAL ESTIMATION FOR GAUSSIAN DISTRIBUTION 91 failures is fixed. Numerous articles in the literature have dealt with inference under Type-I and Type-II censoring for various parametric families of distributions; see [1] for details. Conventional Type-I and Type-II censoring schemes do not allow removal of units at points other than the terminal point of the experiment. References [2], [3] are some of the earliest studies of a more general censoring scheme: Fix censoring times.at, remove of the remaining units randomly; the experiment terminates at with units still surviving. This is Progressive Type-I right censoring. Reference [2] discusses the estimation of parameters under this scheme for the Gaussian distribution. This paper considers a generalized censoring scheme: Progressive Type-II Censoring: units are placed on life test. The, and the, are fixed prior to the test. At the first failure, units are randomly removed from the remaining surviving units. At the second failure, units are randomly removed from the remaining units. The test continues until the th failure, when all remaining are removed. If, then which corresponds to the complete sample. If, then which corresponds to the conventional Type-II right censoring scheme. References [4] [6] discuss inference for the Weibull and exponential distributions under Progressive Type-II censoring. Reference [6] derives explicit expressions for the BLUE of the parameters of the 1- and 2-parameter exponential distributions. Reference [7] discusses the construction of reliability sampling plans for the lognormal distribution based on progressively censored samples. For an exhaustive list of references and further details on progressive censoring, see [8]. This paper considers progressively Type-II censored data from a Gaussian distribution. Section II discusses the MLE of the parameters. Section III provides explicit estimators by appropriately approximating the likelihood function. Section IV provides the expressions for the observed Fisher information. Section V provides results of a simulation study to evaluate the performance of these approximate estimators, compared to MLE. Section VI computes the coverage probabilities for pivotal quantities based on asymptotic normality, and provides unconditional simulated percentage points of these pivotal quantities. The likelihood function based on the progressively Type-II censored sample is then a constant (2) From (2), the likelihood equations for and are, respectively: Equations (3) and (4) do not yield explicit solutions for and except for : the complete-sample case. For the complete-sample case, (3) and (4) lead to the standard formulae for the MLE of the Gaussian distribution. For all other censoring schemes, (3) and (4) must be solved numerically to obtain the MLE of the 2 parameters. Reference [2] discusses MLE under the assumption of Progressive Type-I censoring. The likelihood equations are identical, with being replaced by the corresponding censoring time. Reference [2] also discusses some numerical methods for obtaining the MLE of the parameters; see also [1]. III. APPROXIMATE MLE In all cases (except for the complete sample) it is the presence of the term that makes the likelihood equations nonlinear. Some approximate solutions are discussed in [9]. The hf can be approximated by a Taylor series around ; is order statistic # of a progressively Type-II censored sample from the standard Gaussian distribution. From [10]: is the corresponding order statistic # of a progressively Type-II censored sample from the distribution. Then [8]: (1) (3) (4) (5) (6) II. MLE FOR THE GAUSSIAN DISTRIBUTION Let the failure-time distribution be Gaussian with pdf (7) Expand around ; then (keeping only the first 2 terms), (8) (9)
92 IEEE TRANSACTIONS ON RELIABILITY, VOL. 52, NO. 1, MARCH 2003 Replace by : Consider term #2 in (10), (10) Replace by : (21) Return to the Taylor expansion: (11) The last 3 terms can be shown to vanish; thus: (22) (12) (13) Using (11) (13), approximate the likelihood equations (3) and (4) by (14) or (23) Equation (14) may be written as which implies Equation (15) may be written as (15) (16) (17) (18) (19) (20) Equation (23) is quadratic in Because, with roots:, only one root is admissible; hence (24) The approximate MLE can easily be obtained from (17) and (24). Now, the logical question: How good are the approximate estimators? It would be of interest to evaluate the efficiency of these estimators compared to the MLE obtained by solving the full likelihood equations numerically. And the approximate solution might provide an excellent starting value for the iterative solution of the likelihood equations. It is important to mention that a similar approximate MLE has been presented in [9]. First, [9] considers progressively Type-I censored samples; then the approximation is developed by considering a random interval that is likely to cover (defined in Section II). The hf of the Gaussian distribution is then linearly approximated by using the 2 endpoints of this random interval. Thus the approximate MLE in [9] is different (not based on properties of uniform order statistics). IV. OBSERVED FISHER INFORMATION Formulas are derived to compute the observed Fisher information for the full and approximate equations. These enable the development of pivotal quantities based on the limiting Gaussian distribution and the examination of the probability coverages through simulation.
BALAKRISHNAN et al.: POINT AND INTERVAL ESTIMATION FOR GAUSSIAN DISTRIBUTION 93 TABLE I MEANS, VARIANCES, COVARIANCES OF THE MLE AND THE APPROXIMATE MLE From the log-likelihood function in (2), derive the observed Fisher Information as (30) (25) Let (26) The observed information matrix can be inverted to obtain the asymptotic variance-covariance matrix of the estimators as Similarly, from the approximate likelihood equations: (27) (28) (29) (31) Similarly,,, can be obtained from the observed Fisher information for the approximate equations. V. SIMULATION RESULTS A simulation study compared the performance of the approximate estimators with the MLE. Type-II censored samples were generated progressively from the standard Gaussian distribution using the algorithm in [10]. The approximate MLE were computed from (24) and (17). The MLE of the parameters were obtained by solving the nonlinear equations (3) and (4) using
94 IEEE TRANSACTIONS ON RELIABILITY, VOL. 52, NO. 1, MARCH 2003 TABLE II 95% AND 90% COVERAGE PROBABILITIES FOR THE MLE AND APPROXIMATE MLE the IMSL nonlinear equation solver, in which the approximate MLE were used as starting values for the iterations. Table I provides the average values of the estimates, their variances, and covariance. The values of variances and covariances determined from the observed Fisher information matrix were averaged, and those values are presented in Table I for comparison. All the averages were computed over 1000 simulations. The MSE of these estimators are comparable to the variances of the BLUE in [8]. Table I shows that the approximate MLE and the MLE are almost identical in terms of both bias and variance. The approximate MLE are almost as efficient as the MLE for all samplesizes and censoring schemes. As the effective sample proportion increases, the bias and variance of the estimators reduce appreciably. For a fixed and, one can determine the censoring scheme that is most efficient. For almost all choices, the seems to provide the smallest bias and variance for the estimates. Even though it is difficult to analytically-justify the optimality of this censoring scheme, extensive simulations show that: for all choices of and, this particular censoring scheme is far superior to all others considered, including traditional right-censoring schemes. Also, a study [8] for the BLUE reveals a similar result for the optimal censoring scheme. VI. COVERAGE PROBABILITIES To compute -confidence intervals, or to conduct tests of hypotheses, for the location and scale parameters, one must construct pivotal quantities, viz, a statistic whose distribution is -independent of any unknown parameters. The asymptotic -normality of gives the asymptotic distribution of (32) to be standard Gaussian. The are pivotal quantities because their distributions do not depend on the unknown location and scale parameters. Monte Carlo simulations were used to find the probability coverages of which are approximately 90% and 95%, respectively. This process was repeated for the approximate estimators as well. Table II provides results of the simulation for the MLE and the approximate estimators. When is unknown, the probability coverages are extremely unsatisfactory, especially when the effective sample fraction ( ) is small. If is known, then the coverage probabilities for are close to the required levels. In most practical situations, however, is unknown; hence using the Gaussian approximation for the corresponding pivotal quantity is not advisable. The distributions of the pivotal quantities are extremely skewed. Table III provides the unconditional fractional points for these pivotal quantities determined through simulation. For small sample sizes, the fractional points are very different from what might be anticipated if the distribution were Gaussian. These simulated fractional points allow one to construct -confidence intervals for and. For example, let
BALAKRISHNAN et al.: POINT AND INTERVAL ESTIMATION FOR GAUSSIAN DISTRIBUTION 95 TABLE III (2.5, 97.5) AND (5.0, 95.0) PERCENTAGE POINTS OF THE PIVOTAL QUANTITIES BASED ON MLE AND ON APPROXIMATE MLE and denote the lower and upper fractional points determined through simulation for the pivotal quantity, then [10] N. Balakrishnan and R. A. Sandhu, A simple simulational algorithm for generating progressive type-ii censored samples, Amer. Statist., vol. 49, pp. 229 230, 1995. forms a % -confidence interval for when is unknown. REFERENCES [1] A. C. Cohen, Truncated and Censored Samples: Theory and Applications: Marcel Dekker, 1991. [2], Progressively censored samples in life testing, Technometrics, vol. 5, pp. 327 329, 1963. [3], Life testing and early failure, Technometrics, vol. 8, pp. 539 549, 1966. [4] N. R. Mann, Exact three-order-statistic confidence bounds on reliable life for a Weibull model with progressive censoring, J. Amer. Statist. Assoc., vol. 64, pp. 306 315, 1969. [5], Best linear invariant estimation for Weibull parameters under progressive censoring, Technometrics, vol. 13, pp. 521 533, 1971. [6] R. Viveros and N. Balakrishnan, Interval estimation of parameters of life from progressively censored data, Technometrics, vol. 36, pp. 84 91, 1994. [7] U. Balasooriya and N. Balakrishnan, Reliability sampling plans for lognormal distribution, based on progressively-censored samples, IEEE Trans. Rel., vol. 49, pp. 199 203, 2000. [8] N. Balakrishnan and R. Aggarwala, Progressive Censoring: Theory, Methods, and Applications: Birkhauser, 2000. [9] M. L. Tiku, W. Y. Tan, and N. Balakrishnan, Robust Inference: Marcel Dekker, 1986. N. Balakrishnan is a Professor of Statistics at McMaster University, Hamilton, Canada. He received the B.Sc. in 1976 and the M.Sc. in 1978 in Statistics from the University of Madras, Tamil Nadu, and the Ph.D. in 1981 in Statistics from the Indian Institute of Technology, Kanpur. His areas of interest include order statistics, estimation theory, robust inference, and classification analysis. N. Kannan is an Associate Professor of Statistics at the University of Texas at San Antonio, USA. She received the M.A. in 1988 in Mathematics from the University of Pittsburgh, and the Ph.D. in 1992 from Penn State University. Her areas of interest include statistical signal processing, reliability theory, survival analysis, and multivariate analysis. C. T. Lin is a Professor in the Department of Mathematics, Tamkang University, Taiwan. She received the Ph.D. in 1993 from Florida State University. Her areas of interest include reliability and applications of scan statistics. H. K. T. Ng is an Assistant Professor in the Department of Statistical Science, Southern Methodist University, Dallas, TX, USA. He received the B.Sc. in 1997 and M.Phil. in 1999 in Statistics from the Chinese University of Hong Kong, and the M.Sc. in 2000 and Ph.D. in 2002 in Statistics from McMaster University. His research interests include order statistics, reliability theory, and analysis of censored data.