zbMATH — the first resource for mathematics

Type-I hybrid censoring of multiple samples. (English) Zbl 1432.62329
Summary: We consider inference based on multiple samples subject to type-I hybrid censoring. The original data is supposed to be obtained as failure times from $$k \geq 2$$ independent type-I censored sequential $$r$$-out-of-$$n$$ systems. For the mean $$\vartheta$$ of exponentially distributed lifetimes, we derive the distribution of the maximum likelihood estimator $$\hat{\vartheta}$$ given the condition that at least one failure has been observed in the $$k$$ samples. As special cases, we get results for particular multi-sample hybrid censoring models, that is, for multi-sample type-I hybrid censoring and multi-sample type-I progressive hybrid censoring, respectively. As a tool, we need expressions for the convolutions of B-splines in terms of iterative divided differences. The resulting expressions are used to determine the exact density functions efficiently as well as to construct exact confidence intervals for $$\vartheta$$. Furthermore, we propose two approximated two-sided confidence intervals as an alternative for larger sample sizes. The results are illustrated by data as well as by simulations.

MSC:
 62N01 Censored data models 90C90 Applications of mathematical programming
StInt
Full Text:
References:
 [1] Kamps, U., (A Concept of Generalized Order Statistics. A Concept of Generalized Order Statistics, Teubner Skripten Zur Mathematischen Stochastik (1995), Teubner: Teubner Stuttgart) · Zbl 0851.62035 [2] Kamps, U., A concept of generalized order statistics, J. Statist. Plann. Inference, 48, 1-23 (1995) · Zbl 0838.62038 [3] Cramer, E.; Kamps, U., Sequential $$k$$-out-of-$$n$$ systems, (Balakrishnan, N.; Rao, C. R., Handbook of Statistics: Advances in Reliability, Vol. 20 (2001), Elsevier: Elsevier Amsterdam), 301-372, chap. 12 · Zbl 0988.62027 [4] Cramer, E., Sequential Order Statistics. Wiley StatsRef: Statistics Reference Online (2016), John Wiley & Sons, Ltd. [5] Burkschat, M.; Cramer, E.; Górny, J., Type-I censored sequential $$k$$-out-of-$$n$$ systems, Appl. Math. Model., 40, 8156-8174 (2016) · Zbl 07163007 [6] Epstein, B., Truncated life tests in the exponential case, Ann. Math. Stat., 25, 555-564 (1954) · Zbl 0058.35104 [7] Chen, S.-M.; Bhattacharyya, G. K., Exact confidence bounds for an exponential parameter under hybrid censoring, Comm. Statist. Theory Methods, 16, 2429-2442 (1987) · Zbl 0628.62097 [8] Gupta, R. D.; Kundu, D., Hybrid censoring schemes with exponential failure distribution, Comm. Statist. Theory Methods, 27, 3065-3083 (1998) · Zbl 1008.62679 [9] Childs, A.; Chandrasekar, B.; Balakrishnan, N.; Kundu, D., Exact likelihood inference based on Type-I and Type-II hybrid censored samples from the exponential distribution, Ann. Inst. Statist. Math., 55, 319-330 (2003) · Zbl 1049.62021 [10] Kundu, D.; Joarder, A., Analysis of Type-II progressively hybrid censored data, Comput. Statist. Data Anal., 50, 2509-2528 (2006) · Zbl 1284.62605 [11] Kundu, D., On hybrid censored Weibull distribution, J. Statist. Plann. Inference, 137, 2127-2142 (2007) · Zbl 1120.62081 [12] Balakrishnan, N.; Kundu, D., Hybrid censoring: Models, inferential results and applications, Comput. Statist. Data Anal., 57, 166-209 (2013) · Zbl 1365.62364 [13] Cramer, E., Comments on ‘Hybrid censoring: Models, inferential results and applications’ by N. Balakrishnan and D. Kundu, Comput. Statist. Data Anal., 57, 1995, 201-204 (2013) [14] Cramer, E.; Balakrishnan, N., On some exact distributional results based on Type-I progressively hybrid censored data from exponential distributions, Stat. Methodol., 10, 128-150 (2013) · Zbl 1365.62061 [15] Rastogi, M. K.; Tripathi, Y. M., Inference on unknown parameters of a Burr distribution under hybrid censoring, Statist. Papers, 54, 619-643 (2013) · Zbl 1307.62059 [16] Balakrishnan, N.; Cramer, E., (The Art of Progressive Censoring: Applications to Reliability and Quality. The Art of Progressive Censoring: Applications to Reliability and Quality, Statistics for Industry and Technology (2014), Springer: Springer New York) · Zbl 1365.62001 [17] Balakrishnan, N.; Cramer, E.; Iliopoulos, G., On the method of pivoting the CDF for exact confidence intervals with illustration for exponential mean under life-test with time constraints, Statist. Probab. Lett., 89, 124-130 (2014) · Zbl 1288.62046 [18] Wu, M.; Shi, Y.; Sun, Y., Inference for accelerated competing failure models from Weibull distribution under Type-I progressive hybrid censoring, J. Comput. Appl. Math., 263, 423-431 (2014) · Zbl 1381.62267 [19] Arabi Belaghi, R.; Noori Asl, M., Estimation based on progressively type-I hybrid censored data from the Burr XII distribution, Statist. Papers, 60, 411-453 (2019) · Zbl 1419.62027 [20] Górny, J., A New Approach to Hybrid Censoring (2017), RWTH Aachen University: RWTH Aachen University Aachen, Germany, (Ph.D.thesis) [21] Noori Asl, M.; Arabi Belaghi, R.; Bevrani, H., Classical and Bayesian inferential approaches using Lomax model under progressively type-I hybrid censoring, J. Comput. Appl. Math., 343, 2006, 397-412 (2018) · Zbl 1392.62292 [22] Bai, X.; Shi, Y.; Liu, Y.; Liu, B., Reliability estimation of multicomponent stress – strength model based on copula function under progressively hybrid censoring, J. Comput. Appl. Math., 344, 100-114 (2018) · Zbl 1460.62163 [23] Górny, J.; Cramer, E., From B-spline representations to gamma representations in hybrid censoring, Statist. Papers (2019), in press · Zbl 1432.62330 [24] Cramer, E.; Kamps, U., Sequential order statistics and $$k$$-out-of-$$n$$ systems with sequentially adjusted failure rates, Ann. Inst. Statist. Math., 48, 2019, 535-549 (1996) · Zbl 0925.62424 [25] Cramer, E.; Kamps, U., Maximum likelihood estimation with different sequential $$k$$-out-of-$$n$$ systems, (Kahle, W.; von Collani, E.; Franz, J.; Jensen, U., Advances in Stochastic Models for Reliability, Quality and Safety (1998), Birkhäuser Boston: Birkhäuser Boston Boston, MA), 101-111 · Zbl 0947.62016 [26] Cramer, E.; Kamps, U., Sequential $$k$$-out-of-$$n$$ systems with Weibull components, Economic Quality Control, 13, 227-239 (1998) · Zbl 0959.62090 [27] Cramer, E.; Kamps, U., Estimation with sequential order statistics from exponential distributions, Ann. Inst. Statist. Math., 53, 307-324 (2001) · Zbl 0999.62080 [28] Schenk, N.; Burkschat, M.; Cramer, E.; Kamps, U., Bayesian estimation and prediction with multiply Type-II censored samples of sequential order statistics from one- and two-parameter exponential distributions, J. Statist. Plann. Inference, 141, 1575-1587 (2011) · Zbl 1204.62042 [29] Bhattacharyya, G. K., Inferences under two-sample and multi-sample situations, (Balakrishnan, N.; Basu, A. P., The Exponential Distribution (1995), Gordon and Breach: Gordon and Breach Amsterdam), 93-118 [30] Balakrishnan, N.; Cramer, E.; Kamps, U.; Schenk, N., Progressive type II censored order statistics from exponential distributions, Statistics, 35, 537-556 (2001) · Zbl 1008.62051 [31] Burkschat, M.; Kamps, U.; Kateri, M., Estimation in a model of sequential order statistics with ordered hazard rates, (Choudhary, P. K.; Nagaraja, C. H.; Ng, H. K.T., Ordered Data Analysis, Modeling and Health Research Methods: In Honor of H. N. Nagaraja’s 60th Birthday (2015), Springer: Springer Cham), 105-119 · Zbl 1341.62074 [32] Kateri, M.; Kamps, U.; Balakrishnan, N., A meta-analysis approach for step-stress experiments, J. Statist. Plann. Inference, 139, 2907-2919 (2009) · Zbl 1168.62092 [33] Kateri, M.; Kamps, U.; Balakrishnan, N., Multi-sample simple step-stress experiment under time constraints, Stat. Neerl., 64, 77-96 (2010) [34] Bedbur, S.; Kamps, U.; Kateri, M., Meta-analysis of general step-stress experiments under repeated Type-II censoring, Appl. Math. Model., 39, 2261-2275 (2015) · Zbl 1443.62318 [35] Volterman, W.; Balakrishnan, N., Exact nonparametric confidence, prediction and tolerance intervals based on multi-sample Type-II right censored data, J. Statist. Plann. Inference, 140, 3306-3316 (2010) · Zbl 1204.62081 [36] Volterman, W.; Balakrishnan, N.; Cramer, E., Exact nonparametric meta-analysis for multiple independent doubly Type-II censored samples, Comput. Statist. Data Anal., 56, 1243-1255 (2012) · Zbl 1241.62071 [37] Volterman, W.; Balakrishnan, N.; Cramer, E., Exact meta-analysis of several independent progressively type-II censored data, Appl. Math. Model., 38, 949-960 (2014) · Zbl 1428.62192 [38] Balakrishnan, N.; Volterman, W.; Zhang, L., A meta-analysis of multisample Type-II censored data with parametric and nonparametric results, IEEE Trans. Reliab., 62, 2-12 (2013) [39] Bedbur, S.; Johnen, M.; Kamps, U., Inference from multiple samples of Weibull sequential order statistics, J. Multivariate Anal., 169, 381-399 (2019) · Zbl 1409.62098 [40] Proschan, F., Theoretical explanation of observed decreasing failure rate, Technometrics, 5, 375-383 (1963) [41] Nelson, W. B., Applied Life Data Analysis (2004), John Wiley & Sons Inc.: John Wiley & Sons Inc. Hoboken, New Jersey · Zbl 1054.62109 [42] Casella, G.; Berger, R. L., Statistical Inference (2002), Duxbury: Duxbury Australia [43] Hahn, G. J.; Meeker, W. Q.; Escobar, L. A., Statistical Intervals: A Guide for Practitioners (2017), John Wiley & Sons: John Wiley & Sons New York · Zbl 1395.62002 [44] van Bentum, T.; Cramer, E., Stochastic monotonicity of MLEs of the mean for exponentially distributed lifetimes under sequential hybrid censoring, Statist. Probab. Lett., 148, 1-8 (2019) · Zbl 1450.62126 [45] de Boor, C., On calculating with B-splines, J. Approx. Theory, 6, 50-62 (1972) · Zbl 0239.41006 [46] de Boor, C., B(asic)-spline basics, (Piegl, L., Fundamental Developments of Computer Aided Geometric Modeling (1993), Academic Press: Academic Press London) [47] de Boor, C., Divided differences, Surveys in Approximation Theory, 1, 1993, 46-69 (2005) · Zbl 1071.65027 [48] de Boor, C., A Practical Guide to Splines (2001), Springer: Springer Berlin, New York · Zbl 0987.65015 [49] Schumaker, L. L., Spline Functions: Basic Theory (2007), Cambridge University Press: Cambridge University Press Cambridge, UK · Zbl 1123.41008 [50] Cramer, E.; Burkschat, M.; Górny, J., On the exact distribution of the MLEs based on Type-II progressively hybrid censored data from exponential distributions, J. Stat. Comput. Simul., 86, 2036-2052 (2016) [51] Golparvar, L.; Parsian, A., Inference on proportional hazard rate model parameter under Type-I progressively hybrid censoring scheme, Comm. Statist. Theory Methods, 45, 7258-7274 (2016) · Zbl 1349.62479 [52] Górny, J.; Cramer, E., Exact likelihood inference for exponential distributions under generalized progressive hybrid censoring schemes, Stat. Methodol., 29, 2016, 70-94 (2016) · Zbl 07035799 [53] Górny, J.; Cramer, E., Modularization of hybrid censoring schemes and its application to unified progressive hybrid censoring, Metrika, 81, 173-210 (2018) · Zbl 1459.62182 [54] Górny, J.; Cramer, E., Type-I hybrid censoring of uniformly distributed lifetimes, Comm. Statist. Theory Methods, 48, 412-433 (2019) [55] Górny, J.; Cramer, E., Exact inference for a new flexible hybrid censoring scheme, J. Indian Soc. Probab. Statist., 19, 169-199 (2018) [56] Górny, J.; Cramer, E., A volume based approach to establish b-spline based expressions for density functions and its application to progressive hybrid censoring, J. Korean Stat. Soc., 38, 340-355 (2019) · Zbl 1428.62105 [57] Folland, G. B., Real Analysis: Modern Techniques and their Applications (1999), John Wiley & Sons, Inc: John Wiley & Sons, Inc New York · Zbl 0924.28001 [58] Strøm, K., On convolutions of B-splines, J. Comput. Appl. Math., 55, 1-29 (1994) · Zbl 0830.65006 [59] Bărbosu, D., Two dimensional divided differences revisited, Creative Mathematics and Informatics, 17, 1995, 1-7 (2008) · Zbl 1199.41002 [60] Pop, O. T.; Bărbosu, D., Two dimensional divided differences with multiple knots, Analele Stiint. Univ. Ovidius Constanta, 17, 181-190 (2009) · Zbl 1199.41012 [61] Roozegar, R.; Jafari, A. A., A concise formula for two dimensional divided differences with multiple knots, J. Math. Ext., 7, 63-69 (2013) · Zbl 1314.41026 [62] Cramer, E.; Kamps, U., Marginal distributions of sequential and generalized order statistics, Metrika, 58, 293-310 (2003) · Zbl 1042.62048 [63] Burkschat, M.; Lenz, B., Marginal distributions of the counting process associated with generalized order statistics, Comm. Statist. Theory Methods, 38, 2089-2106 (2009) · Zbl 1167.62047 [64] Balakrishnan, N.; Iliopoulos, G., Stochastic monotonicity of the MLE of exponential mean under different censoring schemes, Ann. Inst. Statist. Math., 61, 753-772 (2009) · Zbl 1332.62384 [65] Balakrishnan, N.; Iliopoulos, G., Stochastic monotonicity of the MLEs of parameters in exponential simple step-stress models under Type-I and Type-II censoring, Metrika, 72, 89-109 (2010) · Zbl 1189.62160 [66] Volterman, W.; Arabi Belaghi, R.; Balakrishnan, N., Joint records from two exponential populations and associated inference, Comput. Statist., 33, 549-562 (2018) · Zbl 1417.62127 [67] Shaked, M.; Shanthikumar, J. G., Stochastic Orders (2007), Springer: Springer New York [68] Cramer, E.; Górny, J.; Laumen, B., Multi-sample progressive Type-I censoring of exponentially distributed lifetimes (2019), submitted for publication [69] Sundberg, R., Comparison of confidence procedures for type I censored exponential lifetimes, Lifetime Data Anal., 7, 393-413 (2001) · Zbl 1116.62411 [70] Cohen, A. C., MLEs under censoring and truncation and inference, (Balakrishnan, N.; Basu, A. P., The Exponential Distribution: Theory, Methods, and Applications (1995), Gordon and Breach: Gordon and Breach Newark, NJ), 33-51 [71] Singh, N., A simple and asymptotically optimal test for the equality of $$K(\geq 2)$$ exponential distributions based on type II censored samples, Comm. Statist. Theory Methods, 14, 1615-1625 (1985) · Zbl 0595.62012 [72] Thiagarajah, K.; Paul, S. R., Testing for the equality of scale parameters of $$K(\geq 2)$$ exponential populations based on complete and type II censored samples, Comm. Statist. Simulation Comput., 19, 891-902 (1990) · Zbl 0850.62217
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.