Abstract
The likelihood ratio test is one of the commonly used procedures for hypothesis testing. Several results on likelihood ratio test have been discussed for testing the scale parameter of an exponential distribution under complete and censored data; however, all of them are based on approximations of the involved null distributions. In this paper, we first derive the exact distribution of the likelihood ratio statistic for testing the scale parameter of an exponential distribution based on a time-constrained life-testing experiment. We also obtain the asymptotic distribution, which is useful in the case of a large sample size. We then discuss the derivation of its power function. Next, we consider the likelihood ratio test of \(\theta _2=\gamma _0\theta _1\) when data are obtained from two exponential distributions based on a time-constrained life-testing experiment. Here again, we derive both exact and asymptotic distributions of the likelihood ratio statistic and then use them to determine the reject region as well as the power function. Monte Carlo simulations are then carried out to evaluate the performance of the inferential methods developed here. Finally, some examples are used to illustrate all the inferential results.
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References
Arnold BC, Balakrishnan N, Nagaraja HN (1992) A First Course in Order Statistics. Wiley, New York
Balakrishnan N, Basu AP (1996) The Exponential Distribution: Theory, Methods, and Applications. Gordon and Breach, Philadelphia, PA
Balakrishnan N, Cohen AC (1991) Order Statistics and Inference: Estimation Methods. Academic Press, Boston, MA
Balakrishnan N, Iliopoulos G (2009) Stochastic monotonicity of the MLE of exponential mean under different censoring schemes. Ann Inst Stat Math 61:753–772
Balakrishnan N, Zhu X (2016) Exact likelihood-based point and interval estimation for Laplace distribution based on Type-II right censored samples. J Stat Comput Simul 86:29–54
Bartholomew DJ (1963) The sampling distribution of an estimate arising in the life testing. Technometrics 5:361–374
Bhattacharyya GK, Mehrotra GK (1981) On testing equality of two exponential distributions under combined Type-II censoring. J Am Stat Assoc 76:886–894
Cantor AB, Knapp RG (1985) A test of the equality of survival distributions based on paired observations from conditionally independent exponential distributions. IEEE Trans Reliab 34:342–346
Cox DR (1959) The analysis of exponentially distributed life-times with two types of failure. J R Stat Soc Series B 21:411–421
Cole KN, Nagarsenker BN, Nagarsenker PB (1987) A test for equality of exponential distributions based on Type-II censored samples. IEEE Trans Reliab 36:94–97
Epstein B (1958) The exponential distribution and its role in life testing. Technical Report No.2, Department of Mathematics, Wayne State University, Detroit, MI, USA
Knapp RG, Gross AJ, Cantor AB (2007) A likelihood ratio test of the equality of paired survival data with censoring. Biometrics 28:665–672
Kharrati-Kopaei M, Malekzadeh A (2019) On the exact distribution of the likelihood ratio test for testing the homogeneity of scale parameters of several two-parameter exponential distributions: complete and censored samples. Metrika 82: 409–427
Nagarsenker BN, Nagarsenker PB (1985) Distribution of LRT for testing the equality of several 2-parameter exponential distributions. IEEE Trans Reliab 34:65–68
Page ES (1955) A test for a change in a parameter occurring at an unknown point. Biometrics 42:523–527
Paulson E (1941) On certain likelihood-ratio tests associated with the exponential distribution. Ann Inst Stat Math 12:301–306
Perng SK (1978) A test for equality of two exponential distributions. Stat Neerl 32:93–102
Proschan F (1963) Theoretical explanation of observed decreasing failure rate. Technometrics 5:375–383
Shetty BN, Joshi PC (1989) Likelihood ratio test for testing equality of location parameters of two exponential distributions from doubly censored samples. Commun Stat Theory Methods 18:2063–2072
Singh N (1983) The likelihood ratio test for the equality of location parameters of \(k(\ge 2)\) exponential populations based on Type-II censored samples. Technometrics 25:193–195
Singh N (1986) An exact test for comparing location parameters of \(k\) exponential distributions with unequal scales based on Type-II censored data. Technometrics 28:157–164
Singh N, Narayan P (1983) The likelihood ratio test for the equality of \(k(\ge 3)\) two-parameter exponential populations based on Type-II censored samples. J Stat Comput Simul 18:287–297
Stehlik M, Wagner H (2011) Exact likelihood ratio testing for homogeneity of exponential distribution. Commun Stat Simul Comput 40:663–684
Xiong C (1999) Inferences on a simple step-stress model with type-II censored exponential data. IEEE Trans Reliab 47:142–146
Zhu X, Balakrishnan N (2016) Exact inference for Laplace quantile, reliability and cumulative hazard functions based on Type-II censored data. IEEE Trans Reliab 65:164–178
Zhu X, Balakrishnan N, Zhou Y (2020) Exact likelihood-ratio tests for a simple step-stress cumulative exposure model with censored exponential data. Methodol Comput Appl Probab 22:497–509
Funding
The first author was supported by National Natural Science Foundation of China–Young Scientists Fund [No. 11801459], Jiangsu Science and Technology Programme–The Young-Scholar Programme [No. BK20180241] and the Research Development Fund of Xi’an-Jiaotong Liverpool University [No. RDF-17-01-20].
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Zhu , X., Balakrishnan, N. & So, HY. Exact One- and Two-Sample Likelihood Ratio Tests based on Time-Constrained Life-Tests from Exponential Distributions. Methodol Comput Appl Probab 24, 2009–2028 (2022). https://doi.org/10.1007/s11009-021-09907-0
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DOI: https://doi.org/10.1007/s11009-021-09907-0
Keywords
- Asymptotic distribution
- Exact distribution
- Exponential distribution
- Likelihood ratio test
- Time-constrained life-testing experiment