## 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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## 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* (2021). https://doi.org/10.1007/s11009-021-09907-0

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### Keywords

- Asymptotic distribution
- Exact distribution
- Exponential distribution
- Likelihood ratio test
- Time-constrained life-testing experiment

### Mathematics Subject Classification

- 62E15
- 62F03
- 62N01
- 62N05