# Bootstrap Confidence Intervals for Common Signal-to-noise Ratio of Two-parameter Exponential Distributions

### Abstract

Signal-to-noise ratio (SNR) is a reciprocal of coefffficient of variation. The SNR is a measure of mean relative to the variability. Confidence procedures for common SNR of two-parameter exponential distributions were developed using generalized confidence interval (GCI) approach, large sample (LS) approach, adjusted method of variance estimates recovery (Adjusted MOVER) approach, and bootstrap approaches based on standard bootstrap (SB) and parametric bootstrap (PB). The performances of all approaches are measured by coverage probability and average length. Simulation studies show that all approaches have the coverage probabilities below the nominal confidence level of 0.95 when the common SNR is negative value. However, the coverage probabilities of all approaches are greater than the nominal confidence level of 0.95 when the common SNR is positive value. Moreover, the LS and AM approaches are the conservative confidence intervals. In addition, the GCI and PB approaches provide the confidence intervals with coverage probabilities close to the nominal confidence level of 0.95 when the sample sizes are large and the common SNR is positive value. The GCI and PB approaches are recommended to estimate the confidence intervals for the common SNR of two-parameter exponential distributions. Finally, all proposed approaches are employed in the data of the survival days of lung cancer patients for a demonstration.### References

J.F. Lawless, Prediction intervals for the two-parameter exponential distribution, Technometrics, vol. 19, pp. 469–472,

A. Baten, and A. Kamil, Inventory management systems with hazardous items of two-parameter exponential distribution, Journal of Social Sciences, vol. 5, pp. 183–187, 2009.

C. Petropoulos, New classes of improved confidence intervals for the scale parameter of two-parameter exponential

distribution, Statistical Methodology, vol. 8, pp. 401–410, 2011.

L. Jiang, and A.C.M. Wong, Interval estimations for the two-parameter exponential distribution, Journal of Probability

and Statistics, vol.2012, pp. 1–8, 2012.

W. Thangjai, and S. Niwitpong, Confidence intervals for the weighted coefficients of variation of two-parameter

exponential distributions, Cogent Mathematics, vol.4, pp. 1–16, 2017.

L. Saothayanun, and W. Thangjai, Confidence intervals for the signal to noise ratio of two-parameter exponential

distribution, Studies in Computational Intelligence, vol.760, pp. 255–265, 2018.

W. Thangjai, S. Niwitpong, and S. Niwitpong, Simultaneous confidence intervals for all differences of means of

two-parameter exponential distributions, Studies in Computational Intelligence, vol. 760, pp. 298–308, 2018.

W. Thangjai, and S. Niwitpong, Simultaneous confidence intervals for all differences of coefficients of variation of

two-parameter exponential distributions, Thailand Statistician, vol. 18, pp. 135–149, 2020.

C. Chesneau, H.S. Bakouch, and M.N. Khan, A weighted transmuted exponential distribution with environmental

applications, Statistics, Optimization and Information Computing, vol. 8, pp. 36–53, 2020.

K.K. Sharma, and H. Krishna, Asymptotic sampling distribution of inverse coefficient of variation and its applications,

IEEE Transactions on Reliability, vol.43, pp. 630–633, 1994.

F. George, and B.M.G. Kibria, Confidence intervals for estimating the population signal to noise ratio: a simulation

study, Journal of Applied Statistics, vol.39, pp. 1225–1240, 2012.

A.N. Albatineh, B.M.G. Kibria, and B. Zogheib, Asymptotic sampling distribution of inverse coefficient of variation

and its applications: Revisited, International Journal of Advanced Statistics and Probability, vol.2, pp. 15–20, 2014.

A.N. Albatineh, I. Boubakari, and B.M.G. Kibria, New confidence interval estimator of the signal to noise ratio based

on asymptotic sampling distribution, Communications in Statistics - Theory and Methods, vol.46, pp. 574–590, 2017.

S. Niwitpong, Confidence intervals for functions of signal-to-noise ratios of normal distributions, Studies in

Computational Intelligence, vol.760, pp. 196–212, 2018.

W. Thangjai, and S. Niwitpong, Confidence intervals for the signal-to-noise ratio and difference of signal-to-noise

ratios of log-normal distribution, Stats, Vol.2, pp. 164–173, 2019.

W. Thangjai, and S. Niwitpong, Confidence intervals for the signal-to-noise ratio and difference of signal-to-noise

ratios of gamma distributions, Advance and Applications in Mathematical Sciences, vol.18, pp. 503–520, 2019.

W. Thangjai, and S. Niwitpong, Confidence intervals for common signal-to-noise ratio of several log-normal

distributions, Iranian Journal of Science and Technology, Transactions A: Science, vol. 44, pp. 99–107, 2020.

S. Weerahandi, Generalized confidence intervals, Journal of the American Statistical Association, vol. 88, pp. 899–905,

L. Tian, Inferences on the common coefficient of variation, Statistics in Medicine, vol. 24, pp. 2213–2220, 2005.

L. Tian, and J. Wu, Inferences on the common mean of several log-normal populations: the generalized variable

approach, Biometrical Journal, vol. 49, pp. 944–951, 2007.

R.D. Ye, T.F. Ma, and S.G. Wang, Inferences on the common mean of several inverse Gaussian populations,

Computational Statistics and Data Analysis, vol. 54, pp. 906–915, 2010.

C.K. Ng, Inference on the common coefficient of variation when populations are lognormal: A simulation-based

approach, Journal of Statistics: Advances in Theory and Applications, vol. 11, pp. 117–134, 2014.

G.Y. Zou, and A. Donner, Construction of confidence limits about effect measures: A general approach, Statistics in

Medicine, vol. 27, pp. 1693–1702, 2008.

G.Y. Zou, J. Taleban, and C.Y. Hao, Confidence interval estimation for lognormal data with application to health

economics, Computational Statistics and Data Analysis, vol. 53, pp. 3755–3764, 2009.

W. Thangjai, S. Niwitpong, and S. Niwitpong, Confidence intervals for the common mean of several normal

populations, Robustness in Econometrics, vol. 692, pp. 321–331, 2017.

J. Chachi, Bootstrap approach to the one-sample and two-sample test of variances of a fuzzy random variable,

Statistics, Optimization and Information Computing, vol. 5, pp. 188–199, 2017.

F.A. Graybill, and R.B. Deal, Combining unbiased estimators, Biometrics, vol. 15, pp. 543–550, 1959.

V. Maurya, A. Goyal, and A.N. Gill, Simultaneous testing for the successive differences of exponential location

parameters under heteroscedasticity, Statistics and Probability Letters, vol. 81, pp. 1507–1517, 2011.

*Statistics, Optimization & Information Computing*,

*10*(3), 858-872. https://doi.org/10.19139/soic-2310-5070-931

- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).