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Exact Likelihood-Based Point and Interval Estimation for Lifetime Characteristics of Laplace Distribution Based on a Time-Constrained Life-Testing Experiment

  • Xiaojun Zhu
  • N. Balakrishnan
Chapter

Abstract

In this paper, we first derive explicit expressions for the MLEs of the location and scale parameters of the Laplace distribution based on a Type-I right-censored sample arising from a time-constrained life-testing experiment by considering different cases. We derive the conditional joint MGF of these MLEs and use them to derive the bias and MSEs of the MLEs for all the cases. We then derive the exact conditional marginal and joint density functions of the MLEs and utilize them to develop exact conditional CIs for the parameters. We also briefly discuss the MLEs of reliability and cumulative hazard functions and the construction of exact CIs for these functions. Next, a Monte Carlo simulation study is carried out to evaluate the performance of the developed inferential results. Finally, some examples are presented to illustrate the point and interval estimation methods developed here under a time-constrained life-testing experiment.

Acronyms and Abbreviations

CDF

Cumulative density function

CI

Confidence interval

K–M curve

Kaplan–Meier curve

i.i.d.

Independent and identically distributed

MGF

Moment generating function

MLE

Maximum likelihood estimator

MSE

Mean square error

PDF

Probability density function

P–P plot

Probability–probability plot

Q–Q plot

Quantile–quantile plot

SE

Standard error

Notation

n

Sample size

r

Number of smallest order statistics observed in the Type-II censored sample

\(X_{i:n}\)

The i-th-ordered failure time from a sample of size n

L

Likelihood function

f(t)

Probability density function

R(t)

Reliability or survival function

F(t)

Cumulative distribution function

\(F_\varGamma (t)\)

Cumulative distribution function of a gamma variable

\(S_\varGamma (t)\)

Reliability function of a gamma variable

\(\varLambda (t)\)

Cumulative hazard function

\(E(\cdot )\)

Expectation

\(Var(\cdot )\)

Variance

\(Cov(\cdot , \cdot )\)

Covariance

\(E(\sigma )\)

Exponential distribution with scale parameter \(\sigma \)

\(\varGamma (\alpha ,\beta )\)

Gamma distribution with shape parameter \(\alpha \) and scale parameter \(\beta \)

\(\varGamma (t,\cdot ,\cdot )\)

The CDF of the gamma distribution

\(L(\mu ,\sigma )\)

Laplace distribution with location parameter \(\mu \) and scale parameter \(\sigma \)

q

Quantile of the standard L(0, 1)

\(Q_\alpha \)

100\(\alpha \%\) quantile

Notes

Acknowledgements

The authors express their sincere thanks to the editor and anonymous reviewers for their useful comments and suggestions on an earlier version of this manuscript which led to this improved one.

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Copyright information

© Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMcMaster University HamiltonHamiltonCanada

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