Exact prediction intervals for order statistics from the Laplace distribution based on the maximum-likelihood estimators. (English) Zbl 1367.62073
Summary: In this work we construct exact prediction intervals for order statistics from the Laplace (double exponential) distribution. We consider both the one- and two-sample prediction cases. The intervals are based on certain pivotal quantities that employ the corresponding maximum-likelihood predictors and the predictive maximum-likelihood estimators of the unknown parameters. Similar to G. Iliopoulos and N. Balakrishnan [J. Stat. Plann. Inference 141, No. 3, 1224–1239 (2011; Zbl 1206.62032)], we express the distributions of the pivotal quantities as mixtures of ratios of linear combinations of independent standard exponential random variables. Since these distributions are in closed form we solve numerically the corresponding equations and obtain their exact quantiles. Tables containing selected quantiles of the pivotal quantities are provided. Numerical examples are also given for illustration purposes.
MSC:
| 62F25 | Parametric tolerance and confidence regions |
| 62F10 | Point estimation |
| 62G30 | Order statistics; empirical distribution functions |
| 62N01 | Censored data models |
Keywords:
Laplace distribution; exact prediction intervals; predictive likelihood function; maximum-likelihood estimators; ratios of linear combinations of exponential random variablesCitations:
Zbl 1206.62032References:
| [1] | DOI: 10.1007/BF02925273 · Zbl 0911.62019 · doi:10.1007/BF02925273 |
| [2] | Balakrishnan N, Statistical theory and applications: papers in honor of herbert A. David pp 145– (1995) |
| [3] | DOI: 10.1080/00401706.1973.10489120 · doi:10.1080/00401706.1973.10489120 |
| [4] | DOI: 10.2307/1268356 · Zbl 0312.62031 · doi:10.2307/1268356 |
| [5] | DOI: 10.1016/S0167-7152(96)00010-7 · Zbl 0898.62036 · doi:10.1016/S0167-7152(96)00010-7 |
| [6] | DOI: 10.1007/s001840000092 · Zbl 1093.62525 · doi:10.1007/s001840000092 |
| [7] | DOI: 10.1016/0026-2714(95)00073-9 · doi:10.1016/0026-2714(95)00073-9 |
| [8] | DOI: 10.1007/978-1-4612-0173-1 · doi:10.1007/978-1-4612-0173-1 |
| [9] | DOI: 10.1016/j.jspi.2010.09.024 · Zbl 1206.62032 · doi:10.1016/j.jspi.2010.09.024 |
| [10] | Balakrishnan N, Prediction intervals for Laplace distribution based on censored samples (1994) |
| [11] | DOI: 10.1007/BF02481119 · Zbl 0585.62045 · doi:10.1007/BF02481119 |
| [12] | DOI: 10.1016/j.spl.2008.12.005 · Zbl 1158.62323 · doi:10.1016/j.spl.2008.12.005 |
| [13] | DOI: 10.1137/1.9780898719062 · Zbl 1172.62017 · doi:10.1137/1.9780898719062 |
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