On the estimation of the exponential distribution of parameters. (English. Russian original) Zbl 1343.93083
J. Math. Sci., New York 216, No. 4, 569-576 (2016); translation from Sovrem. Mat. Prilozh. 94 (2014).
Summary: In the present paper, the problem of estimation of the exponential distribution of parameters is investigated.
In Sec. 1, the ordinary exponential distribution is considered, and the parameter \(\lambda\) is estimated by means of censored observations by the pseudomaximal likelihood method. It is shown that the estimator is asymptotically consistent and effective.
In Sec. 2, the problem of estimation of parameters of the truncated exponential distribution is considered using the maximal likelihood method. The existence and uniqueness of the solution corresponding to the likelihood equation are shown.
The practical application of the obtained results with the aid of computer realization is given. In particular, a sample of size \(n = 1000\) is selected, which is distributed by the truncated exponential law. The sample mean \( \overline{x} = 1.3435\) and the solution \(\theta^\ast = 2.004\) of the corresponding likelihood control are obtained.
In Sec. 1, the ordinary exponential distribution is considered, and the parameter \(\lambda\) is estimated by means of censored observations by the pseudomaximal likelihood method. It is shown that the estimator is asymptotically consistent and effective.
In Sec. 2, the problem of estimation of parameters of the truncated exponential distribution is considered using the maximal likelihood method. The existence and uniqueness of the solution corresponding to the likelihood equation are shown.
The practical application of the obtained results with the aid of computer realization is given. In particular, a sample of size \(n = 1000\) is selected, which is distributed by the truncated exponential law. The sample mean \( \overline{x} = 1.3435\) and the solution \(\theta^\ast = 2.004\) of the corresponding likelihood control are obtained.
MSC:
| 93E10 | Estimation and detection in stochastic control theory |
| 93E12 | Identification in stochastic control theory |
| 93E03 | Stochastic systems in control theory (general) |
Keywords:
estimation of the exponential distribution of parameters; censored observations; pseudomaximal likelihood methodReferences:
| [1] | Faris M. Al-Athari, “Estimation of the mean of truncated exponential distribution,” J. Math. Stat., 4, No. 4, 284-288 (2008). · Zbl 1184.62023 |
| [2] | G. Kulldorf, Contribution to the Theory of Estimation from Grouped and Partially Grouped Samples, Wiley, New York (1962). |
| [3] | E.L. Lehmann, Theory of Point Estimation, Springer-Verlag, New York (1997). · Zbl 0870.62018 |
| [4] | E. Nadaraya, M. Patsatsia, and G. Sokhadze, “On the maximum pseudo-likelihood estimations of distribution parameters by grouped observations with censoring,” Proc. Sukhumi State Univ., 7, 33-43 (2009). |
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