UMVUEUniformly Minimum Variance Unbiased Estimate
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In the following theorem, we obtain the UMVUE of R(t).
We conclude from (14) that the UMVUE of R(t) can be obtained simply integrating f(x; a, [alpha], [lambda], [[theta].bar]) from t to [infinity].
From the arguments similar to those adopted in proving Theorem 5, it can be shown that the UMVUE of "P" is given by
If we look at the proofs of Theorem 5 and Theorem 6, we observe that the UMVUE of the sampled pdf is used to obtain the UMVUES of R(t) and "P" Thus, we have established interrelationship between the two estimation problems.
Thus, the UMVUE of [alpha] is more efficient than its MLE.
We consider the problems of estimating R(t) and "P." Uniformly minimum variance unbiased estimators (UMVUES) and maximum likelihood estimators (MLES) are derived.
In Section 3, we derive the UMVUES of R(t) and "P." In Section 4, we obtain the MLES of R(t) and "P." In Section 5, analysis of a simulated data has been presented for illustrative purposes.
The following lemma provides the UMVUES of the powers of a.
(ii) In the literature, the researchers have derived the UMVUES of "P" for the case when X and Y follow the same distribution (maybe with different parameters).
(i) All the comments made under Remarks 1 for UMVUES are tenable for MLES.