Abstract
We provide sharp upper and lower mean-variance bounds on the expectations of trimmed means of $$k$$ th record values from general family of distributions. Also we improve these bounds in the case of non-trimmed means for parent distributions with decreasing density or decreasing failure rate. They can be viewed as bounds on the bias of approximation of expectation of the parent population by mean or trimmed mean of record values. The results are illustrated with numerical examples.
Highlights
Let {Xn, n ≥ 1} be a sequence of i.i.d. random variables with continuous common cummulative distribution function F with finite mean μ = F−1(u) du and finite variance σ 2 = 1 F−1(u) − μ 2 du
For exhaustive review on results concerning bounds on expectations of kth record values and their functions valid in general and restricted families of distributions we refer to Section 1 of Goroncy and Rychlik (2011)
From the numerical results presented in the previous section we see that the smallest upper bias is obtained for r = 1 i.e. if we consider the mean Sn(k) of kth record values
Summary
. ., but only kth record values and we want to estimate the mean μ of the parent distribution F. The bounds obtained this way are not optimal since the equality in the above inequality holds iff Yr(k) = Yn(k) which holds with probability 0 From numerical computations it appears that the smallest upper error is commited if r = 1. For exhaustive review on results concerning bounds on expectations of kth record values and their functions valid in general and restricted families of distributions we refer to Section 1 of Goroncy and Rychlik (2011). The key step in application of these results is to determine the shapes of the functions to be projected
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