Improved estimators of the distribution function based on lower record values

In this paper, we define different types of estimators for the distribution function, namely preliminary test (PT), shrinkage PT (SPT), Stein type (S), and Thompson shrinkage (TS) estimators based on lower record observations and their inter record times. Their asymptotic distributional bias and mean square error are explicitly derived. The superiority conditions of the new proposed estimators over the existing estimator of distribution function are also obtained. It is shown that about the neighborhood of the null hypothesis <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$F_0$$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <msub> <mi>F</mi> <mn>0</mn> </msub> </math> </EquationSource> </InlineEquation>, the PTE is superior to the SE in the sense of having higher asymptotic relative efficiency. Further for the reasonable values of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$\alpha $$</EquationSource> <EquationSource Format="MATHML"> <math xmlns:xlink="http://www.w3.org/1999/xlink"> <mi mathvariant="italic">α</mi> </math> </EquationSource> </InlineEquation> the newly proposed SPT estimator uniformly dominates the non-parametric maximum like likelihood estimators in the literatures. A table is also given to be more specifics along the exhibited theoretical results for practical sake. Some graphical representations are given as proofs of our assertions. A simulation study is also carried out for some life time distribution, to examine the accuracy of the proposed estimators with a limited sample size. The results show that combination of the parametric and nonparametric estimators will give more efficient estimators. This study is finally concluded by applying the theoretic results to a real data set. Copyright Springer-Verlag Berlin Heidelberg 2015