Improved estimation of the scale parameter, the hazard rate parameter and the ratio of the scale parameters in exponential distributions: An integrated approach

We give new classes of Strawderman-type improved estimators for the scale parameter σ2σ2 and the hazard rate parameter 1/σ11/σ1 of the exponential distributions E(μ2,σ2)E(μ2,σ2) and E(μ1,σ1)E(μ1,σ1) under the entropy loss. We then use these estimators to construct improved estimators for the ratio ρ=σ2/σ1ρ=σ2/σ1. The sampling framework we consider integrates important life-testing schemes separately studied in the literature so far, namely, (i) i.i.d. sampling, (ii) Type-II censoring, (iii) progressive Type-II censoring and adaptive progressive Type-II censoring and (iv) record values data. Furthermore, we establish simple identities connecting the risk functions of the estimators of σ2σ2 and 1/σ11/σ1 and those of ρρ that have a direct impact on studying the risk behavior of the latter estimators. Finally, we indicate that no matter which of the above life-testing schemes is employed for the estimation of σ2σ2, 1/σ11/σ1 or ρρ, the corresponding improved estimator, which may be of Stein-type or Brewster and Zidek-type or Strawderman-type, will offer the same improvement over the usual estimator as long as the number of observed complete failure times is the same for each scheme. Our results unify and extend existing results on the estimation of exponential scale parameters in one or two populations.