1954/06/01 by Benjamin Epstein, B. Epstein, M. Sobel +1 · 5 citations
Mathematics · #Statistical Distribution Estimation and Applications #Statistical Methods and Bayesian Inference
paper · pdf · doi:10.1214/aoms/1177728793
A life test on N items is considered in which the common underlying distribution of the length of life of a single item is given by the density p(x; θ, A) = \begincases(1)/(θ) e-(x-A)/θ,\quadfor x \geqq A
0,\quadotherwise\endcases where θ > 0 is unknown but is the same for all items and A \geqq 0. Several lemmas are given concerning the first r out of n observations when the underlying p.d.f. is given by (1). These results are then used to estimate θ when the N items are divided into k sets Sj (each containing nj > 0, items, ∑kj=1 nj = N) and each set Sj is observed only until the first rj failures occur (0 < rj \leqq nj). The constants rj and nj are fixed and preassigned. Three different cases are considered: 1. The nj items in each set Sj have a common known Aj (j = 1, 2, ⋯, k). 2. All N items have a common unknown A. 3. The nj items in each set Sj have a common unknown Aj (j = 1, 2, ⋯, k). The results for these three cases are such that the results for any intermediate situation (i.e. some Aj values known, the others unknown) can be written down at will. The particular case k = 1 and A = 0 is treated in [2]. The constant A in (1) can be interpreted in two different ways: (i) A is the minimum life, that is life is measured from the beginning of time, which is taken as zero. (ii) A is the "time of birth", that is life is measured from time A. Under interpretation (ii) the parameter θ, which we are trying to estimate, represents the expected length of life.