Some Theorems Relevant to Life Testing from an Exponential Distribution

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 equation*1 p (x;, A) = cases1 e^- (x-A) /, x A \\ 0, * where > 0 is unknown but is the same for all items and A 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 when the N items are divided into k sets Sⱼ (each containing nⱼ > 0, items, ᵏ₉=₁ nⱼ = N) and each set Sⱼ is observed only until the first rⱼ failures occur (0 < rⱼ 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, which we are trying to estimate, represents the expected length of life.