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 \begin{equation*}\tag{1} p(x; \theta, A) = \begin{cases}\frac{1}{\theta} e^{-(x-A)/\theta},\quad\text{for} x \geqq A \\ 0,\quad\text{otherwise}\end{cases}\end{equation*} where $\theta > 0$ 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 $\theta$ when the $N$ items are divided into $k$ sets $S_j$ (each containing $n_j > 0,$ items, $\sum^k_{j=1} n_j = N)$ and each set $S_j$ is observed only until the first $r_j$ failures occur $(0 < r_j \leqq n_j).$ The constants $r_j$ and $n_j$ are fixed and preassigned. Three different cases are considered: 1. The $n_j$ items in each set $S_j$ have a common known $A_j (j = 1, 2, \cdots, k).$ 2. All $N$ items have a common unknown $A.$ 3. The $n_j$ items in each set $S_j$ have a common unknown $A_j (j = 1, 2, \cdots, k).$ The results for these three cases are such that the results for any intermediate situation (i.e. some $A_j$ 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 $\theta,$ which we are trying to estimate, represents the expected length of life.