One-sided Estimating and Testing Problems from Grouped Samples

Let x₀＜ｘ₁＜…＜x_k-1＜x_k be dividing points of a support set of the distribution function F（x-θ）, by n_i denote the number of observations falling in the interval (x_i-1, x_i and said n_i’s to be grouped samples.
In this paper, we give some conditions of existence and uniqueness of the MLE of θ for grouped samples under a restriction θ≥θ₀ with some given θ₀ The one-sided testing problem under the restriction is also considered in this paper.