Restricted minimum volume confidence region for Pareto distribution

Abstract

Under some restriction, we establish the minimum volume confidence region for parameters of Pareto distribution, which can be applied to complete samples and, as well as left, right or doubly censored samples. It is not only computationally convenient, but also almost as accurate as the best confidence region in the literature, the computation of which is difficult in the double or left censoring case.

Appendices

Appendix A: Proof of Theorem 1

Proof

Let $C^T$ be any level $1-\alpha$ confidence set of $\theta$, satisfying ${\int }_{\theta \in C^t}~dt\le r_k(\theta )|C^\theta |$. Then for any $\theta \in \Theta$,

$$\begin{aligned} 1-\alpha \le P[\theta \in C^T]=\underset{\theta \in C^t}{\int }[f(t,\theta )-kp(\theta )]dt +kp(\theta )\underset{\theta \in C^t}{\int }dt. \end{aligned}$$

It follows from $P[\theta \in C^T_k]=1-\alpha$ that

$$\begin{aligned} 0\le & {} P[\theta \in C^T]-P[\theta \in C^T_k]\\= & {} d_k(\theta )+kp(\theta )\big [\underset{\theta \in C^t}{\int }dt -\underset{\theta \in C^t_k}{\int }dt\big ]\\\le & {} kp(\theta )r_k(\theta )\big [|C^\theta |-|C_k^\theta |\big ], \end{aligned}$$

where

$$\begin{aligned} d_k(\theta )= & {} \left( \underset{\theta \in C^t}{\int }dt-\underset{\theta \in C^t_k}{\int }dt\right) [f(t,\theta )-kp(\theta )]dt\\= & {} \left( \underset{\theta \in C^t\cap \overline{C^t_k}}{\int }- \underset{\theta \in C^t_k\cap \overline{C^t}}{\int }\right) [f(t,\theta )-kp(\theta )]dt\le 0 \end{aligned}$$

by definition of $C^T_k$, and $\overline{C}$ denotes the complementary set of C. Thus, $|C_k^\theta |\le |C^\theta |$ for $\theta \in \Theta$, which implies that $|C^T_k|\le |C^T|$. The proof is complete. $\square$

Appendix B: R code for computing $k_{\alpha },\ k_1,\ k_2,\ k_1(x),\ k_2(x)$ in (4) and (5)