The efficiency of Buehler confidence limits

Abstract

The Buehler 1−α upper confidence limit is as small as possible, subject to the constraints that (a) its coverage probability never falls below 1−α and (b) it is a non-decreasing function of a designated statistic T. We provide two new results concerning the influence of T on the efficiency of this confidence limit. Firstly, we extend the result of Kabaila (Statist. Probab. Lett. 52 (2001) 145) to prove that, for a wide class of Ts, the T which maximizes the large-sample efficiency of this confidence limit is itself an approximate 1−α upper confidence limit. Secondly, there may be ties among the possible values of T. We provide the result that breaking these ties by a sufficiently small modification cannot decrease the finite-sample efficiency of the Buehler confidence limit.

Introduction

Suppose that the distribution of the data Y is indexed by β∈B and that the scalar parameter of interest is θ=θ(β). Also suppose that Y is a discrete random vector taking values in a known countable set $\text{Y}$. This paper is concerned with the construction of a 1−α $\text{(0<α<}\text{1}\text{2}\text{)}$ upper confidence limit for θ i.e. a statistic u(Y) satisfying $\text{P(θ⩽u(Y)}\text{|}\text{β)⩾1−α}$ for all β. Such limits are appropriate when θ is a measure of loss. One method of finding an upper confidence limit is to invert a test of H₀: θ=θ′ against H₁: θ<θ′ using the test statistic t(θ′,Y) which tends to be smaller under H₁ than under H₀. Define the P-value

$$\text{g(θ′,y)=}\text{sup}\text{β∈B:θ(β)=θ′}\text{P(t(θ′,Y)⩽t(θ′,y)}\text{|}\text{β)}\text{for}\text{each}\text{y∈}\text{Y}\text{.}$$

A confidence set for θ with coverage probability at least 1−α is {θ(β):g(θ,y)>α}. Thus a 1−α upper confidence limit for θ is provided by the formula

$$\text{sup}\text{{θ(β):g(θ,y)>α}}$$

for observed data y. Suppose, for the moment, that t(θ,Y) depends on θ. For example, $\text{t(θ,Y)=(}\text{Θ}\text{̂}\text{−θ)/(}\text{standard}\text{error}\text{of}\text{Θ}\text{̂}\text{)}$, where $\text{Θ}\text{̂}$ is an estimator of θ. Assuming (as do Bolshev and Loginov (1966)) that g(θ,y) is a decreasing continuous function of θ for each $\text{y∈}\text{Y}$, the solution for θ of g(θ,y)=α provides a 1−α upper confidence limit for θ. However, this assumption is not tenable for the discrete data case which we are considering. This is because g(θ,y), when considered as a function of θ for fixed y, has “jump” and “drop” discontinuities (Kabaila, 2002b).

Henceforth, we restrict attention to t(θ,Y) a function of Y, which we denote by T=t(Y). Formula (1) now becomes the following:

$$\text{sup}\text{{θ(β):P(t(Y)⩽t(y)}\text{|}\text{β)>α,β∈B}.}$$

Under mild regularity conditions, g(θ,y) is a continuous function of θ for every $\text{y∈}\text{Y}$ (see e.g. Kabaila and Lloyd, 1997). Furthermore, for a wide class of models (including binomial and poisson), parameters of interest θ and designated statistics T, g(θ,y) is a non-increasing function of θ for each $\text{y∈}\text{Y}$ (Harris and Soms, 1991; Kabaila, 2002a). The combination of g(θ,y) being continuous and non-increasing in θ greatly eases the computation of (1).

Formula (2) was presented by Buehler (1957). He also provided the insight that the 1−α upper confidence limit given by this formula is as small as possible subject to the constraint that it is a non-decreasing function of the designated statistic T. For proofs see Bolshev (1965) (for the case of no nuisance parameters), Jobe and David (1992) and Lloyd and Kabaila (2003). This confidence limit is commonly called the Buehler confidence limit.

The efficiency of a Buehler 1−α upper confidence limit is measured by how probabilistically small it is. Kabaila (2001) proves that, for T a modified Wald upper confidence limit for θ with nominal coverage $\text{1−γ}\text{(0<γ⩽}\text{1}\text{2}\text{)}$, maximum large sample efficiency is achieved when γ=α. It is reasonable to expect that a similar result will hold for the class $\text{A}$ of statistics that are asymptotically equivalent to these designated statistics. For example, the set of signed root likelihood ratio upper confidence limits with nominal coverage $\text{1−γ}\text{(0<γ⩽}\text{1}\text{2}\text{)}$ belongs to $\text{A}$.

We provide two new results on how T should be chosen so as to maximize the efficiency of the Buehler confidence limit. Firstly, not all of the designated statistics T which have been proposed in the literature belong to $\text{A}$. For example, neither the simple upper confidence limit for θ which was suggested as a reasonable choice for T by Buehler (1957) nor its natural generalization, as described by Kabaila and Lloyd (2002), belong to this class. In Section 2 we describe a class of statistics that includes both these designated statistics and $\text{A}$. This class is much wider than $\text{A}$. We then extend the result of Kabaila (2001). We prove that the T belonging to this much wider class that maximizes the large-sample efficiency of the Buehler 1−α upper confidence limit is asymptotically equivalent to a modified Wald upper confidence limit with nominal coverage 1−α.

Secondly, there may be ties among the possible values of t(Y). In Section 3 we provide the result that breaking these ties by a sufficiently small modification cannot decrease the finite-sample efficiency of the Buehler confidence limit.