Transport and Telecommunication
Vol.3, N 1, 2002
CONSTRUCTING SHORTEST-LENGTH CONFIDENCE INTERVALS Konstantin N. Nechval Transport and Telecommunication Institute Lomonosov Street 1, LV-1019 Riga, Latvia Ph: (+371)-7100650, Fax: (+371)-7100660, E-mail:
[email protected]
Nicholas A. Nechval & Edgars K. Vasermanis Department of Mathematical Statistics, University of Latvia Raina Blvd 19, LV-1050 Riga, Latvia Ph: (+371)-7034702, Fax: +371-7034702, E-mail:
[email protected]
Valery Ya. Makeev Department of Computer Control, Transport and Telecommunication Institute Lomonosov Street 1, LV-1019 Riga, Latvia Ph: (+371)-7100650, Fax: (+371)-7100660, E-mail:
[email protected]
Abstract In this paper, we present an approach to invariant confidence intervals that emphasizes pivotal quantities. We consider confidence interval problems that are invariant under a group of transformation G such that the induced group G acts transitively on the parameter space. The purpose of this paper is to give a technique for deriving confidence intervals with a minimum length property. Examples illustrating the use of this technique are given. Key words: Confidence interval, Shortest length, Technique for constructing
1. Introduction In many problems of statistical inference the experimenter is interested in constructing a family of sets that contain the true (unknown) parameter value with a specified (high) probability. If X, for example, represents the length of life of a piece of equipment, the experimenter is interested in a lower bound θL for the mean θ of X. Since θL=θL(X) will be a function of the observations, one cannot ensure with probability 1 that θL(X)≤θ. All that one can do is to choose a number 1−α that is close to 1 so that Pθ{θL(X)≤θ}≥1−α for all θ. Problems of this type are called problems of confidence estimation. Let Pθ, θ∈Θ⊆Rk, be the set of probability distributions of an rv X. A family of subsets S(x) of Θ, where S(x) depends on the observation x but not on θ, is called a family of random sets. If, in particular, Θ⊆R and S(x) is an interval (θL(x), θU(x)), where θL(x) and θU(x) are functions of x alone (and not θ), we call S(X) a random interval with θL(X) and θU(X) as lower and upper bounds, respectively. θL(X) may be −∞, and θU(X) may be +∞. In a wide variety of inference problems one is not interested in estimating the parameter or testing some hypothesis concerning it. Rather, one wishes to establish a lower or an upper bound, or both, 95
Transport and Telecommunication
Vol.3, N 1, 2002
for the real-valued parameter. For example, if X is the time to failure of a piece of equipment, one is interested in a lower bound for the mean of X. If the rv X measures the toxicity of a drug, the concern is to find an upper bound for the mean. Similarly, if the rv X measures the nicotine content of a certain brand of cigarettes, one is interested in determining an upper and a lower bound for the average nicotine content of these cigarettes. In this paper we are interested in the problem of confidence estimation, namely, that of finding a family of random sets S(x) for a parameter θ such that, for a given α, 0
