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Some Problems in Interval Estimation

E. C. Fieller
Journal of the Royal Statistical Society. Series B (Methodological)
Vol. 16, No. 2 (1954), pp. 175-185
Published by: Wiley for the Royal Statistical Society
Stable URL: http://www.jstor.org/stable/2984043
Page Count: 11
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Some Problems in Interval Estimation
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Abstract

The object of this paper is to propose for discussion of the following topic: b1, b2, ... are unbiased estimates of β1, β2 ..., distributed normally with variances and covariances jointly estimated, with f degrees of freedom and independently of b1, b2, ..., as $v_{11}, v_{12}, v_{22}, \dots$, and the functions Fr(α) do not involve the parameters βr. What can we say about the roots of the equation in (β,α) = β1 F1 (α) + β2 F2 (α) + ... = 0? Numerical examples are discussed in detail to illustrate the problems of Ldetermining the fiducial distributions of (i) the root of a simple equation (i.e., a ratio), (ii) the roots of a quadratic equation with variable coefficients. The solutions proposed are based on a consideration of the region of the (α, t2) plane lying above the curve {F(b, α)}2 = t2 V{F(b,α)} = t2 Σ vij Fi(α) Fj(α).

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