Abstract
A general expression for the UMP unbiased tests of the hypotheses \(H_1:\theta \le \theta _0\) and \(H_4:\theta =\theta _0\) in the exponential family.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
This problem is discussed in Section 3 of Hodges and Lehmann (1954).
- 2.
The relationship \(W=Y/(1+Y)\) shows the F and beta distributions to be equivalent. Tables of these distributions are discussed in Chapters 24 and 26 of Johnson, Kotz and Balakrishnan (1995. Vol. 2). Critical values of F are tabled by Mardia and Zemroch (1978), who also provide algorithms for the associated computations.
- 3.
A comparison of these limits with those obtained from the equal-tails test is given by Scheffé (1942); some values of \(C_1\) and \(C_2\) are provided by Ramachandran (1958).
- 4.
The literature on regression is enormous and we treat the simplest model. Some texts on the subject include Weisberg (1985), Atkinson and Riani (2000) and Chatterjee Hadi and Price (2000).
- 5.
A method for obtaining the size of this effect was developed by Neyman, and tables have been computed on its basis by Fix. This work is reported by Bennett (1957).
- 6.
- 7.
They also do not occur when the posterior distribution of \(\Theta \) is discrete.
- 8.
For a closely related result. see Odén and Wedel (1975).
- 9.
The problem is simplified here, not just due to the assumption of normality, but also by the assumption that the distribution of \(X_i - u_i\) does not depend on \(\mu _i\), and similarly for the \(Y_i\). In general, when observations are paired approximately according to covariates, then pairs cannot be treated as if they were sampled from a population. A recent treatment is provided in Bai, Romano, and Shaikh (2019).
- 10.
See, for example, Billingsley (1995), p. 417.
- 11.
Actually, all that is needed is that \(f_1,\ldots ,f_c\in \mathcal{F}\), where \(\mathcal F\) is any family containing all normal distributions.
- 12.
A systematic account of this distribution can be found in in Owen (1985) and Johnson Kotz and Balakrishnan (1995).
- 13.
For additional information concerning inference in inverse Gaussian distributions, see Folks and Chhikara (1978) and Johnson, Kotz and Balakrishnan (1994, volume 1).
- 14.
A similar conclusion holds in the problem of constructing a confidence interval for the ratio of normal means (Fieller’s problem), as discussed in Koschat (1987). For problems where it is impossible to construct confidence intervals with finite expected length , see Gleser and Hwang (1987).
- 15.
For the corresponding result concerning one-sided confidence bounds, see Madansky (1962).
- 16.
The distribution of R is reviewed by Johnson and Kotz (1970, Vol. 2, Section 32) and Patel and Read (1982).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Lehmann, E.L., Romano, J.P. (2022). Unbiasedness: Applications to Normal Distributions; Confidence Intervals. In: Testing Statistical Hypotheses. Springer Texts in Statistics. Springer, Cham. https://doi.org/10.1007/978-3-030-70578-7_5
Download citation
DOI: https://doi.org/10.1007/978-3-030-70578-7_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-70577-0
Online ISBN: 978-3-030-70578-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)