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Register a new userConfidence intervals for the normal mean utilizing prior information
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Added by
Paul Kabaila
Publication date
2007
fields
Mathematical Statistics
and research's language is
English
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Consider X_1,X_2,...,X_n that are independent and identically N(mu,sigma^2) distributed. Suppose that we have uncertain prior information that mu = 0. We answer the question: to what extent can a frequentist 1-alpha confidence interval for mu utilize this prior information?
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We consider a linear regression model with regression parameter beta=(beta_1,...,beta_p) and independent and identically N(0,sigma^2) distributed errors. Suppose that the parameter of interest is theta = a^T beta where a is a specified vector. Define the parameter tau=c^T beta-t where the vector c and the number t are specified and a and c are linearly independent. Also suppose that we have uncertain prior information that tau = 0. We present a new frequentist 1-alpha confidence interval for theta that utilizes this prior information. We require this confidence interval to (a) have endpoints that are continuous functions of the data and (b) coincide with the standard 1-alpha confidence interval when the data strongly contradicts this prior information. This interval is optimal in the sense that it has minimum weighted average expected length where the largest weight is given to this expected length when tau=0. This minimization leads to an interval that has the following desirable properties. This interval has expected length that (a) is relatively small when the prior information about tau is correct and (b) has a maximum value that is not too large. The following problem will be used to illustrate the application of this new confidence interval. Consider a 2-by 2 factorial experiment with 20 replicates. Suppose that the parameter of interest theta is a specified simple effect and that we have uncertain prior information that the two-factor interaction is zero. Our aim is to find a frequentist 0.95 confidence interval for theta that utilizes this prior information.
We consider a linear regression model with regression parameter beta =(beta_1, ..., beta_p) and independent and identically N(0, sigma^2)distributed errors. Suppose that the parameter of interest is theta = a^T beta where a is a specified vector. Define the parameter tau = c^T beta - t where the vector c and the number t are specified and a and c are linearly independent. Also suppose that we have uncertain prior information that tau = 0. Kabaila and Giri (2009c) present a new frequentist 1-alpha confidence interval for theta that utilizes this prior information. This interval has expected length that (a) is relatively small when the prior information about tau is correct and (b) has a maximum value that is not too large. It coincides with the standard 1-alpha confidence interval (obtained by fitting the full model to the data) when the data strongly contradicts the prior information. At first sight, the computation of this new confidence interval seems to be infeasible. However, by the use of the various computational devices that are presented in detail in the present paper, this computation becomes feasible and practicable.
The recent paper Simple confidence intervals for MCMC without CLTs by J.S. Rosenthal, showed the derivation of a simple MCMC confidence interval using only Chebyshevs inequality, not CLT. That result required certain assumptions about how the estimator bias and variance grow with the number of iterations $n$. In particular, the bias is $o(1/sqrt{n})$. This assumption seemed mild. It is generally believed that the estimator bias will be $O(1/n)$ and hence $o(1/sqrt{n})$. However, questions were raised by researchers about how to verify this assumption. Indeed, we show that this assumption might not always hold. In this paper, we seek to simplify and weaken the assumptions in the previously mentioned paper, to make MCMC confidence intervals without CLTs more widely applicable.
Consider a two-treatment, two-period crossover trial, with responses that are continuous random variables. We find a large-sample frequentist 1-alpha confidence interval for the treatment difference that utilizes the uncertain prior information that there is no differential carryover effect.
In the Gaussian linear regression model (with unknown mean and variance), we show that the standard confidence set for one or two regression coefficients is admissible in the sense of Joshi (1969). This solves a long-standing open problem in mathematical statistics, and this has important implications on the performance of modern inference procedures post-model-selection or post-shrinkage, particularly in situations where the number of parameters is larger than the sample size. As a technical contribution of independent interest, we introduce a new class of conjugate priors for the Gaussian location-scale model.
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