Corrected confidence intervals based on the signed root transformation for multi-parameter sequentially designed experiments
Introduction
Consider a two-parameter model in which where is the response variable, is a design variable, θ is the parameter of interest, η is a nuisance parameter, and f is a density of a known form. Here, and are compact spaces. The model is adaptive in the sense that The above model is very general. For example, it includes some well-known nonlinear models and also the exponential family studied by Weng and Woodroofe (2000). Surveys of adaptive nonlinear models are given by Ford et al. (1989) and Chaudhuri and Mykland (1993).
A one-parameter adaptive normal nonlinear regression model was studied by Woodroofe (1991) and Coad and Woodroofe (2005). Using a Bayesian approach and a Taylor series, Woodroofe (1991) obtained asymptotic expansions for sampling distributions. Coad and Woodroofe (2005) used Stein's (1986) identity instead of a Taylor series to obtain corrected confidence intervals for θ. The latter were applied to two well-known nonlinear models with known variance. The aim of the present paper is to extend these ideas to the above two-parameter model and to apply the results to a variety of examples, which include both adaptive nonlinear models and sequential testing problems.
Asymptotic posterior normality has been studied by many authors. In the multi-parameter case, Sweeting (1992) gave three conditions which ensure asymptotic posterior normality in a wide range of situations. Weng (2003) showed that one of these conditions can be avoided in the one-parameter case and that a more flexible approach is possible based on a version of Stein's identity. This approach is extended by Weng and Tsai (2008) to the multi-parameter case and comparisons with earlier work are provided of the conditions required. In particular, an example is given for which their conditions are satisfied, but one of Sweeting's (1992) conditions fails to hold.
It is well known that the likelihood function is not affected by the adaptive nature of the model or by the use of a stopping time; see, for example, Berger and Wolpert (1984). Thus, the log-likelihood function based on is Write and for the maximum likelihood estimators of θ and η, respectively, and for the restricted maximum likelihood estimator of η when θ is fixed. Then, letting , if is an interior point of the parameter space, we have and, therefore, and Further suppose that in for all as , where κij is not necessarily the partial derivative of κ.
Let and so that Next, letandThen Zn is the first component of the bivariate signed root transformation (e.g. Bickel and Ghosh, 1990), which is asymptotically standard normal as under modest conditions, and thus may be treated as a first approximation to a pivotal quantity.
For suitable families of stopping times t depending on a parameter , the main aim of this work is to find data-dependent quantities and such thatis asymptotically standard normal to third order in the very weak sense of Woodroofe (1986). This means thatas , for large classes of functions h and twice continuously differentiable densities ξ on , whereand ϕ denotes the standard normal density. The basic idea is to consider Bayesian models and derive expansions which are valid for a large class of prior distributions. Note that the Bayesian approach is a device and that the correction terms and will not depend on the prior. In effect, Bayesian mathematics is used to draw frequentist conclusions. Woodroofe, 1986, Woodroofe, 1989 calls relations of the form (4) “very weak expansions” and writes very weakly. He argues that very weak expansions are strong enough to support a frequentist interpretation of confidence intervals, essentially because parameter values will vary among users according to a density ξ in repeated applications of the confidence procedure.
In Section 2, Stein's identity is used to obtain asymptotic expansions for the marginal posterior distribution of Zn. Very weak expansions for are presented in Section 3 and the approximately pivotal quantities are constructed. In Section 4, several examples are described and the accuracy of the approximations presented in Section 3 is assessed by simulation in Section 5. An indication of how the approach extends to higher dimensions is briefly discussed in Section 7. The main results and achievements are stated in Section 7, together with details of possible extensions to the work. An Appendix shows how the third-order correction term is calculated.
Section snippets
Expansions for marginal posterior distributions
Consider a Bayesian model in which θ and η have a joint prior density ξ on . Let denote expectation in the Bayesian model in which θ and η are replaced with random variables and η, and let denote conditional expectation given the data . Then The approach here is to obtain asymptotic expansions for the posterior expectations and then to integrate them with respect to the marginal distribution of the data.
If and η have joint density ξ, then
Very weak expansions
Let be a family of stopping times depending on a parameter and suppose that in for almost every as , where ρ is a continuous function on . The case of nonrandom sample size is not excluded here, since then : see 4.2 Normal nonlinear regression models, 4.3 Logistic model. In this section, third-order very weak expansions for Zt are obtained, and an approximately pivotal quantity is constructed by standardizing.
First observe that, from (9)
Sequential testing
Suppose that , are independent random variables with density and that , are independent random variables with density , and consider the stopping time where . Then we know that in -probability for all as . Two sequential testing problems will be considered. These are the normal model of Robbins and Siegmund (1974) and the Poisson model studied by Weng and Woodroofe (2000). Example 1 Normal model
General
In order to assess the accuracy of the approximations presented in Section 4, a simulation study based on 10,000 replications was carried out, for selected values of the design parameters. The results are reported separately for the sequential testing problems, the normal nonlinear regression models and the logistic model. In each case, the coverage probabilities are reported for both the first-order pivot, Zt, and the corrected pivot, .
Sequential testing
We first consider Example 1. Monte Carlo results are
Extension to higher dimensions
Now suppose that there are nuisance parameters, so that , and write for the vector of partial derivatives with respect to η. Then we can extend the approach which has been developed for a single nuisance parameter to higher dimensions. This is achieved by first writing down the joint posterior density of Zn and for , and then by finding the marginal posterior density of Zn as before. Of course, all of the resulting integrals will now be p-dimensional.
To
Main results and achievements
We have shown how to construct corrected confidence intervals for the parameter of interest for data from a sequentially designed experiment when there is a nuisance parameter. The model considered is very general, and includes some well-known nonlinear models and also the exponential family. The accuracy of the approximations has been assessed by simulation for a variety of examples, which include both adaptive nonlinear models and sequential testing problems.
The corrected confidence intervals
Acknowledgments
Part of this work was carried out while the author was a visiting scholar at the University of Michigan during August and September 2005, and in receipt of Overseas Travel Grant EP/D034450/1 from the UK Engineering and Physical Sciences Research Council. He is indebted to Professor M.B. Woodroofe for formulating the approach developed in this paper and for his comments on the work. The author also wishes to thank Professor B. Qaqish for his help with Example 6, and a referee and an Associate
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