Suppose that \(y = X \beta + \varepsilon\) is a random \(n\)-vector of responses, \(X\) is a known \(n \times p\) matrix with linearly independent columns, \(\beta\) is an unknown parameter \(p\)-vector, and \(\varepsilon \sim N(0, \, \sigma^2 \, I)\), with \(\sigma^2\) assumed known. Suppose that the parameter of interest is \(\theta = a^{\top} \beta\) and that there is uncertain prior information that \(\tau = c^{\top} \beta - t = 0\), where \(a\) and \(c\) are specified linearly independent nonzero \(p\)-vectors and \(t\) is a specified number. This package computes a confidence interval, with minimum coverage \(1 - \alpha\), for \(\theta\) that utilizes the uncertain prior information that \(\tau = 0\) through desirable expected length properties.
Define \(\hat{\beta}\) to be the least squares estimator of \(\beta\). Then \(\hat{\theta} = a^{\top} \hat{\beta}\) and \(\widehat{\tau} = c^{\top} \hat{\beta} - t\) are the least squares estimators of \(\theta\) and \(\tau\), respectively. Also define \(v_{\theta} = \text{Var}(\hat{\theta})/\sigma^2\), \(v_{\tau} = \text{Var}(\hat{\tau})/\sigma^2\), \(\gamma = \tau / (\sigma \, v_{\tau}^{1/2})\) and \(\hat{\gamma} = \hat{\tau}/(\sigma \, v_{\tau}^{1/2})\). The \(1 - \alpha\) confidence interval for \(\theta\) that utilizes uncertain prior information about \(\tau\) (CIUUPI) has the form \[ \left[ \hat{\theta} - v_{\theta}^{1/2} \, \sigma \, b(\hat{\gamma}) - v_{\theta}^{1/2} \, \sigma \, s(\hat{\gamma}), \, \hat{\theta} - v_{\theta}^{1/2} \, \sigma \, b(\hat{\gamma}) + v_{\theta}^{1/2} \, \sigma \, s(\hat{\gamma}) \right], \]
where \(b\) is an odd continuous function that takes the value \(0\) for \(|x| \geq 6\), and \(s\) is an even continuous function that takes the value \(z_{1-\alpha/2}\) for all \(|x| \geq 6\), where \(z_{1-\alpha/2}\) is the \(1 - \alpha/2\) quantile of the standard normal distribution. The values of \(b(x)\) and \(s(x)\) for \(x \in [-6,6]\) are determined by \((b(1), b(2), \dots, b(5), s(0), s(1), \dots, s(5))\) through either natural (default) or clamped cubic spline interpolation.
The CIUUPI is found by computing the value of \((b(1), b(2), \dots, b(5), s(0), s(1), \dots, s(5))\) so that the confidence interval has minimum coverage probability \(1 - \alpha\) and desirable expected length properties. This constrained optimization is carried out using a similar methodology to Kabaila and Mainzer (2017), Section 2.1, and using the slsqp function in the nloptr package.
This confidence interval has the following three practical applications. Firstly, if \(\sigma^2\) has been accurately estimated from previous data then it may be treated as being effectively known. Secondly, for sufficiently large \(n - p\) (\(n-p \ge 30\), say) if we replace the assumed known value of \(\sigma^2\) by its usual estimator in the formula for the confidence interval then the resulting interval has, to a very good approximation, the same coverage probability and expected length properties as when \(\sigma^2\) is known. Thirdly, some more complicated models can be approximated by the linear regression model with \(\sigma^2\) known when certain unknown parameters are replaced by estimates.
The function bsciuupi is used to obtain the vector \((b(1), b(2), \dots, b(5), s(0), s(1), \dots, s(5))\). Once this vector is obtained, the functions \(b\) and \(s\) can be evaluated using bsspline.
Define the scaled expected length of the CIUUPI to be the expected length of the CIUUPI divided by the expected length of the standard interval for \(\theta\), given by \[ \left[ \hat{\theta} - z_{1-\alpha/2} \, v_{\theta}^{1/2} \, \sigma, \, \hat{\theta} + z_{1-\alpha/2} \, v_{\theta}^{1/2} \, \sigma \right]. \]
For given \(\alpha\), \(a\), \(c\) and \(X\), the coverage probability and scaled expected length of the CIUUPI are even functions of the unknown parameter \(\gamma\). The coverage probability of the CIUUPI can be evaluated using cpciuupi and the scaled expected length of the CIUUPI can be evaluated using selciuupi.
Note that \(\rho = \text{Cor}(\hat{\theta}, \hat{\tau}) = a^{\top} (X^{\top} X)^{-1} c / (v_{\theta} v_{\tau})^{1/2}\) can be specified instead of \(a\), \(c\) and \(X\) in the above functions.
For given \(\alpha\), \(a\), \(X\), \(\sigma\) and \(y\) we can obtain the standard confidence interval for \(\theta\) using the cistandard function. If, in addition, we have \(c\) and \(t\) (used to determine \(\tau\)), we can estimate \(\hat{\gamma}\). For given \(\alpha\), \(a\), \(c\), \(t\), \(X\), \(\sigma\) and \(y\), we can obtain the CIUUPI, using the ciuupi function.
Kabaila, P. and Mainzer, R. (2017). Confidence intervals that utilize uncertain prior information about exogeneity in panel data. URL https://arxiv.org/pdf/1708.09543.pdf.