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.
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