here is the lengthy calculation of the joint 100(1-alpha)% confidence region for para

To provide a clear response, I would need more specific details about the parameters and data you're working with, such as the type of statistical model, distribution, or context in which the joint confidence region is being calculated. However, I can outline the general steps for calculating a joint confidence region for parameters.

1. Break Down the Problem

1. Identify the parameters of interest (e.g., means, variances, regression coefficients).
2. Determine the sample size and the data distribution.
3. Define the confidence level (e.g., 100(1-α)%).

2. Relevant Concepts

1. Use the relevant statistical theory:
   • For normally distributed data, use the multivariate normal distribution.
   • For regression coefficients, use the covariance matrix.
2. Determine the critical value associated with the confidence level from statistical tables (e.g., t-distribution or z-distribution).

3. Analysis and Detail

1. Calculate the estimate of the parameters.
2. Compute the covariance matrix of the estimates.
3. Construct the confidence region by finding the appropriate bounds: $\hat{\theta} \pm z_{\alpha/2} \times SE(\hat{\theta})$ where $\hat{\theta}$ is the parameter estimate and $SE(\hat{\theta})$ is the standard error.
4. For multivariate cases, the joint confidence region can be expressed as: $(\hat{\theta} - \mathbf{c})' \mathbf{V}^{-1} (\hat{\theta} - \mathbf{c}) \leq \chi^2_{k, \alpha}$ where $\mathbf{V}$ is the covariance matrix, and $\chi^2_{k, \alpha}$ is the critical value from the chi-squared distribution with $k$ degrees of freedom.

4. Verify and Summarize

1. Check calculations for accuracy.
2. Ensure that the confidence region captures the parameters at the desired confidence level.
3. Summarize the findings in terms of the calculated region and its implications.

Final Answer

Without specific data or details about the parameters, this outline provides a general approach for calculating joint confidence regions in statistics. If you can provide the specific context or values, a more tailored solution can be generated.