Asymptotic Theory & Testing

If an estimator is "weakly consistent", then:
1. It will also be mean-square consistent
2. Its asymptotic distribution will be the same as that of the corresponding MLE
3. The probability limit of the estimator is equal to the true value of the parameter being estimated
4. It will also be mean-square consistent if it is unbiased in finite samples
If an estimator is "strongly consistent" then:
1. It is also "weakly consistent"
2. The estimator converges "almost surely" to the true value of the parameter
3. Both A and B
4. It must also be asymptotically efficient
If an estimator is "Fisher consistent", then:
1. It will also be mean square consistent
2. It will satisfy Slutsky's Theorem
3. It will also be weakly consistent
4. Both B and C
The "regularity conditions" associated with MLE are:
1. Conditions that the Hessian matrix of the log-likelihood function should satisfy
2. Conditions that the underlying population density function must satisfy
3. Conditions that MLE's generally satisfy
4. Conditions that ensure that the "likelihood equations" have a unique solution
If the usual "regularity conditions" are satisfied, then MLE's are:
1. Weakly consistent & asymptotically unbiased
2. A hypothetical, but not always attainable, minimum value for the variance of any estimator for this parameter
3. The smallest value for the that can be achieved by any unbiased estimator for this parameter
4. The value of Fisher's "Information Measure"
When estimating a scalar parameter, the Cramer-Rao lower bound is:
1. The smallest variance that that any MLE can achieve
2. A hypothetical, but not always attainable, minimum value for the variance of any estimator for this parameter
3. The smallest value for the that can be achieved by any unbiased estimator for this parameter
4. The value of Fisher's "Information Measure"
Fisher's Information Matrix:
1. Is obtained by taking the negative of the expectation of the Hessian matrix of the log-likelihood function
2. Is the Cramer-Rao lower bound for the covariance matrix of an unbiased estimator
3. Has an inverse that is the Cramer-Rao lower bound for the covariance matrix of an unbiased estimator
4. Both 'A' and 'C' above
When testing restrictions on the parameters, the Wald, LM and LRT tests are "asymptotically equivalent" in the sense that:
1. The asymptotic distributions of the associated test statistics are the same if the null hypothesis is false
2. The asymptotic distributions of the associated test statistics are all standard normal if the null hypothesis is true
3. The test statistics will all take the same value if the model is correctly specified
4. The asymptotic distributions of the associated test statistics are the same if the null hypothesis is true
The LM test might be preferred over the Wald or LR tests if:
1. It is easier to estimate the model by MLE once the restrictions under test are imposed on the model
2. It is easier to estimate the model by MLE if the restrictions under test are NOT imposed on the model
3. The restrictions under test are non-linear functions of the parameters
4. Both A and C
The main motivation for the LRT is that:
1. In the case of a "point" null hypothesis and a "point" alternative hypothesis, the test is consistent
2. In the case of a "composite" null hypothesis and a "composite" alternative hypothesis, the test is UMP
3. In the case of a "point" null hypothesis and a "point" alternative hypothesis, the test is UMP
4. It is easy to compute, even for complex hypotheses

Asymptotic Theory & Testing

Multiple Choice Questions