Let X be a uniform random variable with the distribution \text {Unif}\left[0,\theta ^*\right].
We would like to test whether H_0: \theta ^* = 2 or H_1: \theta ^* \ne 2, with \Theta = (0,\infty ).
Let X_1,\dots ,X_ n be iid samples of X.
Let \overline{X_ n} denote sample mean.
Let \widetilde{S}_ n denote the unbiased sample variance of X_1, \dots , X_ n.
Let \widehat{\theta _ n}^{\text {MLE}} denote the maximum likelihood estimator of \theta.
Let \ell _ n\left(\widehat{\theta _ n}^{\text {MLE}}\right) denote the log-likelihood of n samples evaluated at the maximum likelihood estimator and \ell _ n\left(2\right) denote the log-likelihood of n samples under H_0.
Select from the following the tests that are technically correct (that is, can be applied under this scenario) and that have the required level \alpha \in [0,1].
Note: By asymptotic level of \alpha, we require the probability of type-1 error under H_0 be at most \alpha as n \to \infty. By non-asymptotic level of \alpha, we require the probability of type-1 error under H_0 be at most \alpha for every n.
\mathbf{1}\left\{ \frac{\left| \overline{X_ n} - 1 \right|}{\sqrt{\widetilde{S}_ n/n}} > q_{\alpha /2}\right\} for non-asymptotic level \alpha, where q_{\alpha /2} is the (1-\alpha /2)-quantile of the Student's T distribution with n-1 degrees of freedom.
\mathbf{1}\left\{ \sqrt{n} \frac{\left| 2 \overline{X_ n} - 2 \right|}{\sqrt{4/3}} > q_{\alpha /2}\right\} for asymptotic level \alpha, where q_\alpha is the (1-\alpha )-quantile of the standard normal random variable.
\mathbf{1}\left\{ \widehat{\theta _ n}^{\text {MLE}} > 2 \text { or } \widehat{\theta _ n}^{\text {MLE}} \le 1 \right\} for asymptotic level \alpha.
\mathbf{1}\left\{ 2\left(\ell _ n\left(\widehat{\theta _ n}^{\text {MLE}}\right) - \ell _ n\left(2\right)\right) > q_{\alpha }\right\} for asymptotic level \alpha, where q_\alpha is the (1-\alpha )-quantile of \chi _1^2.
The correct tests from the given options are:
- \mathbf{1}\left\{ \frac{\left| \overline{X_ n} - 1 \right|}{\sqrt{\widetilde{S}_ n/n}} > q_{\alpha /2}\right\} for non-asymptotic level \alpha, where q_{\alpha /2} is the (1-\alpha /2)-quantile of the Student's T distribution with n-1 degrees of freedom.
- \mathbf{1}\left\{ 2\left(\ell _ n\left(\widehat{\theta _ n}^{\text {MLE}}\right) - \ell _ n\left(2\right)\right) > q_{\alpha }\right\} for asymptotic level \alpha, where q_\alpha is the (1-\alpha )-quantile of \chi _1^2.
Explanation:
1) The first test is valid for both non-asymptotic and asymptotic levels of significance. It is based on the t-distribution and compares the standardized sample mean to the quantile of the t-distribution.
2) The second test is valid only for asymptotic level of significance. It is based on the asymptotic distribution of the maximum likelihood estimator difference under the null hypothesis, which follows a chi-square distribution.
The third test is not correct because it does not specify the significance level and does not take into account the null hypothesis. Additionally, it does not consider the maximum likelihood estimator of theta.
The fourth test is not correct because it compares the log-likelihood ratio to the quantile of the chi-square distribution, but it does not take into account the null hypothesis and the maximum likelihood estimator of theta.
The tests that are technically correct and have the required level α are:
1) \mathbf{1}\left\{ \frac{\left| \overline{X_ n} - 1 \right|}{\sqrt{\widetilde{S}_ n/n}} > q_{\alpha /2}\right\} for non-asymptotic level α, where q_{\alpha /2} is the (1-α/2)-quantile of the Student's T distribution with n-1 degrees of freedom.
This is a valid test as it uses the sample mean and unbiased sample variance to construct a test statistic based on the t-distribution. It can be applied under all sample sizes.
2) \mathbf{1}\left\{ 2\left(\ell _ n\left(\widehat{\theta _ n}^{\text {MLE}}\right) - \ell _ n\left(2\right)\right) > q_{\alpha }\right\} for asymptotic level α, where q_α is the (1-α)-quantile of the chi-squared distribution with 1 degree of freedom.
This is a valid test as it compares the log-likelihoods under the null hypothesis and the maximum likelihood estimate. It can be applied asymptotically as the sample size increases.
The other options are not technically correct or do not have the required level α:
3) \mathbf{1}\left\{ \sqrt{n} \frac{\left| 2 \overline{X_ n} - 2 \right|}{\sqrt{4/3}} > q_{\alpha /2}\right\} for asymptotic level α, where q_α is the (1-α)-quantile of the standard normal random variable.
This test is not valid because it incorrectly uses a standard normal distribution for the test statistic instead of the correct distribution (t-distribution).
4) \mathbf{1}\left\{ \widehat{\theta _ n}^{\text {MLE}} > 2 \text { or } \widehat{\theta _ n}^{\text {MLE}} \le 1 \right\} for asymptotic level α.
This test is not valid because it combines two separate events (MLE greater than 2 and MLE less than or equal to 1) into one test statistic. The correct approach would be to use separate indicators for each event.