| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | B |
B |
| deepseekr1 | The question asks which statement is incorrect regarding the comparison between the Box-Pierce Q and Ljung-Box Q* tests for autocorrelation. Here's the analysis: - **Option A**: Correct. Asymptotically, both tests are equivalent because the Ljung-Box adjustment factor \((n+2)/(n-k)\) approaches 1 as \(n \to \infty\), making their statistics equal. - **Option B**: Incorrect. The Ljung-Box Q* test was developed to improve small-sample performance over the Box-Pierce Q test. Thus, claiming that Q has better small-sample properties than Q* is false. - **Option C**: Correct. The Box-Pierce Q test is known to be over-sized (i.e., rejects the null hypothesis more frequently than the nominal significance level) in small samples. - **Option D**: Incorrect in general. While both tests will reject the null with high probability if autocorrelation exists (true alternative), they should not "always reject" under the null hypothesis (zero autocorrelation). As \(n \to \infty\), their rejection rates under the null should approach the nominal significance level (e.g., 5%), not 100%. However, this statement does not directly compare the two tests but misrepresents their asymptotic behavior. The **key distinction** lies in **Option B**, which directly compares the tests' small-sample properties. Since Ljung-Box Q* is superior in small samples, stating that Box-Pierce Q is better is unequivocally incorrect. While Option D is also incorrect, it pertains to a general asymptotic property rather than a direct comparison between the two tests. The question specifically asks for the incorrect comparison, making **B** the best answer. **Answer: B** |
B |
| gpto1 | B | B |
| GPT-4o | B | B |
| llama3.1-70B-Instruct | B |
B |
| llama3.1-405B-Instruct | B |
B |
| DeepSeek-Coder-V2 | D | B |
| Qwen2-72B-Instruct | B | B |
| baichuan4 | D | B |
| doubao | B The Ljung-Box Q* statistic generally has better small-sample properties than the Box-Pierce Q test. |
B |
| minimax | B. The Q test has better small-sample properties than the Q* | B |
| kimi | D | B |
| wenxinyiyan | B. The Q test has better small-sample properties than the Q*. Explanation: The Box-Pierce Q statistic and the Ljung-Box Q* statistic are both used to test for linear dependence in time series data, specifically to test the null hypothesis that the autocorrelation coefficients at lags greater than zero are all zero. A. Asymptotically, the values of the two test statistics will be equal. This is correct because as the sample size increases, the difference between the two statistics becomes negligible. B. The Q test has better small-sample properties than the Q*. This is incorrect. In fact, the Ljung-Box Q* statistic was developed as a modification of the Box-Pierce Q statistic to address some of its issues, particularly in small samples. The Q* test often provides better small-sample properties than the Q test. C. The Q test is sometimes over-sized for small samples. This is correct. In small samples, the Q test can sometimes have a type I error rate (false positive rate) that is higher than the specified significance level, meaning it rejects the null hypothesis more often than it should. D. As the sample size tends towards infinity, both tests will show a tendency to always reject the null hypothesis of zero autocorrelation coefficients. This is correct, assuming that the true autocorrelation coefficients are not all zero. As the sample size increases, both tests gain more power to detect deviations from the null hypothesis, leading to more frequent rejections. However, if the null hypothesis is true, then both tests should asymptotically maintain the correct type I error rate. |
B |
| zhipu | The correct answer is: D. As the sample size tends towards infinity, both tests will show a tendency to always reject the null hypothesis of zero autocorrelation coefficients. Explanation: - A is correct because asymptotically (as the sample size grows very large), the Box-Pierce Q statistic and the Ljung-Box Q* statistic tend to converge in value. - B is correct because the Box-Pierce Q statistic generally has better small-sample properties compared to the Ljung-Box Q* statistic. - C is correct because the Box-Pierce Q statistic can be over-sized (i.e., it tends to reject the null hypothesis more often than it should) in small samples. - D is incorrect because as the sample size tends towards infinity, the tests do not necessarily always reject the null hypothesis of zero autocorrelation coefficients. They are designed to have the correct size (i.e., the probability of rejecting the null hypothesis when it is true converges to the chosen significance level) as the sample size increases. Therefore, D is the incorrect statement. |
B |
| gpto1 | B | B |