Answer: Only I

**Explanation:** **Statement I is correct:** The Q-statistic (both Box-Pierce and Ljung-Box variants) measures the degree to which autocorrelations deviate from zero and tests whether white noise is present in a dataset. It's used to test the null hypothesis that a series is white noise. **Statement II is incorrect:** The Box-Pierce Q-statistic represents the **sum** of squared autocorrelations, not the weighted sum. The formula for Box-Pierce Q-statistic is: $$Q_{BP} = n \sum_{k=1}^{m} \hat{\rho}_k^2$$ where n is the sample size, m is the number of lags, and $\hat{\rho}_k$ is the autocorrelation at lag k. **Statement III is incorrect:** The Ljung-Box Q-statistic represents the **weighted** sum of squared autocorrelations, not the simple sum. The formula for Ljung-Box Q-statistic is: $$Q_{LB} = n(n+2) \sum_{k=1}^{m} \frac{\hat{\rho}_k^2}{n-k}$$ The Ljung-Box statistic is a modified version that has better small-sample properties than the Box-Pierce statistic. Since only Statement I is correct, the correct answer is **A. Only I**.

Author: Nikitesh Somanthe