为什么sample是低估

问题如下：

An analyst working for a large investment firm is evaluating the performance of various investment funds as part of a report to the firm's portfolio managers. One component of the evaluation is the assessment of the statistical moments of the entire population of fund returns. Since the analyst has access to a very large dataset of historical returns, the analyst decides to estimate sample moments such as the sample mean return, sample variance, and sample skewness using a smaller subset of historical returns data. Which of the following would be correct for the analyst to conclude about the sample variance?

选项：

A.An unbiased estimator cannot be constructed for the sample variance.

B.The sample variance is biased when it is formulated as a function of the population size decreased by 1.

C.The sample variance slightly underestimates the population variance.

D.The sample variance diverges from the population variance as the size of the sample increases.

解释：

Explanation: C is correct. The sample variance depends on the estimator of the mean, or the sample mean. Estimation of the mean consumes a degree of freedom, and the sample mean tends to resemble the sample of data a little too well. This slight “overfitting” produces a slight underestimation of the population variance.

A is incorrect. Because the bias of the sample variance is known, an unbiased estimator for the variance can be constructed as:

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Note that this procedure divides by n − 1 instead of n, and the resulting estimator is unbiased.

B is incorrect. As explained above in A, the correction of dividing by n − 1, where n is the sample size, makes the sample variance unbiased.

D is incorrect. The sample variance converges to the population variance as the sample size n increases. As the sample size grows larger, the sample variance becomes a more accurate estimator of the population variance because the limit of the bias goes to 0.

Learning Objective: Explain the difference between a population moment and a sample moment.

Reference: Global Association of Risk Professionals. Quantitative Analysis. New York, NY: Pearson, 2023, Chapter 5, Sample Moments [QA-5].