Discussion
Suppose \(\{X_{1}, X_{2}, ......, X_{n}\}\) are independent and identically distributed random variables with mean \(\mu\) and variance \(\sigma^{2}\). Suppose \(S_{n}^{2} = \dfrac{1}{n - 1}\sum_{i=1}^{n} (X_{i} - \bar{X}_{n})^{2}\). Then which of the following statements is not true ?
Answer: A
The statement that is not true is \(S_{n}\) is unbiased for \(\sigma\).
\(S_{n}^{2}\) with divisor \(n - 1\) is unbiased for \(\sigma^{2}\), but the square root is strictly concave, so by Jensen's inequality
\[ E(S_{n}) = E\sqrt{S_{n}^{2}} \lt \sqrt{E S_{n}^{2}} = \sigma \]
unless \(S_{n}\) is constant. So \(S_{n}\) underestimates \(\sigma\) on average.
The other three are true: B is the reason for the \(n - 1\) divisor, \(S_{n}^{2} \to \sigma^{2}\) in probability by the law of large numbers, and \(S_{n} \to \sigma\) then follows because the square root is continuous. Unbiasedness is not preserved by non-linear maps, but consistency is.
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