Part BCSIR NET December 2023sufficient-statistic-by-expansion

Sufficient statistic by expansion

For n ≥ 2, let $ε_{1}, \dots, ε_{n}$ be $i . i . d . N (0, σ^{2})$ and $Y_{i} = i α + i^{2} α^{2} + ε_{i}$ , where $σ > 0$ and $α \in R$ are unknown. Which of the following is a jointly minimal sufficient statistic for $(α, σ)$ ?

A. $(\sum Y_{i}^{2}, \sum$ iY $_{i}, \sum i^{2} Y_{i})$
B. $(\sum Y_{i}^{2}, \sum$ iY $_{i}, \sum i^{2} Y_{i}^{2})$
C. $(\sum$ iY $_{i}, \sum i^{2} Y_{i}^{2})$
D. $(\sum Y_{i}, \sum$ iY $_{i})$

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p-values, size, power and error types say specific things. This was not one of them.

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Related counterexample: A sufficient statistic is complete

Sufficient statistic by expansion

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