Mathematics · Statistical Inference

Let X 1 , X 2 , ..., X n be independent and identically distributed (i.i.d.) N(0, σ 2 ) random variables. Which of the following statistics are sufficient for σ 2 ? I. T 1 (X 1 , ..., X n ) = (X 1 , ..., X n ) II. T 2 (X 1 , ..., X n ) = (X 1 2 , ..., X n 2 ) III. T 3 (X 1 , ..., X n ) = ∑ i=1 n X i 2 IV. T 4 (X 1 , ..., X n ) = (X 1 2 + X 2 2 , X 3 2 + ... + X n 2 )