Let we have $X_1,X_2,...,X_n$ random sample of size $n$ from normal population with mean $\theta$ and variance $\sigma^2$. Define $T(x)=(X_3,....,X_{n-8})$. It is required to test the sufficiency of the estimator $T(x)$. Our teacher told us that the sample will always be sufficient. So, is that mean $T(x)$ will be sufficient since this is nothing but Samples. Also, will $T(x)=X_1$ be sufficient since $X_1$ is part of the sample. I have just began learning these things in inference. Please clear my doubt regarding the sufficiency of the samples. My main doubt is in which order the samples will be sufficient. So we have to take the whole sample together to have sufficiency or samples taken in any order will be sufficient? I am really confused.
I don't want to use the factorization theorem or don't want to use the definition of Sufficiency. Right now, I just wanted to make sure I understand the meaning of Sufficiency in regard of samples. Thanks in advance
Any sub-sample of $(X_1,...,X_n)$ will be an insufficient statistic. The intuition is if the sample $X_1,...,X_n$ is iid, then the information (Fisher's information) in this sample is $n$ times the information in one observation, i.e., $n\mathcal{I}_X(\theta)$. Hence, if you are taking a function $T(.)$ (including the identity) function of $k$ (such that $k< n$) observations, the information in $T(X)$ will be $\le k\mathcal{I}_X(\theta)$, which is strictly less thnn $n\mathcal{I}_X(\theta)$.