# How to prove or disprove that $T(X_{1},X_{2}) = X_{1} + X_{2}$ is a sufficient statistic

Let $$X_{1},X_{2},\ldots,X_{n}$$ be random sample from a population whose distribution is given by $$X\sim\text{Bernoulli}(\theta)$$, $$0 < \theta < 1$$.

a. Show that $$T(x) = \displaystyle\sum_{i=1}^{n}X_{i}$$ is sufficient for $$\theta$$.

b. In the same context, consider that $$T = X_{1} + X_{2}$$. What is the distribution of $$T$$? Is $$T$$ a sufficient statistic?

MY ATTEMPT

a. To begin with, let us determine the likelihood function for this sample \begin{align*} L(\textbf{x}|\theta) = \prod_{i=1}^{n}\theta^{x_{i}}(1-\theta)^{1-x_{i}} = \theta^{\sum x_{i}}(1-\theta)^{n - \sum x_{i}} = h(\textbf{x})g_{\theta}(T(\textbf{x})) \end{align*}

Therefore, according to the factorization theorem, $$T(\textbf{x})$$ is sufficient for $$\theta$$.

b. As it is known, the sum of independent Bernoulli random variables is a Binomial random variable. Therefore $$T = X_{1} + X_{2} \sim \text{Binomial}(2,\theta)$$.

Then I get stuck. Can someone help me out? Thanks in advance!

• (a) trivially implies the answer to the second part of (b) by setting $n=2.$ – whuber May 17 '19 at 15:12
• On the other hand, if $n>2$ then the answer by @Vishaal answer is helpful. I think this may be the intention of the question, but it is unclear with the information you have given us. – knrumsey May 17 '19 at 17:34

All you need to show is that $$P(X_1=x_1,X_2=x_2,...X_n=x_n|X_1+X_2 = t )$$ depends on $$\theta$$ to prove that $$T=X_1+X_2$$ is NOT a sufficient statistic.

• Could you explain what part of the question this answers and why it answers it? – whuber May 17 '19 at 15:13
• @Whuber I have answered "Is $X_1 + X_2$ sufficient?" ... That was the last question where the OP got stuck. I interpreted the question like this, out of a random sample following bernoulli($\theta$)of size N if we know the sum of the first 2 random variables does it contain sufficient information about the parameter $\theta$. – Vishaal Sudarsan May 17 '19 at 15:34
• Unfortunately, that answer is wrong. It contradicts the answer to question (1). – whuber May 17 '19 at 16:07
• According to me.. Question(1) is whether sum of all samples is sufficient of not. Question(2) is, In the presence of N samples, whether the sum of first 2 sufficient or not. I've updated my answer. Please check it once more. – Vishaal Sudarsan May 17 '19 at 16:24
• In that case you're correct. But I understood the question as we have already observed n samples, now before discarding the sample we would like to compute a sufficient statistic for $\theta$, so is $X_1 +X_2$ a sufficient statistic?. I interpreted the question in this way because in Question(2) it says "In the same context" so I assumed the OP wants to say all n samples have been observed like it has been observed for Question(1). – Vishaal Sudarsan May 17 '19 at 17:13