Question

What is the sufficient statistic for $\theta$ if the sample arises from a beta distribution in which $\alpha =\beta =\theta >0?$

Short answer

The sufficient statistic for $\theta$ in this Beta distribution is $\sum _{i=1}^n{X}_i$

Step-by-step solution

Familiarizing with Beta Distribution

Firstly, it's important to understand that Beta distribution is a family of continuous probability distributions defined on the interval [0, 1] parameterized by two positive shape parameters, denoted by $\alpha$ and $\beta$, that appear as exponents of the random variable and control the shape of the distribution.

Recognize Distribution Parameters

In this scenario, the Beta distribution has equal parameters $\alpha =\beta =\theta >0$. Therefore this exercise involves the case when the two shape parameters are identical, resulting in a symmetrical distribution.

Apply Concept of Sufficient Statistics

A statistic is sufficient for a parameter if it includes all the information in the sample about the parameter. Here, we are interested in finding a sufficient statistic for $\theta$, the sample consisting of observations $X_1,X_2,...,X_n$ from a Beta distribution.

Determine the Sufficient Statistic

From the properties of the Beta distribution, we know that the sum of the sample observations, $\sum _{i=1}^n{X}_i$, is a sufficient statistic for $\theta$. This is because it contains all necessary information to estimate $\theta$, making the individual $X_i$'s irrelevant once the sum is known.