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Chapter 7: Problem 8

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

Short Answer

Expert verified
The sufficient statistic for \(\theta\) in this Beta distribution is \(\sum_{i=1}^{n} X_i\)

Step by step solution

01

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.
02

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.
03

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.
04

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.

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Most popular questions from this chapter

Let \(Y_{1}

Let \(Y_{1}

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from \(N\left(\theta_{1}, \theta_{2}\right)\). (a) If the constant \(b\) is defined by the equation \(P(X \leq b)=0.90\), find the mle and the MVUE of \(b\). (b) If \(c\) is a given constant, find the mle and the MVUE of \(P(X \leq c)\).

Let the pdf \(f\left(x ; \theta_{1}, \theta_{2}\right)\) be of the form $$\exp \left[p_{1}\left(\theta_{1}, \theta_{2}\right) K_{1}(x)+p_{2}\left(\theta_{1}, \theta_{2}\right) K_{2}(x)+S(x)+q_{1}\left(\theta_{1}, \theta_{2}\right)\right], \quad a

Let \(X_{1}, X_{2}, \ldots, X_{n}\) denote a random sample of size \(n>1\) from a distribution with pdf \(f(x ; \theta)=\theta e^{-\theta x}, 00 .\) Then \(\sum_{1}^{n} X_{i}\) is a sufficient statistic for \(\theta\). Prove that \((n-1) / Y\) is the MVUE of \(\theta\).
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