Question

If \(X_{1}, X_{2}\) is a random sample of size 2 from a distribution having pdf \(f(x ; \theta)=(1 / \theta) e^{-x / \theta}, 0

Short answer

The joint pdf of the sufficient statistics \(Y_{1}\) and \(Y_{2}\) is \(g(y_{1}, y_{2} ; \theta) = (1 / \theta^{2}) e^{- y_{1} / \theta}, 0 < y_{2} < y_{1} < \infty\). \(Y_{2}\) is an unbiased estimator of \(\theta\) with variance \(\theta^{2}\). The conditional expectation of \(Y_{2}\) given \(Y_{1}\) is \(y_{1} / 2\), and the variance of this estimator is \(\theta^{2} / 12\).

Step-by-step solution

Find joint pdf

The joint pdf of \(X_{1}, X_{2},\) given by \(f(x_{1},x_{2} ; \theta)=(1 / \theta^{2}) e^{-(x_{1}+x_{2}) / \theta}, 0<x_{1},x_{2}<\infty\). If we let \(Y_{1}=X_{1}+X_{2}\), and \(Y_{2}=X_{2}\), then we can find the joint pdf of \(Y_{1}\) and \(Y_{2}\) by changing variables, giving \(g(y_{1},y_{2} ; \theta)=(1 / \theta^{2}) e^{-y_{1} / \theta}, 0<y_{2}<y_{1}<\infty\).

Prove \(Y_{2}\) is unbiased estimator

An estimator is unbiased if its expected value is equal to the parameter it estimates. The expected value of \(Y_{2}\) can be calculated as: \(E(Y_{2}) = \int_{0}^{\infty}y_{2}*g(y_{2} ; \theta)dy_{2} = \theta\). So it is an unbiased estimator of \(\theta\). The variance of \(Y_{2}\) can be calculated as: \(Var(Y_{2}) = E(Y_{2}^2) - (E(Y_{2}))^2\), and we get \(Var(Y_{2}) = \theta^{2}\). Therefore, \(Y_{2}\) is an unbiased estimator of \(\theta\) with variance \(\theta^{2}\).

Compute Conditional Expectation and Its Variance

Finally, compute \(E(Y_{2}|Y_{1}= y_{1}) = \varphi(y_{1})\) as the integral of \(y_{2}*g(y_{1}, y_{2} ; \theta)\) with respects to the variable \(y_{2}\) over the range 0 to \(y_{1}\). Solving this gives us \[\varphi(y_{1}) = y_{1} / 2\]. The variance of \(\varphi(Y_{1})\) is the expectation of \(\varphi^2(Y_{1})\) minus the square of the expectation of \(\varphi(Y_{1})\), calculated using the pdf of \(Y_{1}\). Evaluting this gives us \(Var(\varphi(Y_{1})) = \theta^{2} / 12\).

Key concepts

Sufficient Statistic

A sufficient statistic is a metric that captures all the relevant information from a sample regarding a parameter. In the context of our problem, the sum of the two sampled values, denoted as Y1=X1+X2, is considered a sufficient statistic for the parameter θ. This means that Y1 encompasses all the necessary information needed to estimate θ effectively.

To confirm the sufficiency, we often use a factorization theorem. The theorem states that if the joint probability density function of the sample can be factorized into two parts – one involving only the sample and the statistic, and the other involving only the parameter and the statistic, then the statistic is sufficient. Here, since the joint pdf factors into a function of Y1 and θ, Y1 serves as a sufficient statistic.

Unbiased Estimator

An unbiased estimator is one whose expected value is equal to the true value of the parameter it is estimating. In simpler terms, if you were to repeatedly estimate the parameter using your unbiased estimator across numerous samples, the average of those estimates would converge to the actual parameter value.

In our exercise, Y2 has been shown to be an unbiased estimator for θ. This is because the expected value of Y2 equals θ, fulfilling the unbiased condition. The computation of its expected value through integration confirms that Y2 consistently estimates θ without any systematic error, making it a reliable metric for inference.

Conditional Expectation

Conditional expectation refers to the expected value of a random variable given that another variable is known. It's a way of understanding the average outcome of a random process when certain information is given.

The exercise demonstrates this through the calculation of E(Y2|Y1=y1), which is the expected value of Y2 when Y1 is known to be y1. Symbolically, this is often expressed as a function, &varphi(y1), which represents the conditional expectation of Y2. Knowing the distribution of your sufficient statistic Y1, allows one to determine this function and thus understand how the expected value of Y2 changes with Y1.

Variance of an Estimator

The variance of an estimator measures the spread of its sampling distribution, or in other words, how much the estimator's values differ from one sample to another. A smaller variance means that the estimator is consistently close to the true parameter value across different samples.

In our provided problem, the variance of the estimator Y2 is θ2, which indicates how spread out the estimates will be around the true value of θ. Moreover, the variance of the conditional expectation &varphi(Y1) has also been calculated, with a resulting variance of θ2/12. This signifies the expected variability in the conditional estimates given different values of the sufficient statistic Y1, reflecting the precision of &varphi(Y1) as an estimator in the presence of this additional information.