91Ó°ÊÓ

The pdf for the first order statistic \(Y_1\) from a sample of size \(n\) is given by the formula \(f_{Y_1}(y) = n \cdot [1 - F(y)]^{n-1} \cdot f(y), y > \theta \), where \(F(y)\) is the cumulative distribution function of the original distribution. The cumulative distribution function \(F(y)\) of the original distribution can be computed as \(F(y) = \int_{\theta}^{y} f(x) dx = 1 - e^{-(y-\theta)}\). Now substituting \(F(y)\) and \(f(y)\) in the formula gives \(f_{Y_1}(y) = n \cdot e^{-(n-1)(y-\theta)}, y > \theta \).

The Neyman Factorization Theorem can now be used to show that \(Y_1\) is a sufficient statistic for \(\theta\). According to the theorem, a statistic \(T(X)\) is sufficient for \(\theta\) if the joint pdf \(f(x_1, x_2, ..., x_n;\theta)\) can be written as the product of two functions, the first being a function \(g(T(x),\theta)\) of both \(T(x)\) and \(\theta\), and the second \(h(x)\) depending only on \(x\). The joint pdf in this case can be written as the product of the individual pdfs, i.e. \(f(x_1, x_2, ..., x_n;\theta) = e^{-(nY_1-n\theta)} \cdot e^{-(n-n)}\). We can observe that the joint pdf has the required form as per Neyman Factorization Theorem and hence \(Y_1\) is a sufficient statistic.

Showing completeness can be achieved by showing if \(E[g(T)]=0\) for all \(\theta\), then \(g(T)=0\) with probability 1. We can use the pdf of \(Y_1\) to compute the expectation. Let \(g(T)=g(Y_1)\) be a function such that \(E[g(Y_1)]=0\) for all \(\theta\). Then we must solve: \(0=\int_{\theta}^{\infty} g(y) \cdot n \cdot e^{-(n-1)(y-\theta)} dy\) for all \(\theta\). If and only if \(g(y) = 0\) for \(y > \theta\), will the integral be zero for every \(\theta\). Hence, \(Y_{1}\) is a complete statistic.

Lehmann-Scheffe Theorem states that if a statistic is complete and sufficient, then any unbiased estimator based on that statistic is the unique MVUE. From the previous steps, we know that \(Y_{1}\) is complete and sufficient. Now we need to find an unbiased estimator \(g(Y_{1})\). Unbiased estimator has the property that its expected value equals the parameter. Here, the unbiased estimator would be \(\theta\). Now we need to calculate \(E[Y_{1}]\) using \(Y_{1}\)‘s pdf. On calculating the expectation, we get \(E[Y_{1}] = \theta+ \frac{1}{n}\), hence it's a biased estimator because expectation does not equal \(\theta\). Therefore, we introduce a function of \(Y_{1}\), \(g(Y_{1}) = Y_{1} - \frac{1}{n}\) which is the unbiased estimator since \(E[g(Y_{1})] = E[Y_{1}] - \frac{1}{n} = \theta\). This function of \(Y_{1}\) is the MVUE for \(\theta\).