Exam in Statistics 20 Aug 1999

Let \(X_1,X_2,\ldots ,X_n\) be independent random variables. We assume that \(X_i\) has a Poisson\((\lambda _i)\) distribution. We wish to test \[ H_0:\lambda _1=\lambda _2=\cdots =\lambda _n \] against the alternative \[ H_A:H_0\text { is not true.} \]

1. Show that the likelihood ratio test and the \(\chi ^2\) tests are equivalent under \(H_0\). (15 points)
2. Explain why and how the normal as well as the \(\chi ^2\) distributions can be used to approximate the critical values. (5 points)

Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed random variables with a continuous distribution function \(F\). The ordered observations are \(X_{1,n}\leq X_{2,n}\leq \cdots \leq X_{n,n}\). Let \[ T_n=\sum _{i=1}^{n-1}\left (F(X_{i+1,n})-F(X_{i,n})\right )^2. \] Show that there are numerical sequences \(a_n\) and \(b_n\) such that \((T_n-a_n)/b_n\) converges in distribution to a standard normal random variable. (20 points)

Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed normal \(N(0,\sigma ^2)\) random variables. We wish to test \[ H_0:\sigma \leq \sigma _0 \] against \[ H_A:\sigma >\sigma _0. \]

1. Find the uniformly most powerful test of size \(\alpha \). (10 points)
2. Compute the rejection region as a function of \(\alpha \). (10 points)

Consider the following model: \[ y_i=f(x_i)+\epsilon _i,\qquad 1\leq i\leq n, \] where \[ f(t)=a_0+a_1t+\cdots +a_pt^p. \] The parameters \(a_0,a_1,\ldots ,a_p\) are unknown. The errors \(\epsilon _1,\epsilon _2,\ldots ,\epsilon _n\) are independent normal \(N(0,\sigma ^2)\) random variables. It is assumed that \(x_i\in [0,1]\) and known. We observe \((y_i,x_i)\), \(1\leq i\leq n\).

1. Estimate the parameters \(a_0,a_1,\ldots ,a_p\) and \(\sigma ^2\). (15 points)
2. Get the distribution of the estimator for \(\sigma ^2\). (5 points)

Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed random variables with distribution function \(F\). We assume that \(f=F'\) has a bounded second derivative. \(K(t)\) is a non-negative function satisfying \[ K(t)=K(-t)\quad \text {(symmetric),} \] \[ \int _{-\infty }^{\infty }K(t)\,dt=1 \quad \text {and}\quad K(t)=0\text { if }t\notin [-1,1]. \] The sequence \(h=h(n)\) is positive and goes to 0. We estimate \(f\) with \[ \hat {f}_n=\frac {1}{nh}\sum _{1\leq i\leq n}K\left (\frac {t-X_i}{h}\right ). \]

1. Compute \(B_n\), the bias of \(\hat {f}_n\). (10 points)
2. Compute the limit of \(B_n/h^2(n)\) as \(n\to \infty \). (10 points)

Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed random variables uniform on \([\theta ,\theta +1]\).

1. Find a sufficient statistic for \(\theta \). (5 points)
2. Is your statistic complete? (5 points)
3. Find the maximum likelihood estimator for \(\theta \). (10 points)

Let \(X_1,X_2,\ldots ,X_n\) and \(Y_1,Y_2,\ldots ,Y_m\) be independent random samples of independent observations. The distribution of \(X_i\) is normal \(N(\mu _1,\sigma _1^2)\) and the distribution of \(Y_i\) is normal \(N(\mu _2,\sigma _2^2)\). The values of \(\sigma _1\) and \(\sigma _2\) are known. We wish to test \(H_0:\mu _1=\mu _2\) against \(H_A:\mu _1\neq \mu _2\). Show that the likelihood ratio test and the t-test are equivalent under \(H_0\). (20 points)

Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed normal \(N(\mu ,\sigma ^2)\) random variables. Let \[ U_n=\frac {2}{n(n-1)} \sum _{1\leq i<j\leq n}I\{|X_i-X_j|>1\}. \]

1. Compute \(EU_n\). (10 points)
2. Assume that \(\sigma \) is known. We wish to test \(H_0:\mu =\mu _0\) (\(\mu _0\) is given) against the alternative that \(H_0\) is not true. Can we use \(U_n\) to test \(H_0\) against this alternative? (5 points)
3. Now \(\mu \) and also \(\sigma \) are unknown. We wish to test the null-hypothesis that \(\sigma =\sigma _0\) against the alternative that \(H_0\) is not true. Can we use \(U_n\) now? (5 points)