PhD Qualifying Exam in Statistics 0.4 August, 2007

Let \(X_1,X_2,\ldots ,X_n\) and \(Y_1,Y_2,\ldots ,Y_m\) be two independent random samples. We assume that \(X_1,X_2,\ldots ,X_n\) are independent identically distributed exponential \((\lambda )\) random variables and \(Y_1,Y_2,\ldots ,Y_m\) are independent identically distributed exponential \((\mu )\) random variables. We want to test \(H_0:\lambda =\mu \) against the alternative \(H_A:\lambda \ne \mu \).

(i) Find \(T_{m,n}\) the likelihood ratio test. (7 points)

(ii) Prove that \(-2\log T_{m,n}\) converges in distribution under the null hypothesis when \(\min (m,n)\to \infty \). (7 points)

(iii) Show that the likelihood ratio test is asymptotically consistent and prove that the power goes to 1 as \(\min (m,n)\to \infty \). (7 points)

Let \(X_1,X_2,\ldots ,X_n\) and \(Y_1,Y_2,\ldots ,Y_m\) be two independent random samples. We assume that \(X_1,X_2,\ldots ,X_n\) are independent identically distributed normal \((\mu _1,\sigma ^2)\) random variables and \(Y_1,Y_2,\ldots ,Y_m\) are independent identically distributed normal \((\mu _2,\sigma ^2)\) random variables (\(\sigma ^2\) is unknown). We want to test \(H_0:\mu _1=\mu _2\) against the alternative \(H_A:\mu _1\ne \mu _2\).

Show that the likelihood ratio test and the two sample t-test are equivalent. (7 points)

Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed random variables with distribution function \(F\). We use \[ \hat f_n(t)=\frac {1}{nh}\sum _{1\leq i\leq n} K\left (\frac {t-X_i}{h}\right ) \] to estimate \(f=F'\).

We assume that the following conditions are satisfied: \[ \int _{-1}^{1}K(u)\,du=1, \qquad K(t)=0,\quad \text {if } t\notin [-1,1]. \] \(F\) is three times differentiable and the derivatives are uniformly bounded on the real line \[ h=h(n)\to 0. \]

(i) Compute the bias of \(\hat f_n(t)\) for any \(t\) and any \(n\). (7 points)

(ii) Prove that \(\hat f_n(t)\) is asymptotically unbiased for any \(t\) as \(n\to \infty \). (7 points)

(iii) Compute the mean square error of \(\hat f_n(t)\) (7 points)

Let \(X\) be a random variable with distribution function \(F\). Prove that \(X\) is symmetric around 0 (i.e. \(F(x)=1-F(-x)\) for all \(x\)) if and only if the characteristic function of \(X\) is real. (7 points)

Let \(X_1\) and \(X_2\) be independent standard normal random variables. Find the necessary and sufficient condition for the independence of \(Y_1=a_1X_1+a_2X_2\) and \(Y_2=b_1X_1+b_2X_2\). (7 points)

Let Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed random variables with distribution function \[ F(t)=\left (\frac {t}{\theta }\right )^2 \quad \text {if } 0\leq t\leq \theta . \]

(i) Find the maximum likelihood estimator for \(\theta \). (7 points)

(ii) Compute the limit distribution of the maximum likelihood estimator. (7 points)

Let Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed random variables with \(\E X_1=\mu \), \(\operatorname {Var} X_1=\sigma ^2\). We assume that \(\E X_4<\infty \). The random variable \[ \hat \sigma _n^2=\frac {1}{2n}\sum _{2\leq i\leq n}(X_i-X_{i-1})^2 \] is suggested to estimate \(\sigma ^2\).

(i) Establish some basic properties of \(\hat \sigma _n^2\). (7 points)

(ii) Is \(\hat \sigma _n^2\) a better estimator than the sample variance? Justify your answer. (7 points)

Let \((X_1,\ldots ,X_m)\) be a multinomial random vector with parameters \(N\) and \(p_1,p_2,\ldots ,p_m\). Note that \(N=X_1+\cdots +X_m\) and \(p_1+p_2+\cdots +p_m=1\). Find the maximum likelihood estimators for \(p_1,p_2,\ldots ,p_m\). (7 points)

Let \(X\) and \(Y\) be independent random variables with density function \(f\) and \(g\). Assume that \(Y>0\). Show that the function \[ h(t)=\int _{0}^{\infty } z f(zt)g(z)\,dz \] can be chosen as the density of \(Z=X/Y\). (7 points)