PhD qualifying exam in statistics-2015, Summer

Let \(X_1,X_2,\ldots ,X_n\) be independent, random variables with density functions \[ h(t;\theta _i)= \begin {cases} \dfrac {1}{\theta _i}e^{-t/\theta _i}, & \text {if } 0\leq t<\infty ,\\ 0, & \text {if } -\infty <t<0. \end {cases} \] We wish to test \(H_0:\theta _1=\theta _2=\cdots =\theta _n\) against the alternative that \(H_0\) is not true.

(a) Compute \(\lambda \), the generalized likelihood ratio and explain the application of the generalized likelihood test.

(b) Provide a large sample approximation for \(-2\log \lambda \) under \(H_0\).

Let \(X_1,X_2,\ldots ,X_n\) be independent, identically distributed random variables with density function \[ h(t;\theta )= \begin {cases} \dfrac {1}{\theta }, & \text {if } 0\leq t\leq \theta ,\\ 0, & \text {if } t\notin [0,\theta ]. \end {cases} \] We wish to test \(H_0:\theta =1\) against the alternative \(H_A:\theta <1\). We reject \(H_0\) if \[ X_{n,n}=\max _{1\leq i\leq n}X_i\leq 1-\frac {2}{n}. \] (a) Compute the significance level.

(b) Compute the power function.

Let \(X_1,X_2,\ldots ,X_n\) be independent, identically distributed random variables with density function \[ h(t;\sigma )=\frac {1}{(2\pi \sigma ^2)^{1/2}} e^{-(t-1)^2/(2\sigma ^2)}. \] We wish to test \(H_0:\sigma =\sigma _0\) against \(H_A:\sigma \ne \sigma _0\).

(a) Find a test, using the generalized likelihood ratio \(\lambda \).

(b) Provide a large sample approximation for \(-2\log \lambda \) under \(H_0\).

Let \(X_1,X_2,\ldots ,X_n\) be independent, identically distributed random variables with density function \[ h(t;\mu )=\frac {1}{(2\pi )^{1/2}}e^{-(t-\mu )^2/2}. \] We wish to test \(H_0:\mu \leq \mu _0\) against \(H_A:\mu >\mu _0\). Find the uniformly most powerful test of size \(\alpha \).

Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed normal \(N(\mu _1,\sigma ^2)\) random variables. Let \(Y_1,Y_2,\ldots ,Y_m\) be independent identically distributed normal \(N(\mu _2,\sigma ^2)\) random variables. The two samples are independent. The parameters \(\mu _1,\mu _2,\sigma ^2\) are unknown. We wish to test \(H_0:\mu _1=\mu _2\). Show that the two sample t-test and the likelihood ratio test are equivalent.

Let \(X_1,X_2\) be independent and identically distributed exponential\((\lambda )\) random variables, i.e. the common density is \[ f(t)= \begin {cases} 0, & \text {if } t<0,\\ \dfrac {1}{\lambda }\exp (-t/\lambda ), & \text {if } t\geq 0. \end {cases} \] Find the joint density function of \[ Y_1=X_1+X_2 \quad \text {and}\quad Y_2=\frac {X_1}{X_1+X_2}. \]

We assume that the number of people entering a store follows a Poisson distribution with parameter \(\lambda \). Every customer spends money in the store, independently of each other according to a uniform distribution on \([10,100]\). We observe \(X\), the total amount of money spent in the store on a given day.

(a) Estimate \(\lambda \) using the method of moments.

(b) Let assume that we have the stores income \(X_1,X_2,\ldots ,X_n\) for \(n\) days and \(X_1,X_2,\ldots ,X_n\) are independent and distributed as \(X\). What would be the moment estimator for \(\lambda \) based on \(X_1,X_2,\ldots ,X_n\)?

Let \(X\) and \(Y\) be independent random variables with densities \(f\) and \(g\), where \[ f(t)= \begin {cases} 0, & \text {if } t<0,\\ \dfrac {1}{2}\exp (-t/2), & \text {if } t\geq 0, \end {cases} \] and \[ g(t)= \begin {cases} 0, & \text {if } t\notin [-1,2],\\ \dfrac {1}{3}, & \text {if } t\in [-1,2]. \end {cases} \] Compute the density of \(X+Y\).

Let \(X_1,X_2,\ldots ,X_n\) be independent and identically distributed random variables with density function \[ f(t)= \begin {cases} 0, & \text {if } t<0,\\ \dfrac {1}{\lambda }\exp (-t/\lambda ), & \text {if } t\geq 0. \end {cases} \] Construct an equal tail \(1-\alpha \) confidence interval for \(\lambda \). Compute the expected value of the length of the confidence interval.

Let \(X_1,X_2,\ldots ,X_n\) be independent and identically distributed random variables with density function \[ f(t)= \begin {cases} 0, & \text {if } t<0,\\ \dfrac {1}{\lambda }\exp (-t/\lambda ), & \text {if } t\geq 0. \end {cases} \] Find the uniformly minimum variance unbiased estimator for \(\theta =1/\lambda \).

Let \(U_1,U_2,\ldots ,U_n\) be independent and identically distributed random variables, uniform on \([0,1]\). Let \(U_{1,n}\leq U_{2,n}\leq \cdots \leq U_{n,n}\) denote the order statistics.

(a) Compute the limit distribution of \(Z=n(U_{3,n}-U_{2,n})\).

(b) Show that \(Z_1=n(U_{3,n}-U_{2,n})\) and \(Z_2=n(U_{4,n}-U_{3,n})\) are asymptotically independent.

Let \(X_1,X_2,\ldots ,X_n\) be independent and identically distributed random variables with distribution function \(F\). We wish to test that \(F\) is a normal distribution function. Provide at least two different methods on to test for the normality of \(F\).