Statistics Prelim Exam 0.8 January 2015

Let \(X_1,X_2,\ldots ,X_n\) be a random sample from the Poisson distribution with parameter \(\lambda >0\). Use the Rao–Blackwell theorem and/or the Lehmann–Scheffé theorem to find a UMVUE of \(\mathbb {P}\{X_1=0\}=e^{-\lambda }\).

Find the maximum likelihood estimators of \(\theta \) and \(\alpha \), based on a random sample \(X_1,X_2,\ldots ,X_n\) from the distribution with density \[ f(x;\theta ,\alpha )=\frac {\alpha }{\theta ^\alpha }x^{\alpha -1}, \qquad 0\leq x\leq \theta . \] Here \(\theta \) and \(\alpha \) are positive.

1000 individuals were classified according to sex and to whether or not they were color-blind, with the following results: male normal 442; female normal 514; male color-blind 38; and female color-blind 6. According to a genetic model, the four relative frequencies should be of the form \(p/2\); \(p^2/2+p(1-p)\); \((1-p)/2\); and \((1-p)^2/2\), respectively. Suppose we want to test whether the data fit the model.

(a) Find the maximum likelihood estimator \(\hat p\) of \(p\).

(b) Describe a size \(\alpha =0.10\) test of the null hypothesis that the data fit the model, assuming that \(\hat p=0.913\). Specify the critical region as precisely as possible, perhaps in terms of a quantile of a certain \(\chi ^2\) distribution.

A scale has two pans. The measurement given by the scale is the difference between the weights on pan 1 and pan 2 plus a random error. Thus, if a weight \(\mu _1\) is put on pan 1 and a weight \(\mu _2\) is put on pan 2, then the measurement is \(Y=\mu _1-\mu _2+\epsilon \). Suppose that \(\mathbb {E}[\epsilon ]=0\), \(\operatorname {Var}(\epsilon )=\sigma ^2\), and in repeated use of the scale observations \(Y_i\) are independent.

Suppose two objects, #1 and #2, have weights \(\beta _1\) and \(\beta _2\). Measurements are then taken as follows:

• Object #1 is put on pan 1 and nothing on pan 2
• Object #2 is put on pan 2 and nothing on pan 1
• Object #1 is put on pan 1 and object #2 is put on pan 2
• Objects #1 and #2 are both put on pan 1 and nothing on pan 2

(a) Let \(\mathbf {Y}=(Y_1,Y_2,Y_3,Y_4)^\prime \) be the vector of observations. Formulate this as a linear model.

(b) Find the least squares estimators of \(\beta _1\) and \(\beta _2\).

(c) Find the variances of the least squares estimators as well as the covariance between them.

Let \(X_1,\ldots ,X_{2n+1}\) be independent lognormal\((\mu ,\sigma ^2)\). That is, \(X_1=e^{Y_1},\ldots ,X_{2n+1}=e^{Y_{2n+1}}\) with \(Y_1,\ldots ,Y_{2n+1}\) being independent \(N(\mu ,\sigma ^2)\). Use a theorem to show that the sample median \(X_{n+1:2n+1}\) is asymptotically normal and find its asymptotic mean and asymptotic variance.

Suppose we have two samples of sizes \(n_1=4\) and \(n_2=6\). We observe \[ (x_1,x_2,x_3,x_4)=(3.7,3.3,4.2,3.1) \] and \[ (y_1,y_2,y_3,y_4,y_5,y_6)=(3.6,5.0,4.5,3.9,5.1,4.7). \] The null hypothesis is that \(F_X=F_Y\) and the alternative hypothesis is that \(X\) is stochastically smaller than \(Y\). Use the Wilcoxon/Mann–Whitney test. Determine the value of the test statistic and evaluate the p-value of the data. Do not use a large-sample approximation.

Let \(X_1,X_2,\ldots ,X_n\) be a random sample from a \(N(\theta ,\sigma ^2)\) distribution (\(\sigma ^2\) known), and let the prior distribution of \(\theta \) be \(N(\mu _0,\sigma _0^2)\), where \(\mu _0\) and \(\sigma _0^2\) are known.

(a) Find the posterior distribution of \(\theta \).

(b) Find the Bayes estimator under squared error loss.

(a) Let \(X_1,X_2,\ldots ,X_n\) be a random sample from \(N(\theta ,\theta ^2)\), where \(\theta >0\). Find a pair of statistics that are jointly sufficient.

(b) Using some linear combination of \(\sum _{i=1}^n X_i^2\) and \(\left (\sum _{i=1}^n X_i\right )^2\), show that the statistics of part (a) are not jointly complete.

Let \(X_1\) and \(X_2\) be independent \(\operatorname {Exponential}(1)\) random variables (density \(e^{-x}\), \(x>0\)). Find the joint density of \(Y_1=X_1/X_2\) and \(Y_2=X_1+X_2\). Find the marginal densities of \(Y_1\) and \(Y_2\) as well.

Let \(X_1,\ldots ,X_n\) be a random sample from \(\operatorname {Uniform}[0,\theta ]\), where \(\theta \) is a positive unknown. Derive the generalized likelihood ratio (GLR) test of \(H_0:\theta =\theta _0\) vs. \(H_a:\theta \ne \theta _0\).