F-Tests and t-Tests

(20 points) Researchers compare three classroom settings and measure a standardized reading score. The observed scores are: \[ \begin {aligned} \text {Group 1: }&18, 21, 20 &&(n_{1} = 3),\\ \text {Group 2: }&23, 22, 24, 21 &&(n_{2} = 4),\\ \text {Group 3: }&26, 28, 27, 29, 25 &&(n_{3} = 5). \end {aligned} \]

Suppose the reading score for the \(i\)th individual in group \(j\) is modeled as \[ Y_{ij} = \mu _{j} + \epsilon _{ij}, \qquad \epsilon _{ij} \sim \mathcal {N}(0, \sigma ^{2}), \] with all errors independent.

Equivalently, this can be written in matrix form as \[ Y = \mathbf {X}\beta + \epsilon . \]

1. Write the model in matrix form by explicitly constructing the design matrix \(\mathbf {X}\), the parameter vector \(\beta \), and the response vector \(Y\).
2. Express the null hypothesis “all three group means are equal” in the form \[ H_{0}: A\beta = c, \] for explicit matrices \(A\) and \(c\).
3. You are told that the residual sum of squares under the full model is \[ RSS = 38.4 \] and under the reduced model (i.e. under \(H_{0}\)) it is \[ RSS_{H_{0}} = 62.1. \] Compute the \(F\)-statistic for testing \(H_{0}\). State the degrees of freedom and give the conclusion at level \(\alpha = 0.05\).
4. Could a \(t\)-test be used instead of an \(F\)-test here? Briefly explain.

(20 points) Researchers study how a child’s reading score \(Y\) depends on home reading time \(x_{1}\) (hours/week) and classroom support level \(x_{2}\) (a small integer score). It is believed that extra home reading time is more effective when classroom support is higher, so the mean is modeled with an interaction between \(x_{1}\) and \(x_{2}\). To make the intercept interpretable, home reading time is centered at \(1\) hour/week.

Consider the linear model \[ Y_{i} = \beta _{0} + \beta _{1}(x_{1,i} - 1) + \beta _{2}(x_{1,i} \times x_{2,i}) + \epsilon _{i}, \qquad i = 1, \ldots , 5, \] where \(\epsilon _{i} \sim \mathcal {N}(0, \sigma ^{2})\) are independent. The observed data consist of: \[ (x_{1}, x_{2}, Y) = (0, 1, 4), (1, 0, 3), (2, 1, 7), (1, 2, 6), (3, 1, 9). \]

A least squares fit yields \(\hat {\beta } = (\hat {\beta }_{0}, \hat {\beta }_{1}, \hat {\beta }_{2})^{\top } = (2.10, 0.90, 1.20)^{\top }\) and \(RSS = 1.50\).

1. Write down the design matrix \(\mathbf {X}\) for this regression model and the parameter vector \(\beta = (\beta _{0}, \beta _{1}, \beta _{2})^{\top }\). Identify the estimated variance \(S^{2}\).
2. We wish to test the joint null hypothesis that there is no effect of the predictors beyond the baseline, \[ H_{0}: \beta _{1} = \beta _{2} = 0. \] Express this hypothesis in matrix form \(A\beta = 0\) for an appropriate matrix \(A\), and state \(q = \operatorname {rank}(A)\).
3. Write down the \(F\)-statistic for testing \(H_{0}\) and state the distribution of \(F\) under \(H_{0}\). (You do not need to evaluate the formula.)
4. Using Scheffé’s method, construct simultaneous 95% confidence intervals for \(\beta _{0}, \beta _{1}, \beta _{2}\). You may express these intervals in terms of \((\mathbf {X}^{\top }\mathbf {X})^{-1}\), but plug in all other quantities.

Consider the linear model

where the \(\epsilon _{i}\) are independent normal random variables with \(\mathbb {E}\epsilon _{i} = 0\) and \(\mathbb {E}\epsilon _{i}^{2} = \sigma ^{2}\). Assume \(\beta \) and \(\sigma ^{2} > 0\) are deterministic but unknown, while the \(x_{i}\) are deterministic and known. Assume also that \(\sum _i x_i \ne 0\). Find the likelihood ratio test for \(\beta = 0\) versus

Let \(X_1,\ldots ,X_n\) be independent and identically distributed \(N(\mu ,\sigma ^2)\), where \(\mu ,\sigma ^2\) are both unknown. We wish to test \(H_0:\sigma >\sigma _0\) against \(H_a:\sigma \leq \sigma _0\). We reject \(H_0\) if \(S^2<c\), where \[ S^2=\frac {1}{n-1}\sum _{i=1}^n (X_i-\bar X)^2, \qquad \bar X=\frac {1}{n}\sum _{i=1}^n X_i. \] Find \(c\) such that the size of the test is \(\alpha \).

Let \(X_1,\ldots ,X_n\) be independent and identically distributed with density function \[ h(t;\theta )=\theta t^{\theta -1}1\{0\leq t\leq 1\} \] for \(\theta >0\) unknown. Use the Neyman-Pearson Lemma to find the uniformly most powerful test of size \(\alpha \) for \(H_0:\theta =\theta _0\) versus \(H_a:\theta =\theta _1\), where \(\theta _1>\theta _0\).

Let \(X_1,\ldots ,X_n\) be independent and identically distributed random variables. The density function of \(X_i\) is \[ g(t;\theta _i)=\theta _i t^{-\theta _i-1}1\{t\geq 1\}. \] Here \(\theta _i>0\) is an unknown parameter. We wish to test \[ H_0:\theta _1=\theta _2=\cdots =\theta _n \] against the alternative that \(H_0\) is not true. Find a test using the generalized likelihood ratio.

Let \(X_1,\ldots ,X_n\) be independent and identically distributed random variables with (unknown) density function \(F\). Let \[ F_0(t)=(1-e^{-t})1\{t>0\}. \] We wish to test \(F(t)=F_0(t)\) for all \(t\). Explain the application of the \(\chi ^2\) goodness-of-fit test.

Let \(X_1,X_2,\ldots ,X_n\) be iid with the same pdf as in question 5. Find the form of the rejection region coming from the generalized likelihood ratio test for \(H_0:\theta =\theta _0\) versus \(H_a:\theta \ne \theta _0\).

Let \(X_1,\ldots ,X_n\sim N(\mu _1,1)\) and \(Y_1,\ldots ,Y_m\sim N(\mu _2,1)\), where all random variables are independent. Derive the likelihood ratio test for \[ H_0:\mu _1^2=\mu _2^2 \quad \text {vs.}H_a:\mu _1^2\ne \mu _2^2. \] What is the distribution of the test statistic?

Suppose a box contains 6 marbles in total. Suppose \(\theta \) of them are white and \(6-\theta \) of them are black. Test \(H_0:\theta =2\) against \(H_a:\theta \ne 2\) as follows: draw two marbles without replacement and reject \(H_0\) if both marbles are the same color, otherwise do not reject. What is the probability of Type II error for each of the alternatives \(\theta =1\) and \(\theta =3\)?

Let \(X_1,\ldots ,X_n\) be iid random variables. The density function of \(X_i\) is \[ g(t;\theta _i)=\theta _i t^{-\theta _i-1}1\{t\geq 1\}. \] Here \(\theta _i>0\) is an unknown parameter. We wish to test \[ H_0:\theta _1=\theta _2=\cdots =\theta _n \] against the alternative that \(H_0\) is not true. Find a test using the generalized likelihood ratio.

A particular gene occurs as one of two alleles (\(A\) and \(a\)), with allele \(A\) having frequency \(\theta \) in the population. In other words, a random copy of the gene is \(A\) with probability \(\theta \) and \(a\) with probability \(1-\theta \). A diploid genotype is just two genes put together (in any particular order) so the probability of each diploid genotype is \(\theta ^2\) for \(AA\), \(2\theta (1-\theta )\) for \(Aa\), and \((1-\theta )^2\) for \(aa\). Suppose we test a random sample of people and find that \(k_1\) of them are \(AA\), \(k_2\) of them are \(Aa\), and \(k_3\) of them are \(aa\). Find the MLE of \(\theta \).

Let \(X\) be discrete taking values in \(\{0,1\}\) and have the pdf \(f(x;\theta )=\theta ^x(1-\theta )^{1-x}\) for \(x\in \{0,1\}\). We test the simple hypothesis \(H_0:\theta =1/4\) agains the alternative composite hypothesis \(H_A:\theta <1/4\) by taking a random sample of size 10 and rejecting \(H_0\) if and only if the observed values \(x_1,\ldots ,x_{10}\) satisfy \[ \sum _{i=1}^{10}x_i\leq 1. \] Find the power function \(K(\theta )\) of this test for \(0\leq \theta \leq 1/4\).

A food processing company packages honey in glass jars. Each jar is supposed to contain 10 fluid ounces of honey. Previous experience suggests that the volume of a randomly selected jar of honey is normally distributed with a known variance of 2 ounces. At a significance level of \(\alpha =.05\) derive the likelihood ratio test for testing the null hypothesis \(H_0:\mu =10\) versus \(H_A:\mu \ne 10\).

A particular gene occurs as one of two alleles (\(A\) and \(a\)), with allele \(A\) having frequency \(\theta \) in the population. In other words, a random copy of the gene is \(A\) with probability \(\theta \) and \(a\) with probability \(1-\theta \). A diploid genotype is just two genes put together (in any particular order) so the probability of each diploid genotype is \(\theta ^2\) for \(AA\), \(2\theta (1-\theta )\) for \(Aa\), and \((1-\theta )^2\) for \(aa\). Suppose we test a random sample of people and find that \(k_1\) of them are \(AA\), \(k_2\) of them are \(Aa\), and \(k_3\) of them are \(aa\). Find the MLE of \(\theta \).

Let \(X\) be discrete taking values in \(\{0,1\}\) and have the pdf \(f(x;\theta )=\theta ^x(1-\theta )^{1-x}\) for \(x\in \{0,1\}\). We test the simple hypothesis \(H_0:\theta =1/4\) agains the alternative composite hypothesis \(H_A:\theta <1/4\) by taking a random sample of size 10 and rejecting \(H_0\) if and only if the observed values \(x_1,\ldots ,x_{10}\) satisfy \[ \sum _{i=1}^{10}x_i\leq 1. \] Find the power function \(K(\theta )\) of this test for \(0\leq \theta \leq 1/4\).

A food processing company packages honey in glass jars. Each jar is supposed to contain 10 fluid ounces of honey. Previous experience suggests that the volume of a randomly selected jar of honey is normally distributed with a known variance of 2 ounces. At a significance level of \(\alpha =.05\) derive the likelihood ratio test for testing the null hypothesis \(H_0:\mu =10\) versus \(H_A:\mu \ne 10\).

Let \(X_1,X_2,\ldots ,X_n\) be iid random variables with density function \[ h(t;\eta )=e^{-(t-\eta )}I\{t\geq \eta \}, \] where \(I\) indicates the indicator function. We want to test \(H_0:\eta =\eta _0\) against \(H_A:\eta \ne \eta _0\). Find a test using the likelihood method and compute the power function.

Let \(X_1,X_2,\ldots ,X_n\) be iid \(N(\mu _1,\sigma _1^2)\) and \(Y_1,Y_2,\ldots ,Y_m\) be iid \(N(\mu _2,\sigma _2^2)\). The \(X\) and the \(Y\) are also independent of each other. Assume that \(\sigma _1^2,\sigma _2^2\) are known while \(\mu _1,\mu _2\) are unknown. We wish to test \(H_0:\mu _1=\mu _2\) against \(H_A:\mu _1\ne \mu _2\). Find a test using the likelihood method. Compute the distribution of the likelihood ratio test.

Suppose that an engineer wishes to compare the number of complaints per week filed by union stewards for two different shifts at a manufacturing plant. It is known that the number of complaints per week on the \(i\)th shift has a Poisson distribution with mean \(\theta _i\) for \(i=1,2\). One hundred independent observations on the number of complaints gave means \(\bar x=20\) for shift 1 and \(\bar y=22\) for shift 2. Using this data test \(H_0:\theta _1=\theta _2\) versus \(H_1:\theta _1\ne \theta _2\) by the likelihood ratio method, with \(\alpha =0.01\). (Hint: First find the likelihood ratio test statistic for \(\lambda \) and then use the distribution of \(-2\log \lambda \).)

Let \(X_1,\ldots ,X_n\) be a random sample with density function \(f(x)=2e^{-2(x-\eta )}\) for \(x>\eta \), and zero otherwise. We wish to test \(H_0:\eta =4\) versus the alternative \(H_A:\eta >4\). We reject \(H_0\) if \(X_{(1)}=\min _{1\leq i\leq n}X_i\geq 4+2/n\).

(a) Compute the significance level of the test.

(b) Compute the power function.

Let \(Y_1<Y_2<\cdots <Y_5\) be the order statistics of a random sample of size \(n=5\) from a distribution with pdf \(f(x;\theta )=e^{-|x-\theta |}/2\), \(-\infty <x<\infty \), with \(\theta \) unknown. Find the likelihood ratio test \(\lambda \) for testing \(H_0:\theta =\theta _0\) versus \(H_1:\theta \ne \theta _0\).

Let \(X_1,\ldots ,X_n\) be a random sample from the \(N(\theta ,1)\) distribution, \(\theta \) unknown. We wish to test \(H_0:\theta =\theta _0\) versus \(H_A:\theta <\theta _0\). Find a test using the generalized likelihood ratio \(\lambda \).

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 \(X_1,X_2,\ldots ,X_n\) 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 \(F\).

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.

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 \(X_1,\ldots ,X_n\) 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. \(H_a:\theta \ne \theta _0\).

Let \(m\) denote the median distance [in 1000 miles] required for a certain brand of automobile tires to wear out. Test to see whether or not \(m\leq 29\), based on the following random sample: \[ 23\quad 20\quad 26\quad 25\quad 48\quad 26\quad 25\quad 24\quad 15\quad 20 \]

The following data are times (in hours) between failures of air conditioning equipment in a particular airplane: \[ 74\quad 57\quad 48\quad 29\quad 502\quad 12\quad 70\quad 21\quad 29\quad 386\quad 59\quad 27\quad 153\quad 26\quad 326. \] Assume that the data are observed values of an i.i.d. random sample from an exponential distribution, \(X_i\sim \operatorname {Exponential}(\theta )\). Test \(H_0:\theta =125\) versus \(H_a:\theta \ne 125\). (A chi-square table is provided.)

A sample of 400 people was asked their degree of support of a balanced budget and their degree of support of public education, with the following results:

Test the hypothesis of independence at \(\alpha =0.05\). (A chi-square table is provided.)

Let \(X_1,\ldots ,X_n\) be a random sample from \(\operatorname {Uniform}[0,\theta ]\). Find the UMP test of size \(\alpha \) of \(H_0:\theta \geq \theta _0\) vs. \(H_a:\theta <\theta _0\) by first deriving a most powerful test of simple hypotheses and then extending it to composite hypotheses.

Let \(X_1,\ldots ,X_n\) be a random sample from \(N(\mu ,\sigma _0^2)\), where \(\sigma _0^2\) is known. Find the generalized likelihood ratio test of size \(\alpha \) of \(H_0:\mu \geq \mu _0\) vs. \(H_a:\mu <\mu _0\).

According to a genetic model the proportions of individuals having the four blood types should be given by O: \(q^2\); A: \(p^2+2pq\); B: \(r^2+2qr\); AB: \(2pr\). Here \(p>0\), \(q>0\), \(r>0\), and \(p+q+r=1\), but the three parameters are unknown. We observe \(n_O\), \(n_A\), \(n_B\), \(n_{AB}\) individuals of the four blood types in a random sample of size \(n\). Describe a goodness-of-fit test of size \(\alpha \) of the stated hypothesis.

Let \(X_1,\ldots ,X_n\) be independent identically distributed Poisson random variables with parameter \(\theta _1\). Let \(Y_1,\ldots ,Y_m\) be independent identically distributed Poisson random variables with parameter \(\theta _2\). We assume that the two samples are independent. We wish to test \(H_0:\theta _1=\theta _2\) against the alternative that \(H_0\) is not true.

a. Derive the likelihood ratio test. (5 points)

b. Provide a large sample approximation for the rejection region. (2 points)

Let \(X_1,\ldots ,X_N\) be independent random variables. The distibution of \(X_i\) is binomial \((n_i,p_i)\). We wish to test \(H_0:p_1=p_2=\cdots =p_N\) against the alternative that \(H_0\) is not true.

a. Derive the likelihood ratio test. (5 points)

b. Provide a large sample approximation for the rejection region. (2 points)

Consider the pdf \(f(x;\theta )=\theta x^{-(\theta +1)}\) (\(x\geq 1\)). Let \(X_i\) be a random sample of size \(n\) with pdf \(f(x;\theta _1)\). Let \(Y_i\) be a random sample of size \(m\) with pdf \(f(x;\theta _2)\). The two sets of random variables are independent. We wish to test \[ H_0:\theta _1=\theta _2 \quad \text {against}\quad H_a:\theta _1\ne \theta _2. \]

6a) (5 points) If one considers just the \(X\)-data, find the MLE \(\hat \theta _1\).

6b) (5 points) Derive a formula for the critical region obtained using the generalized likelihood ratio method.

Let \(X_i\sim \operatorname {EXP}(\theta )\) be a random sample of size \(n\).

7a) (5 points) Find the MGF of \(\frac {2n\bar X}{\theta }\) and identify its distribution.

7b) (10 points) Derive the generalized likelihood ratio test of \(H_0:\theta =\theta _0\) against \(H_a:\theta \ne \theta _0\).

7c) (5 points) The following data are times (in hours) between failures of air conditioning equipment in a particular airplane: 74, 57, 48, 29, 502, 12, 70, 21, 29, 386, 59, 27, 153, 26, 326. Test \(H_0:\theta =125\) against \(H_a:\theta \ne 125\).

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)

Develop carefully the likelihood ratio test for \(H_0:\theta =1\) versus \(H_1:\theta =2\). Discuss the various optimality properties—if any—of your test.

Let \(X_1,\ldots ,X_n\) be independent, identically distributed random variables with density function \[ f(t,\theta _1)= \begin {cases} 0, & \text {if } t\notin [1,\infty ),\\ \theta _1 t^{-\theta _1-1}, & \text {if } 1\leq t<\infty . \end {cases} \] Let \(Y_1,\ldots ,Y_m\) be independent, identically distributed random variables with density function \[ f(t,\theta _2)= \begin {cases} 0, & \text {if } t\notin [1,\infty ),\\ \theta _2 t^{-\theta _2-1}, & \text {if } 1\leq t<\infty . \end {cases} \] We wish to test \(H_0:\theta _1=\theta _2\) against the alternative that \(\theta _1\neq \theta _2\). The two samples are independent.

(a) Find a test using the likelihood ratio.

(b) Provide a large sample approximation for the likelihood ratio test.

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

(a) Find a test using the likelihood ratio.

(b) Provide a large sample approximation for the likelihood ratio test.

Let \(X_1,\ldots ,X_n\) be independent, identically distributed random variables with density function \[ g(t;\theta )= \begin {cases} 0, & \text {if } -\infty <t<\theta ,\\ e^{-(t-\theta )}, & \text {if } \theta \leq t<\infty . \end {cases} \] We wish to test \(H_0:\theta \leq \theta _0\) against \(H_A:\theta >\theta _0\).

(a) Find the uniformly most powerful test of size \(\alpha \).

(b) Compute the power function.

Let \(X_1,X_2,\ldots ,X_n\) be independent, identically distributed normal \(N(\mu ,\sigma ^2)\) random variables. We wish to test \(H_0:\mu =\mu _0\) against \(H_A:\mu \neq \mu _0\) (\(\sigma \) is unknown). Show that the t-test and the likelihood tests are equivalent.

Let \(X_1,X_2,\ldots ,X_n\) be independent, identically distributed uniform on \([0,\theta ]\), \(\theta >0\), random variables. We wish to test \(H_0:\theta \leq \theta _0\) against \(H_A:\theta >\theta _0\). Find the uniformly most powerful test of size \(\alpha \) and compute the power function. (You must explain why your test has this property.)

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

(a) Find the likelihood ratio test.

(b) Show that the likelihood ratio test is distribution free under \(H_0\).

(c) Show that the likelihood ratio test is equivalent with the \(\chi ^2\) test under \(H_0\).

(d) Approximate the critical values of the likelihood ratio test under \(H_0\).

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 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)

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)

Let \(X_1,\ldots ,X_n\) be a random sample from the \(N(\mu ,\sigma ^2)\) distribution, with both \(\mu \) and \(\sigma ^2\) unknown.

(a) Derive the likelihood ratio test for \(H:\sigma ^2=\sigma _0^2\) against all alternatives. Here \(\sigma _0^2\) is a known positive constant. (5 points)

(b) Do the same for \(H:\sigma ^2\ge \sigma _0^2\). (5 points)

Let \(X_1,X_2,\ldots ,X_n\) be independent Poisson random variables with parameters \(\theta _1,\theta _2,\ldots ,\theta _n\). We wish to test \(H_o:\theta _1=\theta _2=\cdots =\theta _n\) against the alternative that \(H_o\) is not true.

(a) Find the likelihood ratio test. (5 points)

(b) Show that the likelihood ratio is asymptotically normal (after centralizing and normalizing) under \(H_o\). (5 points)

Consider the linear model: \(Y_i=\beta +\epsilon _i\), \(1\le i\le n\). Here, \(\epsilon _i\)’s are i.i.d. \(N(0,\sigma ^2)\), and \(\sigma \) and \(\beta \) are unknown.

(a) Find the least squares estimator of \(\beta \);

(b) Find the UMVU estimator for \(\beta \) (prove the optimality);

(c) Find the UMP test for \(H_0:\beta =\beta _0\) versus \(H_1:\beta =\beta _1\);

(d) Discuss–without proofs–how to find the UMP test for \(H_0:\beta >\beta _0\) vs. \(H_1:\beta \le \beta _0\) from tests of the form in part (c) above.

Suppose \(\theta \in \{\theta _0,\theta _1\}\) is an unknown (\(\theta _0\) and \(\theta _1\) are, however, known.) Put some prior, \(\pi \), on \(\theta \). We are to test \(H_0:\theta =\theta _0\) versus \(H_1:\theta =\theta _1\). Our actions are 0 (accept) and 1 (reject). Find the Bayes’ procedure for doing this test, if the loss function, \(\ell (\theta ,a)\), is the 0–1 loss given by: \[ \ell (0,1)=0,\qquad \ell (0,0)=1, \] \[ \ell (1,0)=0,\qquad \ell (1,1)=1. \] Is this procedure minimax?

Suppose \(X_1,\cdots ,X_m\) are independent with \(X_j\sim \operatorname {Binomial}(n_j,p_j)\).

(a) Find the UMVUE’s of \(p_1,\cdots ,p_m\).

(b) Suppose you know that \(p_1=p_2=\cdots =p_m\). Let \(p\) denote this common (but unknown) value. Find the UMVUE of \(p\).

(c) Find the likelihood ratio statistic, \(\lambda \), for \(H_0:p_1=\cdots =p_m\) versus \(H_1:p_i\ne p_j\), for some \(i\) and \(j\).

(d) It can be shown that \(2\log \lambda \) is approximately the same as the usual \(\chi ^2\) statistics. Using this fact, find an approximate test for \(H_0\) vs \(H_1\) above.

Suppose \(\theta \in \{\theta _0,\theta _1\}\) is an unknown (\(\theta _0\) and \(\theta _1\) are, however, known.) Put some prior, \(\pi \), on \(\theta \). We are to test \(H_0:\theta =\theta _0\) versus \(H_1:\theta =\theta _1\). Our actions are 0 (accept) and 1 (reject). Find the Bayes’ procedure for doing this test, if the loss function, \(\ell (\theta ,a)\), is the 0-1 loss given by: \[ \begin {alignedat}{2} \ell (0,1)&=0,\qquad & \ell (0,0)&=1,\\ \ell (1,0)&=0,\qquad & \ell (1,1)&=1. \end {alignedat} \] Is this procedure minimax?

Let \(X_1,X_2,\ldots ,X_n\) be independent identically distributed random variables, uniformly distributed on \([0,\theta ]\), \(\theta >\theta _0\). We want to test \(H_0:\theta =\theta _0\) against \(H_A:\theta >\theta _0\).

(a) Find the uniformly most powerful test. (You must prove your claim.)

(b) Show that the uniformly most powerful test and the likelihood ratio test are equivalent,

(c) Compute the power function of the most powerful test.