Hypothesis Tests

Let $X$ be a sample from a population $P$ and consider testing hypotheses $H_0:P=P_0$ versus $H_1:P=P_1$, where $P_j$ is a known population with probability density $f_j$ with respect to a $\sigma$-finite measure $\nu$, $j=0,1$. Let $\beta (P)$ be the power function of a UMP (uniformly most powerful) test of size $\alpha \in (0,1)$. Show that $\alpha <\beta (P_1)$ unless $P_0=P_1$.

Let $X$ be a sample from a population $P$ and consider testing hypotheses $H_0:P=P_0$ versus $H_1:P=P_1$, where $P_j$ is a known population with probability density $f_j$ with respect to a $\sigma$-finite measure $\nu$, $j=0,1$. For any $\alpha >0$, define

where $0\leq \gamma (\alpha )\leq 1$, $c(\alpha )\geq 0$, $\mathbb {E}_{0}\left[T_\alpha (X)\right]=\alpha$, and $\mathbb E_j$ denotes the expectation with respect to $P_j$. Show that

Let $X$ be a sample from a population $P$ and $P_0$ and $P_1$ be two known populations. Suppose that $T_*$ is a UMP test of size $\alpha \in (0,1)$ for testing $H_0:P=P_0$ versus $H_1:P=P_1$ and that $\beta <1$, where $\beta$ is the power of $T_*$ when $H_1$ is true. Show that $1-T_*$ is a UMP test of size $1-\beta$ for testing $H_0:P=P_1$ versus $H_1:P=P_0$.

Let $(X_1,\ldots ,X_n)$ be a random sample from a population on $\mathbb {R}$ with Lebesgue density $f_\theta$. Let $\theta_0$ and $\theta_1$ be two constants. Find a UMP test of size $\alpha$ for testing $H_0:\theta =\theta_0$ versus $H_1:\theta =\theta_1$ in the following cases:

Let $f_1,\ldots ,f_{m+1}$ be Borel functions on $\mathbb {R}^{p}$ that are integrable with respect to a $\sigma$-finite measure $\nu$. For given constants $t_1,\ldots ,t_m$, let $\mathcal T$ be the class of Borel functions $\phi$ (from $\mathbb {R}^{p}$ to $[0,1]$) satisfying

and $\mathcal T_0$ be the set of $\phi$'s in $\mathcal T$ satisfying

Show that if there are constants $c_1,\ldots ,c_m$ such that

is a member of $\mathcal T_0$, then $\phi_*$ maximizes $\int \phi f_{m+1}\, d\nu$ over $\phi \in \mathcal T_0$. Show that if $c_i\geq 0$ for all $i$, then $\phi_*$ maximizes $\int \phi f_{m+1}\, d\nu$ over $\phi \in \mathcal T$.

Let $f_0$ and $f_1$ be Lebesgue integrable functions on $\mathbb {R}$ and

Show that $\phi_*$ maximizes $\int \phi (x)f_1(x)\, dx$ over all Borel functions $\phi$ on $\mathbb {R}$ satisfying $0\leq \phi (x)\leq 1$ and $\int \phi (x)f_0(x)\, dx=\int \phi_*(x)f_0(x)\, dx$.

Let $F_1$ and $F_2$ be two cumulative distribution functions on $\mathbb {R}$. Show that $F_1(x)\leq F_2(x)$ for all $x$ if and only if $\int g(x)\, dF_2(x)\leq \int g(x)\, dF_1(x)$ for any nondecreasing function $g$.

Let $X$ be an observation with a probability density in the family $\mathcal P=\{ f_\theta :\theta \in \Theta \}$, where $\Theta \subset \mathbb {R}$ is the possible values of the parameter $\theta$.

Let $\Theta \subset \mathbb {R}$ and $\mathcal P=\{ f_\theta (x):\theta \in \Theta \}$ be a family of functions on $\mathcal X\subset \mathbb {R}$ satisfying $f_\theta (x)>0$ for all $\theta \in \Theta$ and $x\in \mathcal X$. Assume that $\frac{\partial^2}{\partial \theta \partial x}\log f_\theta (x)$ exists.

Let $X=(X_1,\ldots ,X_n)$ be a random sample from a distribution on $\mathbb {R}$ with Lebesgue density $f_\theta$, $\theta \in \Theta =(0,\infty )$. Let $\theta_0$ be a positive constant. Find a UMP test of size $\alpha$ for testing $H_0:\theta \leq \theta_0$ versus $H_1:\theta >\theta_0$ when

Suppose that the distribution of $X$ is in a family $\{ f_\theta :\theta \in \Theta \}$ with monotone likelihood ratio in $Y(X)$, where $Y(X)$ has a continuous distribution. Consider the hypotheses $H_0:\theta \leq \theta_0$ versus $H_1:\theta >\theta_0$, where $\theta_0\in \Theta$ is known. Show that the $p$-value of the UMP test is given by $\mathbb {P}_{\theta_0}\left(Y\geq y\right)$, where $y$ is the observed value of $Y$ and $P_\theta$ is the probability corresponding to $f_\theta$.

Let $F$ and $G$ be two known cumulative distribution functions on $\mathbb {R}$ and $X$ be a single observation from the cumulative distribution function $\theta F(x)+(1-\theta )G(x)$, where $\theta \in [0,1]$ is unknown.

Let $(X_1,\ldots ,X_n)$ be a random sample from the uniform distribution on $(\theta ,\theta +1)$, $\theta \in \mathbb {R}$. Suppose that $n\geq 2$.

Let $X=(X_1,\ldots ,X_n)$ be a random sample from the discrete uniform distribution on points $1,\ldots ,\theta$, where $\theta =1,2,\ldots$.

Let $(X_1,\ldots ,X_n)$ be a random sample from the exponential distribution on the interval $(a,\infty )$ with scale parameter $\theta$, where $a\in \mathbb {R}$ and $\theta >0$.

In Exercise 3.11(i), derive a UMP test of size $\alpha \in (0,1)$ for testing $H_0:\theta \leq \theta_0$ versus $H_1:\theta >\theta_0$, where $\theta_0>1$ is known.

Let $(X_1,\ldots ,X_n)$ be a random sample from $\mathscr {N}\left(\theta , 1\right)$. Show that $T=\; \mathbb {1}_{(-c,c)}(\overline{X})$ is a UMP test of size $\alpha \in (0,\frac12)$ for testing $H_0:\left|\theta \right|\geq \theta_1$ versus $H_1:\left|\theta \right|<\theta_1$, where $\overline{X}$ is the sample mean and $\theta_1>0$ is a constant. Provide a formula for determining $c$.

Consider Exercise 6.12 with $H_0:\theta \in [\theta_1,\theta_2]$ versus $H_1:\theta \notin [\theta_1,\theta_2]$, where $0<\theta_1\leq \theta_2<1$ are constants.

Let $X$ be a random variable with probability density $f_\theta$. Assume that $\{ f_\theta :\theta \in \Theta \}$ has monotone likelihood ratio in $X$, where $\Theta \subset \mathbb {R}$. Suppose that for each $\theta_0\in \Theta$, a UMPU test of size $\alpha$ for testing $H_0:\theta =\theta_0$ has the acceptance region $\{ c_1(\theta_0)\leq X\leq c_2(\theta_0)\}$ and is strictly unbiased (i.e., its power is larger than $\alpha$ when $\theta \ne \theta_0$). Show that the functions $c_1(\theta )$ and $c_2(\theta )$ are increasing in $\theta$.

Let $X$ be a random variable from the geometric distribution with mean $p^{-1}$. Find a UMPU test of size $\alpha$ for $H_0:p=p_0$ versus $H_1:p\ne p_0$, where $p_0\in (0,1)$ is known.

Let $X=(X_1,\ldots ,X_n)$ be a random sample from $\mathscr {N}\left(\mu , \sigma^2\right)$ with unknown $\mu$ and $\sigma^2$.

Let $X=(X_1,\ldots ,X_n)$ be a random sample from $\mathscr {N}\left(\mu , \sigma^2\right)$ with unknown $\mu$ and $\sigma^2$ and $t(X)=\sqrt n\overline{X}/S$, where $\overline{X}$ is the sample mean and $S^2$ is the sample variance. For testing $H_0:\mu /\sigma \leq \theta_0$ versus $H_1:\mu /\sigma >\theta_0$, find a test of size $\alpha$ that is UMP among all tests based on $t(X)$, where $\theta_0$ is a known constant.

Let $(X_1,\ldots ,X_n)$ be a random sample from the gamma distribution with unknown shape parameter $\theta$ and unknown scale parameter $\gamma$. Let $\theta_0>0$ and $\gamma_0>0$ be known constants.

Let $X_1$ and $X_2$ be independent observations from the binomial distributions with sizes $n_1$ and $n_2$ and probabilities $p_1$ and $p_2$, respectively, where $n_i$'s are known and $p_i$'s are unknown.

Let $X_1$ and $X_2$ be independently distributed as the negative binomial distributions with sizes $n_1$ and $n_2$ and probabilities $p_1$ and $p_2$, respectively, where $n_i$'s are known and $p_i$'s are unknown.

Let $X_j$, $j=1,2,3$, be independent from the Poisson distributions with means $\lambda_j$, $j=1,2,3$, respectively. Show that there exists a UMPU test of size $\alpha$ for testing $H_0:\lambda_1\lambda_2\leq \lambda_3^2$ versus $H_1:\lambda_1\lambda_2>\lambda_3^2$.

Let $(X_{i1},\ldots ,X_{in_i})$, $i=1,2$, be two independent random samples from $\mathscr {N}\left(\mu_i, \sigma^2\right)$, respectively, where $n_i\geq 2$ and $\mu_i$'s and $\sigma$ are unknown. Show that a UMPU test of size $\alpha$ for $H_0:\mu_1=\mu_2$ versus $H_1:\mu_1\ne \mu_2$ rejects $H_0$ when $\left|t(X)\right|>t_{n_1+n_2-1,\alpha /2}$, where

$\overline{X}_i$ and $S_i^2$ are the sample mean and variance based on $X_{i1},\ldots ,X_{in_i}$, $i=1,2$, and $t_{n_1+n_2-1,\alpha }$ is the $(1-\alpha )$th quantile of the t-distribution $t_{n_1+n_2-1}$. Derive the power function of this test.

Let $(X_{i1},\ldots ,X_{in})$, $i=1,2$, be two independent random samples from $\mathscr {N}\left(\mu_i, \sigma_i^2\right)$, respectively, where $n\geq 1$ and $\mu_i$'s and $\sigma_i$'s are unknown. Show that a UMPU test of size $\alpha$ for testing $H_0:\sigma_2^2=\Delta_0\sigma_1^2$ versus $H_1:\sigma_2^2\ne \Delta_0\sigma_1^2$ rejects $H_0$ when

where $\Delta_0>0$ is a constant, $\int_0^cf_{(n-1)/2,(n-1)/2}(v)\, dv=\alpha /2$ and $f_{a,b}$ is the Lebesgue density of the beta distribution with parameter $(a,b)$.

Suppose that $X_i=\beta_0+\beta_1t_i+\varepsilon_i$, $i=1,\ldots ,n$, where $t_i$'s are fixed constants that are not all the same, $\varepsilon_i$'s are independent and identically distributed as $\mathscr {N}\left(0, \sigma^2\right)$, and $\beta_0$, $\beta_1$, and $\sigma^2$ are unknown parameters. Derive a UMPU test of size $\alpha$ for testing

Let $X$ be a sample from $N_n(Z\beta ,\sigma^2I_n)$, where $\beta \in \mathbb {R}^{p}$ and $\sigma^2>0$ are unknown and $Z$ is an $n\times p$ known matrix of rank $r\leq p<n$. For testing $H_0:\sigma^2\leq \sigma_0^2$ versus $H_1:\sigma^2>\sigma_0^2$ and $H_0:\sigma^2=\sigma_0^2$ versus $H_1:\sigma^2\ne \sigma_0^2$, show that UMPU tests of size $\alpha$ are functions of $\mathrm{SSR}=\left\| X-Z\widehat{\beta }\right\|^2$, where $\widehat{\beta }$ is the least squares estimator of $\beta$, and their rejection regions can be determined using chi-square distributions.

Let $(X_1,\ldots ,X_n)$ be a random sample from a bivariate normal distribution with unknown means $\mu_1$ and $\mu_2$, variances $\sigma_1^2$ and $\sigma_2^2$, and correlation coefficient $\rho$. Let $X_{ij}$ be the $j$th component of $X_i$, $j=1,2$, $\overline{X}_j$ and $S_j^2$ be the sample mean and variance based on $X_{1j},\ldots ,X_{nj}$, and $V=\sqrt{n-2}R/\sqrt{1-R^2}$, where