Confidence Sets

Let $(X_{i1},X_{i2})$, $i=1,\ldots ,n$, be a random sample from the bivariate normal distribution with unknown $\mu_j=\mathbb {E}\left[X_{1j}\right]$, $\sigma_j^2=\operatorname {Var}\left[X_{1j}\right]$, $j=1,2$, and $\sigma_{12}=\operatorname {Cov}\left[ X_{11}, X_{12} \right]$. Let $\theta =\mu_2/\mu_1$ be the parameter of interest ($\mu_1\ne 0$), $Y_i(\theta )=X_{i2}-\theta X_{i1}$, and

where $\overline{Y}(\theta )$ is the average of $Y_i(\theta )$'s and $S_i^2$ and $S_{12}$ are sample variances and covariance based on $X_{ij}$'s. Discuss when Fieller's confidence set for $\theta$,

is a finite interval, the complement of a finite interval, or the whole real line.

Let $X=(X_1,\ldots ,X_n)$ be a random sample from $\mathscr {N}\left(\theta , \theta \right)$ with an unknown $\theta >0$. Find a pivotal quantity and use it to construct a confidence interval for $\theta$.

Let $T$ be a random variable having cumulative distribution function $F$. For any $t\in (0,1)$, show that $\mathbb {P}\left(F(T)<t\right)\leq t$ and $\mathbb {P}\left(F(T-)>1-t\right)\leq t$, where $F(x-)$ denotes the left limit of $F$ at $x$.

Let $(X_1,\ldots ,X_n)$ be a random sample of random variables having Lebesgue density $\frac a\theta \left(\frac x\theta \right)^{a-1}\; \mathbb {1}_{(0,\theta )}(x)$, where $a\geq 1$ is known and $\theta >0$ is unknown.

Let $(X_1,\ldots ,X_n)$ be a random sample from the exponential distribution on $(a,\infty )$ with scale parameter $1$, where $a\in \mathbb {R}$ is unknown.

Let $X$ be a single observation from the uniform distribution on $(\theta -\frac12,\theta +\frac12)$, where $\theta \in \mathbb {R}$ is unknown.

Let $X_1,\ldots ,X_n$ be a random sample of random variables with Lebesgue density $\theta a^\theta x^{-(\theta +1)}\; \mathbb {1}_{(a,\infty )}(x)$, where $\theta >0$ and $a>0$.

Let $X=(X_1,\ldots ,X_n)$ be a random sample from the Weibull distribution with Lebesgue density $\frac a\theta x^{a-1}e^{-x^a/\theta }\; \mathbb {1}_{(0,\infty )}(x)$, where $a>0$ and $\theta >0$ are unknown. Show that $R(X,a,\theta )=\prod_{i=1}^n(X_i^a/\theta )$ is pivotal. Construct a confidence set for $(a,\theta )$ with confidence coefficient $1-\alpha$ by using $R(X,a,\theta )$.

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. Construct a level $1-\alpha$ confidence interval for $\theta$ based on the observation $X$. Find a condition under which the derived confidence interval has confidence coefficient $1-\alpha$.

Let $(X_1,\ldots ,X_n)$ be a random sample of binary random variables with $\mathbb {P}\left(X_i=1\right)=p$. Using the cumulative distribution function of $T=\sum_{i=1}^nX_i$, show that a level $1-\alpha$ confidence interval for $p$ is

where $\alpha_1+\alpha_2=\alpha$, $F_{a,b,\alpha }$ is the $(1-\alpha )$th quantile of the F-distribution $F_{a,b}$, and $F_{a,0,\alpha }$ is defined to be $\infty$.

Let $X$ be a sample of size $1$ from the negative binomial distribution with a known size $r$ and an unknown probability $p\in (0,1)$. Using the cumulative distribution function of $T=X-r$, show that a level $1-\alpha$ confidence interval for $p$ is

where $\alpha_1+\alpha_2$ and $F_{a,b,\alpha }$ is the same as that in the previous exercise.

Let $T$ be a statistic having the noncentral chi-square distribution $\chi_r^2(\theta )$, where the noncentrality parameter $\theta \geq 0$ is unknown and $r$ is a known positive integer. Show that the cumulative distribution function of $T$, $F_\theta (t)$, is nonincreasing in $\theta$ for each fixed $t>0$ and use this result to construct a confidence interval for $\theta$ with confidence coefficient $1-\alpha$.

Repeat the previous exercise when $\chi_r^2(\theta )$ is replaced by the noncentral F-distribution $F_{r_1,r_2}(\theta )$ with unknown $\theta \geq 0$ and known positive integers $r_1$ and $r_2$.

Let $X_{ij}$, $j=1,\ldots ,n_i$, $i=1,\ldots ,m$, be independent random variables having distribution $\mathscr {N}\left(\mu_i, \sigma^2\right)$, $i=1,\ldots ,m$. Let $\overline\mu =n^{-1}\sum_{i=1}^mn_i\mu_i$ and $\theta =\sigma^{-2}\sum_{i=1}^mn_i(\mu_i-\overline\mu )^2$. Construct an upper confidence bound for $\theta$ that has confidence coefficient $1-\alpha$ and is a function of $T=(n-m)(m-1)^{-1}\mathrm{SST}/\mathrm{SSR}$, where

$\overline{X}_{i\cdot }$ is the sample mean based on $X_{i1},\ldots ,X_{in_i}$, and $\overline{X}$ is the sample mean based on all $X_{ij}$'s.

Let $X_i$, $i=1,2$, be independent random variables distributed as the binomial distributions with sizes $n_i$ and probabilities $p_i$, $i=1,2$, respectively, where $n_i$'s are known and $p_i$'s are unknown. Show how to invert the acceptance regions of UMPU tests to obtain a level $1-\alpha$ confidence interval for the odds ratio $\frac{p_2(1-p_1)}{p_1(1-p_2)}$.

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

Let $X=(X_1,\ldots ,X_n)$ be a random sample from $\mathscr {N}\left(\mu , \sigma^2\right)$ with unknown $\mu$ and $\sigma^2$. Discuss how to construct a confidence interval for $\theta =\mu /\sigma$ with confidence coefficient $1-\alpha$ by

Let $(X_1,\ldots ,X_n)$ be a random sample from the uniform distribution on $(\theta -\frac12,\theta +\frac12)$, where $\theta \in \mathbb {R}$. Construct a confidence interval for $\theta$ with confidence coefficient $1-\alpha$.

Let $(X_1,\ldots ,X_n)$ be a random sample from $\mathscr {N}\left(\mu , \sigma^2\right)$ with unknown $\theta =(\mu ,\sigma^2)$. Consider the prior Lebesgue density $\pi (\theta )=\pi_1(\mu \mid \sigma^2)\pi_2(\sigma^2)$, where $\pi_1(\mu \mid \sigma^2)$ is the density of $\mathscr {N}\left(\mu_0, \sigma_0^2\sigma^2\right)$,

and $\mu_0$, $\sigma_0^2$, $a$, and $b$ are known.

Let $(X_1,\ldots ,X_n)$ be a random sample from a distribution on $\mathbb {R}$ with Lebesgue density $\frac1\sigma f(\frac{x-\mu }\sigma )$, where $f$ is a known Lebesgue density and $\mu \in \mathbb {R}$ and $\sigma >0$ are unknown. Let $X_0$ be a future observation that is independent of $X_i$'s and has the same distribution as $X_i$. Find a pivotal quantity $R(X,X_0)$ and construct a level $1-\alpha$ prediction set for $X_0$.

Let $(X_1,\ldots ,X_n)$ be a random sample from a continuous cumulative distribution function $F$ on $\mathbb {R}$ and $X_0$ be a future observation that is independent of $X_i$'s and is distributed as $F$. Suppose that $F$ is increasing in a neighborhood of $F^{-1}(\alpha /2)$ and a neighborhood of $F^{-1}(1-\alpha /2)$. Let $F_n$ be the empirical distribution. Show that the prediction interval $C(X)=[F_n^{-1}(\alpha /2),F_n^{-1}(1-\alpha /2)]$ for $X_0$ satisfies $\lim_n\mathbb {P}\left(X_0\in C(X)\right)=1-\alpha$, where $P$ is the joint distribution of $(X_0,X_1,\ldots ,X_n)$.

Let $X=(X_1,\ldots ,X_n)$ ($n>1$) be a random sample from the exponential distribution on the interval $(\theta ,\infty )$ with scale parameter $\theta$, where $\theta >0$ is unknown.

Let $\theta >0$ be an unknown parameter and $T>0$ be a statistic. Suppose that $T/\theta$ is a pivotal quantity having Lebesgue density $f$ and that $x^2f(x)$ is unimodal at $x_0$ in the sense that $f(x)$ is nondecreasing for $x\leq x_0$ and $f(x)$ is nonincreasing for $x\geq x_0$. Consider the following class of confidence intervals for $\theta$:

Show that if $[b_*^{-1}T,a_*^{-1}T]\in \mathcal C$, $a_*^2f(a_*)=b_*^2f(b_*)>0$, and $a_*\leq x_0\leq b_*$, then the interval $[b_*^{-1}T,a_*^{-1}T]$ has the shortest length within $\mathcal C$.

Let $t_{n-1,\alpha }$ be the $(1-\alpha )$th quantile of the t-distribution $t_{n-1}$ and $z_\alpha$ be the $(1-\alpha )$th quantile of $\mathscr {N}\left(0, 1\right)$, where $0<\alpha <\frac12$ and $n=2,3,\ldots$. Show that

Let $(X_1,\ldots ,X_n)$ be a random sample from $\mathscr {N}\left(\mu , \sigma^2\right)$, $\mu \in \mathbb {R}$ and $\sigma^2>0$.

Let $f$ be a Lebesgue density that is nonzero in $[x_-,x_+]$ and is $0$ outside $[x_-,x_+]$, $-\infty \leq x_-<x_+\leq \infty$.

Let $(X_1,\ldots ,X_n)$ be a random sample from the exponential distribution on $(a,\infty )$ with scale parameter $1$, where $a\in \mathbb {R}$ is unknown. Find a confidence interval for $a$ having the shortest length within the class of confidence intervals $[X_{(1)}+c,X_{(1)}+d]$ with confidence coefficient $1-\alpha$, where $X_{(1)}$ is the smallest order statistic.

Let $(X_1,\ldots ,X_n)$ be a random sample from a distribution with Lebesgue density $\theta x^{\theta -1}\; \mathbb {1}_{(0,1)}(x)$, where $\theta >0$ is unknown.

Let $X$ be a single observation from $\mathscr {N}\left(\theta -1, 1\right)$ if $\theta <0$, $\mathscr {N}\left(0, 1\right)$ if $\theta =0$, and $\mathscr {N}\left(\theta +1, 1\right)$ if $\theta >0$.

Let $X$ be a vector of $n$ observations having distribution $\mathscr {N}\left(Z\beta , \sigma^2I_n\right)$, where $Z$ is a known $n\times p$ matrix of rank $r\leq p<n$, $\beta$ is an unknown $p$-vector, and $\sigma^2>0$ is unknown. Let $\theta =L\beta$, where $L$ is an $s\times p$ matrix of rank $s$ and all rows of $L$ are in $\mathcal R(Z)$,

where $\widehat{\beta }$ is the LSE of $\beta$ and, for each fixed $\theta$, $\widehat{\beta }(\theta )$ is a solution of

Show that $C(X)=\left\{ \theta :W(X,\theta )\leq c_\alpha \right\}$ is an unbiased $1-\alpha$ confidence set for $\theta$, where $c_\alpha$ is the $(1-\alpha )$th quantile of the F-distribution $F_{s,n-r}$.

Let $X=(X_1,\ldots ,X_n)$ be a random sample from the exponential distribution on $(a,\infty )$ with scale parameter $\theta$, where $a\in \mathbb {R}$ and $\theta >0$ are unknown. Find a UMA confidence interval for $a$ with confidence coefficient $1-\alpha$.

Let $X=(X_1,\ldots ,X_n)$ be a random sample from the uniform distribution on $(\theta ,\theta +1)$, where $\theta \in \mathbb {R}$ is unknown. Obtain a UMA lower confidence bound for $\theta$ with confidence coefficient $1-\alpha$.

Let $(X_1,\ldots ,X_n)$ be a random sample from the Poisson distribution with an unknown mean $\theta >0$. Find a randomized UMA upper confidence bound for $\theta$ with confidence coefficient $1-\alpha$.

Let $X$ be a nonnegative integer-valued random variable from a population $P\in \mathcal P$. Suppose that $\mathcal P$ contains discrete probability densities indexed by a real-valued $\theta$ and $\mathcal P$ has monotone likelihood ratio in $X$. Let $U$ be a random variable that has the uniform distribution on $(0,1)$ and is independent of $X$. Show that a UMA lower confidence bound for $\theta$ with confidence coefficient $1-\alpha$ is the solution of the equation

(assuming that a solution exists), where $F_\theta (x)$ is the cumulative distribution function of $X$.

Let $X_1$ and $X_2$ be independent random variables from the exponential distributions on $(0,\infty )$ with scale parameters $\theta_1$ and $\theta_2$, respectively. Show that $[\alpha Y/(2-\alpha ),(2-\alpha )Y/\alpha ]$ is a UMAU confidence interval for $\theta_2/\theta_1$ with confidence coefficient $1-\alpha$, where $Y=X_2/X_1$.

Let $(X_{i1},\ldots ,X_{in_i})$, $i=1,2$, be two independent random samples from $\mathscr {N}\left(\mu_i, \sigma_i^2\right)$, $i=1,2$, respectively, where all parameters are unknown. Let $\theta =\mu_1-\mu_2$, $\overline{X}_i$ and $S_i^2$ be the sample mean and sample variance of the $i$th sample, $i=1,2$.

Let $(X_1,\ldots ,X_n)$ be a random sample of binary random variables with unknown $p=\mathbb {P}\left(X_i=1\right)$.