Unbiasedness

Verify

In Example 1.5, show that $a_{i}^{*}$ minimizes (1.6) for $i=0,1$, and simplify the expression for $a_{0}^{*}$.

Let $X$ take on the values -1, 0, 1, 2, 3 with probabilities $\mathbb {P}\left(X=-1\right)=2 p q$ and $\mathbb {P}\left(X=k\right)=p^{k} q^{3-k}$ for $k=0,1,2,3$.

For a sample of size $n$, suppose that the estimator $T(\mathbf{x})$ of $\tau (\theta )$ has expectation

where $a_{k}$ may depend on $\theta$ but not on $n$.

An alternative proof of the Schwarz inequality is obtained by noting that

so that this quadratic in $\lambda$ has at most one root.

Suppose $X$ is distributed on ( 0,1 ) with probability density $p_{\theta }(x)=(1-\theta )+\theta / 2 \sqrt{x}$ for all $0<x<1,0 \leq \theta \leq 1$. Show that there does not exist an LMVU estimator of $\theta$.

If $\delta$ and $\delta^{\prime }$ have finite variance, so does $\delta^{\prime }-\delta$.

In Example 1.9,

If estimators are restricted to the class of linear estimators, characterization of best unbiased estimators is somewhat easier. Although the following is a consequence of Theorem 1.7, it should be established without using that theorem.

Let $\mathbf{X}_{p \times 1}$ satisfy $\mathbb {E}\left[\mathbf{X}\right]=B \psi$ and $\operatorname {var}(\mathbf{X})=I$, where $B_{p \times r}$ is known, and $\psi_{r \times 1}$ is unknown. A linear estimator is an estimator of the form $a^{\prime } \mathbf{X}$, where $a_{p \times 1}$ is a known vector. We are concerned with the class of estimators

Use Theorem 1.7 to find UMVU estimators of some of the $\eta_{\theta }\left(d_{i}\right)$ in the doseresponse model (1.6.16), with the restriction (1.6.17) (Messig and Strawderman 1993). Let the classes $\Delta$ and $\mathcal{U}$ be defined as in Theorem 1.7.

Show that if $\delta (X)$ is a UMVU estimator of $g(\theta )$, it is the unique UMVU estimator of $g(\theta )$. (Do not assume completeness, but rather use the covariance inequality and the conditions under which it is an equality.)

If $\delta_{1}$ and $\delta_{2}$ are in $\Delta$ and are UMVU estimators of $g_{1}(\theta )$ and $g_{2}(\theta )$, respectively, then $a_{1} \delta_{1}+a_{2} \delta_{2}$ is also in $\Delta$ and is UMVU for estimating $a_{1} g_{1}(\theta )+a_{2} g_{2}(\theta )$, for any real $a_{1}$ and $a_{2}$.

Completeness of $T$ is not only sufficient but also necessary so that every $g(\theta )$ that can be estimated unbiasedly has only one unbiased estimator that is a function of $T$.

Suppose $X_{1}, \ldots , X_{n}$ are iid Poisson $(\lambda )$.

If $T$ has the binomial distribution $b(p, n)$ with $n>3$, use Method 1 to find the UMVU estimator of $p^{3}$.

Let $X_{1}, \ldots , X_{n}$ be iid according to the Poisson distribution $P(\lambda )$. Use Method 1 to find the UMVU estimator of

Let $X_{1}, \ldots , X_{n}$ be distributed as in Example 1.14. Use Method 1 to find the UMVU estimator of $\theta^{k}$ for any integer $k>-n$.

Solve Problem 1.18(b) by Method 2, using the fact that an unbiased estimator of $e^{-\lambda }$ is $\delta =1$ if $X_{1}=0$, and $\delta =0$ otherwise.

In $n$ Bernoulli trials, let $X_{i}=1$ or 0 as the $i$ th trial is a success or failure, and let $T=\Sigma X_{i}$. Solve Problem 1.17 by Method 2, using the fact that an unbiased estimator of $p^{3}$ is $\delta =1$ if $X_{1}=X_{2}=X_{3}=1$, and $\delta =0$ otherwise.

Let $X$ take on the values 1 and 0 with probability $p$ and $q$, respectively, and assume that $1 / 4<p<3 / 4$. Consider the problem of estimating $p$ with loss function $L(p, d)=$ 1 if $\left|d-p\right| \geq 1 / 4$, and 0 otherwise. Let $\delta^{*}$ be the randomized estimator which is $Y_{0}$ or $Y_{1}$ when $X=0$ or 1 where $Y_{0}$ and $Y_{1}$ are distributed as $U(-1 / 2,1 / 2)$ and $U(1 / 2,3 / 2)$, respectively.

If $X_{1}, \ldots , X_{n}$ are iid as $N\left(\xi , \sigma^{2}\right)$ with $\sigma^{2}$ known, find the UMVU estimator of

Solve the preceding problem when $\sigma$ is unknown.

In Example 2.1 with $\sigma$ known, let $\delta =\Sigma c_{i} X_{i}$ be any linear estimator of $\xi$. If $\delta$ is biased, show that its risk $\mathbb {E}\left[(\delta -\xi )^{2}\right]$ is unbounded.

Suppose, as in Example 2.1, that $X_{1}, \ldots , X_{n}$ are iid as $N\left(\xi , \sigma^{2}\right)$, with one of the parameters known, and that the estimand is a polynomial in $\xi$ or $\sigma$. Then, the UMVU estimator is a polynomial in $\bar{X}$ or $S^{2}=\sum \left(X_{i}-\xi \right)^{2}$. The variance of any such polynomial can be estimated if one knows the moments $\mathbb {E}\left[\bar{X}^{k}\right]$ and $\mathbb {E}\left[S^{k}\right]$ for all $k=1,2, \ldots$. To determine $\mathbb {E}\left[\bar{X}^{k}\right]$, write $\bar{X}=Y+\xi$, where $Y$ is distributed as $N\left(0, \sigma^{2} / n\right)$. Show that

In Example 2.1, when both parameters are unknown, show that the UMVU estimator of $\xi^{2}$ is given by $\delta =\bar{X}^{2}-\frac{S^{2}}{n(n-1)}$ where now $S^{2}=\sum \left(X_{i}-\bar{X}\right)^{2}$.

If $X$ is a single observation from $N\left(\xi , \sigma^{2}\right)$, show that no unbiased estimator $\delta$ of $\sigma^{2}$ exists when $\xi$ is unknown.

Let $X_{i}, i=1, \ldots , n$, be independently distributed as $N\left(\alpha +\beta t_{i}, \sigma^{2}\right)$ where $\alpha , \beta$, and $\sigma^{2}$ are unknown, and the $t$ 's are known constants that are not all equal. Find the UMVU estimators of $\alpha$ and $\beta$.

In Example 2.2 with $n=1$, the UMVU estimator of $p$ is the indicator of the event $X_{1} \leq u$ whether $\sigma$ is known or unknown.

Verify Equation (2.14), the density of $\left(X_{1}-\bar{X}\right) / S$ in normal sampling.

Assuming (2.15) with $\sigma =\tau$, determine the UMVU estimators of $\sigma^{2}$ and $(\eta -\xi ) / \sigma$.

Assuming (2.15) with $\eta =\xi$ and $\sigma^{2} / \tau^{2}=\gamma$, show that when $\gamma$ is known:

Show that in the preceding problem with $\gamma$ unknown,

For the model (2.15) find the UMVU estimator of $\mathbb {P}\left(X_{1}<Y_{1}\right)$ when

If $\left(X_{1}, Y_{1}\right), \ldots ,\left(X_{n}, Y_{n}\right)$ are iid according to any bivariate distribution with finite second moments, show that $S_{X Y} /(n-1)$ given by (2.17) is an unbiased estimator of $\operatorname {cov}\left(X_{i}, Y_{i}\right)$.

In a sample size $N=n+k+1$, some of the observations are missing. Assume that $\left(X_{i}, Y_{i}\right), i=1, \ldots , n$, are iid according to the bivariate normal distribution (2.16), and that $U_{1}, \ldots , U_{k}$ and $V_{1}, \ldots , V_{l}$ are independent $N\left(\xi , \sigma^{2}\right)$ and $N\left(\eta , \tau^{2}\right)$, respectively.

For the family (2.22), show that the UMVU estimator of $a$ when $b$ is known and the UMVU estimator of $b$ is known are as stated in Example 2.5.

Show that the estimators (2.23) are UMVU.

For the family (2.22) with $b=1$, find the UMVU estimator of $\mathbb {P}\left(X_{1} \geq u\right)$ and of the density $e^{-(u-a)}$ of $X_{1}$ at $u$.

Find the UMVU estimator of $\mathbb {P}\left(X_{1} \geq u\right)$ for the family (2.22) when both $a$ and $b$ are unknown.

Let $X_{1}, \ldots , X_{m}$ and $Y_{1}, \ldots , Y_{n}$ be independently distributed as $E(a, b)$ and $E(a^{\prime }, b^{\prime })$, respectively.

In the preceding problem, suppose that $b^{\prime }=b$.

In Problem 2.21, suppose that $a^{\prime }=a$.

Let $X_{1}, \ldots , X_{n}$ be iid according to the uniform distribution $U(\xi -b, \xi +b)$. If $\xi , b$ are both unknown, find the UMVU estimators of $\xi , b$, and $\xi / b$.

Let $X_{1}, \ldots , X_{m}$ and $Y_{1}, \ldots , Y_{n}$ be iid as $U(0, \theta )$ and $U\left(0, \theta^{\prime }\right)$, respectively. If $n>1$, determine the UMVU estimator of $\theta / \theta^{\prime }$.

Verify the ML estimators given in (2.24).

In Example 2.6(b), show that

Verify (2.26).

Under the assumptions of Lemma 2.7, show that:

If $T$ is distributed as $b(p, n)$, find an unbiased estimator $\delta (T)$ of $p^{m}(m \leq n)$ by Method 1, that is, using (1.10).

If $Y$ is distributed according to (3.3), use Method 1 of Section 2.1

Consider the scheme in which binomial sampling is continued until at least $a$ successes and $b$ failures have been obtained. Show how to calculate a reasonable estimator of $\log (p / q)$.

If binomial sampling is continued until $m$ successes have been obtained, let $X_{i}$ ( $i= 1, \ldots , m$ ) be the number of failures between the $(i-1)$ st and $i$ th success.

Suppose that binomial sampling is continued until the number of successes equals the number of failures.

Verify Equation (3.7) with the appropriate definition of $N^{\prime }(x, y)$

Consider sequential binomial sampling with the stopping points $(0,1)$ and $(2, y)$, $y=0,1, \ldots$.

In Example 3.4(ii),

Curtailed single sampling. Let $a, b<n$ be three non-negative integers. Continue observation until either $a$ successes, $b$ failures, or $n$ observations have been obtained. Determine the UMVU estimator of $p$.

For any sequential binomial sampling plan, the coordinates $(X, Y)$ of the end point of the sample path are minimal sufficient.

Consider any closed sequential binomial sampling plan with a set $B$ of stopping points, and let $B^{\prime }$ be the set $B \cup \left\{ \left(x_{0}, y_{0}\right)\right\}$ where $\left(x_{0}, y_{0}\right)$ is a point not in $B$ that has positive probability of being reached under plan $B$. Show that the sufficient statistic $T=(X, Y)$ is not complete for the sampling plan which has $B^{\prime }$ as its set of stopping points.

For any sequential binomial sampling plan under which the point $(1,1)$ is reached with positive probability but is not a stopping point, find an unbiased estimator of $p q$ depending only on $(X, Y)$. Evaluate this estimator for

Use (3.3) to determine $A(t, n)$ in (3.11) for the negative binomial distribution with $m=n$, and evaluate the estimators (3.13) of $q^{r}$, and (3.14).

Consider $n$ binomial trials with success probability $p$, and let $r$ and $s$ be two positive integers with $r+s<n$. To the boundary $x+y=n$, add the boundary point $(r, s)$, that is, if the number of successes in the first $r+s$ trials is exactly $r$, the process is stopped and the remaining $n-(r+s)$ trials are not performed.

Generalize the preceding problem to the case that two points $\left(r_{1}, s_{1}\right)$ and $\left(r_{2}, s_{2}\right)$ with $r_{i}+s_{i}<n$ are added to the boundary. Assume that these two points are such that all $n+1$ points $x+y=n$ remain boundary points.

If $X$ has the Poisson distribution $P(\theta )$, show that $1 / \theta$ does not have an unbiased estimator.

If $X_{1}, \ldots , X_{n}$ are iid according to (3.18), the Poisson distribution truncated on the left at 0, find the UMVU estimator of $\theta$ when

Let $X_{1}, \ldots , X_{n}$ be a sample from the Poisson distribution truncated on the left at 0, i.e., with sample space $\mathcal{X}=\left\{ 1,2,3, \ldots \right\}$.

Suppose that $X$ has the Poisson distribution truncated on the right at $a$, so that it has the conditional distribution of $Y$ given $Y \leq a$, where $Y$ is distributed as $P(\lambda )$. Show that $\lambda$ does not have an unbiased estimator.

For the negative binomial distribution truncated at zero, evaluate the estimators (3.13) and (3.14) for $m=1,2$, and 3.

If $X_{1}, \ldots , X_{n}$ are iid $P(\lambda )$, consider estimation of $e^{-b \lambda }$, where $b$ is known.

If $X_{1}, \ldots , X_{n}$ are iid according to the logarithmic series distribution of Problem 1.5.14, evaluate the estimators (3.13) and (3.14) for $n=1,2$, and 3.

For the multinomial distribution of Example 3.8,

In Example 3.9 when $p_{i j}=p_{i+} p_{+j}$, determine the variances of the two unbiased estimators $\delta_{0}=n_{i j} / n$ and $\delta_{1}=n_{i+} n_{+j} / n^{2}$ of $p_{i j}$, and show directly that $\operatorname {var}\left(\delta_{0}\right)>\operatorname {var}\left(\delta_{1}\right)$ for all $n>1$.

In Example 3.9, show that independence of $A$ and $B$ implies that $(n_{1+}, \ldots , n_{I^{+}})$ and $\left(n_{+1}, \ldots , n_{+J}\right)$ are independent with multinomial distributions as stated.

Verify (3.20).