Question

16. Suppose the true average growth µ of one type of plant during a 1-year period is identical to that of a second type, but the variance of growth for the first type is $\sigma^2$, whereas for the second type the variance is $4\sigma^2$. Let $X_1,..., X_m$ be m independent growth observations on the first type [so $E(X_i) = \mu$, $V(X_i) = \sigma^2$], and let $Y_1,..., Y_n$ be n independent growth observations on the second type [$E(Y_i) = \mu$, $V(Y_i) = 4\sigma^2$]. a. Show that the estimator $\hat{\mu} = \delta \bar{X} + (1 - \delta)\bar{Y}$ is unbi- ased for $\mu$ (for $0 < \delta < 1$, the estimator is a weighted average of the two individual sample means). b. For fixed m and n, compute $V(\hat{\mu})$, and then find the value of $\delta$ that minimizes $V(\hat{\mu})$. [Hint: Differentiate $V(\hat{\mu})$ with respect to $\delta$].

          16. Suppose the true average growth µ of one type of plant
during a 1-year period is identical to that of a second
type, but the variance of growth for the first type is $\sigma^2$,
whereas for the second type the variance is $4\sigma^2$. Let
$X_1,..., X_m$ be m independent growth observations on the
first type [so $E(X_i) = \mu$, $V(X_i) = \sigma^2$], and let $Y_1,..., Y_n$
be n independent growth observations on the second
type [$E(Y_i) = \mu$, $V(Y_i) = 4\sigma^2$].
a. Show that the estimator $\hat{\mu} = \delta \bar{X} + (1 - \delta)\bar{Y}$ is unbi-
ased for $\mu$ (for $0 < \delta < 1$, the estimator is a weighted
average of the two individual sample means).
b. For fixed m and n, compute $V(\hat{\mu})$, and then find the
value of $\delta$ that minimizes $V(\hat{\mu})$. [Hint: Differentiate
$V(\hat{\mu})$ with respect to $\delta$].

16. Suppose the true average growth µ of one type of plant
during a 1-year period is identical to that of a second
type, but the variance of growth for the first type is σ^2,
whereas for the second type the variance is 4σ^2. Let
X1,..., Xm be m independent growth observations on the
first type [so E(Xi) = μ, V(Xi) = σ^2], and let Y1,..., Yn
be n independent growth observations on the second
type [E(Yi) = μ, V(Yi) = 4σ^2].
a. Show that the estimator μ̂ = δX̅ + (1 - δ)Y̅ is unbi-
ased for μ (for 0 < δ < 1, the estimator is a weighted
average of the two individual sample means).
b. For fixed m and n, compute V(μ̂), and then find the
value of δ that minimizes V(μ̂). [Hint: Differentiate
V(μ̂) with respect to δ].

Added by Elisa F.

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