Question

An experiment produces random vector $\mathbf{X} = (X_1, \dots, X_k)^T$ with expected value $\boldsymbol{\mu_X} = (\mu_1, \dots, \mu_k)^T$. The $i$'th component of $X$ has variance $\text{Var}(X_i) = \sigma_i^2$. To estimate $\boldsymbol{\mu_X}$, we perform $n$ independent trials such that $\mathbf{X}(i)$ is the sample of $\mathbf{X}$ on trial $i$, and we form the vector mean $$ \mathbf{M}(n) = \frac{1}{n} \sum_{i=1}^n \mathbf{X}(i). $$ (a) Show $\mathbf{M}(n)$ is unbiased by showing $E(\mathbf{M}(n)) = \boldsymbol{\mu_X}$. (b) Show that the sequence of estimates $\mathbf{M}(n)$ is consistent by showing that for any constant $c > 0$, $$ \lim_{n \to \infty} P\left( \max_{j=1, \dots, k} |M_j(n) - \mu_j| \geq c \right) = 0. $$ Hint: Let $A_i = \{ |M_j(n) - \mu_j| \geq c \}$ and apply the union bound to upper bound $P(A_1 \cup A_2 \cup \dots \cup A_k)$. Then apply the Chebyshev inequality.

          An experiment produces random vector $\mathbf{X} = (X_1, \dots, X_k)^T$ with expected value $\boldsymbol{\mu_X} = (\mu_1, \dots, \mu_k)^T$. The $i$'th component of $X$ has variance $\text{Var}(X_i) = \sigma_i^2$. To estimate $\boldsymbol{\mu_X}$, we perform $n$ independent trials such that $\mathbf{X}(i)$ is the sample of $\mathbf{X}$ on trial $i$, and we form the vector mean
$$
\mathbf{M}(n) = \frac{1}{n} \sum_{i=1}^n \mathbf{X}(i).
$$

(a) Show $\mathbf{M}(n)$ is unbiased by showing $E(\mathbf{M}(n)) = \boldsymbol{\mu_X}$.

(b) Show that the sequence of estimates $\mathbf{M}(n)$ is consistent by showing that for any constant $c > 0$,
$$
\lim_{n \to \infty} P\left( \max_{j=1, \dots, k} |M_j(n) - \mu_j| \geq c \right) = 0.
$$

Hint: Let $A_i = \{ |M_j(n) - \mu_j| \geq c \}$ and apply the union bound to upper bound $P(A_1 \cup A_2 \cup \dots \cup A_k)$. Then apply the Chebyshev inequality.

an experiment produces random vector mathbfx x1 dots xkt with expected value boldsymbolmux mu1 dots mukt the ith component of x has variance textvarxi sigmai2 to estimate boldsymbolmu 82132

Added by Joseph K.

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