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I am reviewing a MOOC and realised... I didn't grasp the method of moments, so I did try to get it back "from scratch".

Are the above equations/sentences correct ? In the MOOC the analysis is made on the mapping between $\theta$ and the moments, and then using the inverse of this function, but I find it confusing and prefer to think directly in terms of the moments $\mapsto \theta$ mapping. $\newcommand\array[1]{\begin{bmatrix}#1\end{bmatrix}}$

My interpretation of (generalised) method of moments

Case #1: the estimator can be expressed as the expected value of a function of X

  • $\theta = E[g(X)]$, $G = g(X)$, $\bar G_n = \frac{1}{n} \sum_{i=1}^n g(x_i)$
  • by LLN applied on $g(X)$ we have: $\hat \theta_n := \bar G_n \xrightarrow{(p/d)} \theta := E[G]$
  • by CLT applied to $g(X)$ we have: $\hat \theta_n := \bar G_n \xrightarrow{(d)} \sim N(E[G],\frac{var(G)}{n})$

Case #2: the estimator can be expressed as a function of the expected value of a function of X

  • $\theta = h(E[g(X)])$, $G = g(X)$, $\bar G_n = \frac{1}{n} \sum_{i=1}^n g(x_i)$
  • by LLN applied on $g(X)$ and the CMT we have: $\hat \theta_n := h(\bar G_n) \xrightarrow{(p/d)} \theta := h(E[G])$
  • by CLT applied to $g(X)$ and the delta method we have: $\hat \theta_n := h(\bar G_n) \xrightarrow{(d)} \sim N \left(h(E[G]),\frac{\left( \frac{dh}{dE[G]} \right)^2 var(G)}{n}\right)$

Case #3: the estimators can be expressed as D functions of K multiple expected values of functions of X

  • $\theta = \{h_1(E[g_1(X)],E[g_2(X)],...,E[g_K(X)]), h_2(E[g_1(X)],E[g_2(X)],...,E[g_K(X)]),...,h_D(\cdot)\}$ with $1 \leq K \leq D$
  • $G_k = g_k(X)$, $\bar G_{k,n} = \frac{1}{n} \sum_{i=1}^n g_k(x_i)$, $J(H)$ is the $D \times K$ matrix of the $\frac{dh_d}{dE[g_d(X)]}$ derivatives, and $Cov(G)$ is the $D \times D$ covariance matrix between the $g_d(x)$ random variables
  • by LLN applied on $g_k(X)$ and the CMT we have: $\hat \theta_n := \begin{bmatrix} h_1(\bar G_{1,n}, \bar G_{2,n}, ..., \bar G_{K,n}) \\ h_2(\bar G_{1,n}, \bar G_{2,n}, ..., \bar G_{K,n}) \\ ... \\ h_D(\bar G_{1,n}, \bar G_{2,n}, ..., \bar G_{K,n}) \\ \end{bmatrix} \xrightarrow{(p/d)} \theta := \begin{bmatrix} h_1(E[G_1], E[G_2], ..., E[G_K]) \\ h_2(E[G_1], E[G_2], ..., E[G_K]) \\ ... \\ h_D(E[G_1], E[G_2], ..., E[G_K]) \\ \end{bmatrix}$
  • by CLT applied to $g(X)$ and the delta method we have: $\hat \theta_n := \begin{bmatrix} h_1(\bar G_{1,n}, \bar G_{2,n}, ..., \bar G_{K,n}) \\ h_2(\bar G_{1,n}, \bar G_{2,n}, ..., \bar G_{K,n}) \\ ... \\ h_D(\bar G_{1,n}, \bar G_{2,n}, ..., \bar G_{K,n}) \\ \end{bmatrix} \xrightarrow{(d)} \sim N_D \left( \begin{bmatrix} h_1(E[G_1], E[G_2], ..., E[G_K]) \\ h_2(E[G_1], E[G_2], ..., E[G_K]) \\ ... \\ h_D(E[G_1], E[G_2], ..., E[G_K]) \\ \end{bmatrix} ,\frac{ J(H) ~Cov(G) ~(J(H))' }{n}\right)$
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