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Probability: Difference between revisions
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==Delta Method== | ==Delta Method== | ||
See [https://en.wikipedia.org/wiki/Delta_method Wikipedia] | See [https://en.wikipedia.org/wiki/Delta_method Wikipedia]<br> | ||
Suppose <math>\sqrt{n}(X_n - \theta) \xrightarrow{D} N(0, \sigma^2)</math>.<br> | |||
Let <math>g</math> be a function such that <math>g'</math> exists and <math>g'(\theta) \neq 0</math><br> | |||
Then <math>\sqrt{n}(g(X_n) - g(\theta)) \xrightarrow{D} N(0, \sigma^2 g'(\theta)^2)</math><br> | |||
Multivariate:<br> | |||
<math>\sqrt{n}(B - \beta) \xrightarrow{D} N(0, \Sigma) \implies \sqrt{n}(h(B)-h(\beta)) \xrightarrow{D} N(0, h'(\theta)^T \Sigma h'(\theta))</math><br> | |||
;Notes | |||
* You can think of this like the Mean Value theorem for random variables. | |||
: <math>(g(X_n) - g(\theta)) \approx g'(\theta)(X_n - \theta)</math> | |||
==Limit Theorems== | ==Limit Theorems== | ||
===Markov's Inequality=== | ===Markov's Inequality=== | ||