I want to actually get the confidence interval of gaussian distribution. I want to know how I can use the covariance matrix and check if the obtained mui vector for the multivariate gaussian distribution actually satisfied the confidence interval. I have a mui vector and the actual values to be obtained. How can I use covariance matrix and the actual values plust mui vector to verify if it satisfied the confidence interval

  • $\begingroup$ This post is answering your question. stats.stackexchange.com/questions/7882/… $\endgroup$ – user4581 Jun 5 '12 at 23:48
  • $\begingroup$ I want to know how I can get the standard deviation such that I can check if the true value is within mui+-standard deviation. It's tricky when it comes to covariance matrix $\endgroup$ – user31820 Jun 6 '12 at 0:00
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    $\begingroup$ A confidence region for what? The mean vector? It would be an ellipsoid involving the inverse of the sample covariance matrix. $\endgroup$ – Michael Chernick Jun 6 '12 at 0:02
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    $\begingroup$ Yeah, so if I have a sample lets say x vector. How can I check if that sample lies within the 68 percent region. I mean I can get the standard deviation from the covariance matrix for each variable of the multivariate random vector. Then for each element of the x vector I can check if it lies within the +- standard deviation of the elements of the mui vector. Is this the way to go? $\endgroup$ – user31820 Jun 6 '12 at 0:38
  • $\begingroup$ no you construct the 68% confidence ellipse. Find the contour of constant density that contains 68% of the distribution for the sample mean vector within it. $\endgroup$ – Michael Chernick Jun 6 '12 at 2:54

The quantity $y = (x - \mu)^T \Sigma^{-1} (x-\mu)$ is distributed as $\chi^2$ with $k$ degrees of freedom (where $k$ is the length of the $x$ and $\mu$ vectors). $\Sigma$ is the (known) covariance matrix of the multivariate Gaussian.

When $\Sigma$ is unknown, we can replace it by the sample covariance matrix $S = \frac{1}{n-1} \sum_i (x_i-\overline{x})(x_i-\overline{x})^T$, where $\{x_i\}$ are the $n$ data vectors, and $\overline{x} = \frac{1}{n} \sum_i x_i$ is the sample mean. The quantity $t^2 = n(\overline{x} - \mu)^T S^{-1} (\overline{x}-\mu)$ is distributed as Hotelling's $T^2$ distribution with parameters $k$ and $n-1$.

An ellipsoidal confidence set with coverage probability $1-\alpha$ consists of all $\mu$ vectors such that $n(\overline{x} - \mu)^T S^{-1} (\overline{x}-\mu) \leq T^2_{k,n-k}(1-\alpha)$. The critical values of $T^2$ can be computed from the $F$ distribution. Specifically, $\frac{n-k}{k(n-1)}t^2$ is distributed as $F_{k,n-k}$.

Source: Wikipeda Hotelling's T-squared distribution


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