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Can anybody tell me how to find numerical values for standard errors of the MLEs of Weibull distribution using the uncensored real data set on the breaking stress of carbon fibres (in Gba) reported by Cordeiro GM, Lemonte AJ. The β-Birnbaum–Saunders distribution: an improved distribution for fatigue life modeling. Comput. Stat. Data Anal. 2011; 55: 1445–1461.

The data are (n = 66):

3.70, 2.74, 2.73, 2.50, 3.60, 3.11, 3.27, 2.87, 1.47, 3.11, 3.56, 4.42, 2.41, 3.19, 
3.22, 1.69, 3.28, 3.09, 1.87, 3.15, 4.90, 1.57, 2.67, 2.93, 3.22, 3.39, 2.81, 4.20, 
3.33, 2.55, 3.31, 3.31, 2.85, 1.25, 4.38, 1.84, 0.39, 3.68, 2.48, 0.85, 1.61, 2.79, 
4.70, 2.03, 1.89, 2.88, 2.82, 2.05, 3.65, 3.75, 2.43, 2.95, 2.97, 3.39, 2.96, 2.35, 
2.55, 2.59, 2.03, 1.61, 2.12, 3.15, 1.08, 2.56, 1.80, 2.53

I use Mathematica so kindly give answer for Mathematica users.

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  • $\begingroup$ Table 4 of the cited article provides the MLE and standard errors for the data set quoted here. Do you want to know how find the MLE and standard errors in general? $\endgroup$ – Andrew M Sep 12 '14 at 1:08
  • $\begingroup$ Thanks for reply, where is table 4? Yes, I want to know how to find standard errors of MLE in general? $\endgroup$ – Abdus Saboor Sep 12 '14 at 1:21
  • $\begingroup$ In the the article you cited in your question, @Abdus $\endgroup$ – Andrew M Sep 13 '14 at 17:30
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The normal way to compute the standard error of an MLE is using the Fisher information (which is just the second derivative of the log-likelihood function, evaluated at the location of the MLE). Because of the properties of maximum likelihood, the variance of the MLE is given by a normal distribution centered on the value of the MLE but with variance equal to the inverse Fisher information. For simple distributions, this can often be calculated analytically, and you then simply apply the formula (which will depend on your MLE and your data).

A simple and fairly reliable alternative is to use the bootstrap to estimate the distribution around the MLE and then compute your standard error that way. Let $x_i\in X$ be an element in your data set, and let $n$ be the number of observations in $X$. A bootstrap of your data would be some $Y$ such that each $y_j$ for $1\leq j \leq n$ is an element of $X$ drawn uniformly at random from $X$. In other words, $Y$ contains $n$ values drawn from $X$ with replacement. Note that while $X$ is fixed, $Y$ is a random variable.

Let $\hat{\theta}(X)$ be your MLE on your original data. Then, $\hat{\theta}(Y)$ is the MLE on a bootstrap of your data. And, $\Pr(\hat{\theta}(Y))$ is the distribution of the MLE under the bootstrap. From there, you can compute the variance, standard deviation, and thus standard error of your MLE, which will be asymptotically close to whatever you would calculate using the Fisher information. This approach is attractive because it works even when you cannot compute the Fisher information for a particular model analytically.

If you already have code that computes $\hat{\theta}(X)$, then it should be pretty straightforward to write the code for the bootstrap estimation you need.

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  • $\begingroup$ Nice answer. You might also note that if the MLE is found through a numerical optimization, then often a numerical estimate of the Hessian (matrix of second derivatives) is available (ie, for newton or quasi-newton algorithms). In this case, a cheap and dirty estimate of the information, and hence standard errors, is this numerical Hessian. $\endgroup$ – Andrew M Sep 13 '14 at 17:33

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