Matrix repair

(Nov 25, 2019)

When fitting a statistical model we want two things as a minimum:

  1. The parameter estimates, e.g. the maximum-likelihood estimates (MLEs), and
  2. The estimated variance-covariance matrix, \(\hat V\), for those estimates.

We can get both from the log-likelihood: the MLEs maximise the value of the log-likelihood function, and an approximation for the covariance matrix comes from inverting the negative information matrix, \(\mathcal J\), i.e. the matrix of second partial derivatives evaluated at the MLEs. However, the limitations of computer arithmetic can sometimes get in the way, as shown in the information matrix in Figure 1:

Figure 1. Information matrix, \(\mathcal J\), for a five-parameter model. Only…

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Tags: information matrix, covariance matrix

Lost in translation (reprise)

(Oct 31, 2011)

Late last year I drew up a table of actuarial terms and their translation for statisticians.  I had thought that it was a uniquely actuarial trait to use different names compared to other disciplines.  It turns out that statisticians are almost as guilty.  Table 1 shows some common statistical terms in mortality modelling and their description for non-statisticians.

Table 1. Some statistical terms and their definition for mathematicians and engineers.

Statistical term Notation
Description
hazard function
varies The instantaneous failure rate.
observed information matrix Information matrix The curvature of the log-likelihood function, i.e. the negative of the matrix of second partial derivatives.  This is the same…

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Tags: hazard function, information matrix, score function, log-likelihood

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