# Normal behaviour

One interesting aspect of maximum-likelihood estimation is the common behaviour of estimators, regardless of the nature of the data and model. Recall that the maximum-likelihood estimate, \(\hat\theta\), is the value of a parameter \(\theta\) that maximises the likelihood function, \(L(\theta)\), or the log-likelihood function, \(\ell(\theta)=\log L(\theta)\). By way of example, consider the following three single-parameter distributions:

- The number of events, \(D\), in a binomial(\(n\), \(q\)) distribution.
- The number of events, \(D\), in a Poisson(\(E\mu\)) process, and
- The sum, \(E\), of \(n\) observations from the exponential(\(\lambda\)) distribution.

The response variable \(D\) in (1) and (2) is a non-negative integer, while the response variable \(E\) in (3) is a positive real number. The log-likelihood functions are as follows:

- \(\ell(q|n, d) = (n-d)\log(1-q)+d\log q\)
- \(\ell(\mu|E, d) = d\log\mu-E\mu\)
- \(\ell(\lambda|n, E) = n\log \lambda-E\lambda\)

where the conventional notation is \(\ell({\rm parameter}|{\rm data})\). The close parallel between the log-likelihoods in (2) and (3) is a reminder that the time between events in a Poisson process has an exponential distribution. Although the three distributions are different, the log-likelihood functions have a similar shape around the MLE, as shown in Figure 1.

Figure 1. Log-likelihood functions around maximum-likelihood estimates. Source: own calculations using \(n=1000\) and \(d=200\) for the binomial distribution, \(d=200\) and \(E=900\) for the Poisson distribution and \(n=200\) and \(E=38.47979\) for the exponential distribution.

Figure 1 shows that, regardless of the model, the log-likelihood function has a quadratic shape around the maximum-likelihood estimate. This is not an accident - the maximum-likelihood theorem says that the distribution of a maximum-likelihood estimator is normal (Gaussian) around the MLE. This extends to multivariate models, where the joint maximum-likelihood estimator has a multivariate normal distribution with mean \(\boldsymbol{\theta}\) and covariance matrix \(\boldsymbol{\Sigma}\) (using bold type to signal vectors and matrices). Multivariate likelihoods are trickier to plot, but one approach is to use *profile likelihoods*, i.e. varying each parameter in turn while holding the other parameters constant at their MLE. Some examples are shown in Figure 2.

Figure 2. Profile log-likelihoods for selected parameters from a multi-factor model of pensioner mortality. Source: Richards (2016, page 445).

This is a very handy result for actuaries considering mis-estimation risk: whatever the model specification, joint MLEs have a multivariate normal distribution. The vector mean of the distribution is estimated with \(\boldsymbol{\hat\theta}\), while the covariance matrix is estimated from the inverse of the observed information matrix. We can then assess mis-estimation risk by valuing the liabilities using parameter vectors sampled from this multivariate normal distribution. Richards (2016) presents a methodology for assessing mis-estimation risk in run-off, while Richards (2021) presents a methodology for calculating mis-estimation capital under a value-at-risk (VaR) regime.

**References: **

Richards, S. J. (2016) Mis-estimation risk: measurement and impact, *British Actuarial Journal*, ** 21(3)**, 429-457. doi:10.1017/S1357321716000040. A preprint is available.

Richards, S. J. (2021) A value-at-risk approach to mis-estimation risk, *British Actuarial Journal*, **26**, e13, pages 1–20, doi:10.1017/S1357321721000131. A preprint is available.

## Likelihoods and mis-estimation in Longevitas

Each model in Longevitas has a **Parameters** tab that allows the plotting of profile log-likelihoods.

Longevitas offers two types of mis-estimation assessment:

**Run-off mis-estimation**, i.e. where parameter risk applies across the lifetime of the liabilities being valued. This is appropriate to pricing the transfer of risk, such as with bulk annuities, longevity swaps and reinsurance.**Value-at-risk mis-estimation**, i.e. the sensitivity of liabilities to recalibration risk. This is suitable for Solvency II-style capital assessment.

Users can switch between the two using the configuration option **Mis-Estimation VaR Horizon**.

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