Here we walkthrough an example of using extreme value theory to model large, rare insurance claim events in R. Given some historical claims data, the objective is … Next let’s use meplot in evir to plot sample mean excesses over increasing thresholds. Notice that for k < 0 or k > 0, the density has zero probability above or below, respectively, the upper or lower bound -(1/k). The region contains parameter values that are "compatible with the data". Therefore, we can find the smallest R10 value achieved within the critical region of the parameter space where the negative log-likelihood is larger than the critical value. The support of the GEV depends on the parameter values. Arce copyright 2010 (c) Sharon Walker and theDepartment of Mathematics and To use fmincon, we'll need a function that returns non-zero values when the constraint is violated, that is, when the parameters are not consistent with the current value of R10. That smallest value is the lower likelihood-based confidence limit for R10. 1. In this example, we will illustrate how to fit such data using a single distribution that includes all three types of extreme value distributions as special case, and investigate likelihood-based confidence intervals for quantiles of the fitted distribution. We'll create a wrapper function that computes Rm specifically for m=10. You clicked a link that corresponds to this MATLAB command: Run the command by entering it in the MATLAB Command Window. Here we walkthrough an example of using extreme value theory to model large, rare insurance claim events in R. Given some historical claims data, the objective is to provide an estimate for a size threshold we can set below which, say, 99% of claims occur. The red contours represent the surface for R10 -- larger values are to the top right, lower to the bottom left. We can compare our results with the standard empirically estimated quantile, using quantile() in R, which produces sample quantiles corresponding to the given probabilities. What more must be done the following window dimensions: [- 5, 5] X [- 5, 5] . 4. To find the upper likelihood confidence limit for R10, we simply reverse the sign on the objective function to find the largest R10 value in the critical region, and call fmincon a second time. Naturally Web browsers do not support MATLAB commands. To find the log-likelihood profile for R10, we will fix a possible value for R10, and then maximize the GEV log-likelihood, with the parameters constrained so that they are consistent with that current value of R10. The largest function value from the previous step is the maximum value, and the smallest function value is the minimum value of the function on the given interval. We can plug the maximum likelihood parameter estimates into the inverse CDF to estimate Rm for m=10. find the relative extrema of a function, that is, the relative maximum That is, if you generate a large number of independent random values from a single probability distribution, and take their maximum value, the distribution of that maximum is approximately a GEV. 1. Intro Context EVT Example Discuss from x = 0 to x = 15. Since the function is a polynomial, The function gevfit returns both maximum likelihood parameter estimates, and (by default) 95% confidence intervals. As an alternative to confidence intervals, we can also compute an approximation to the asymptotic covariance matrix of the parameter estimates, and from that extract the parameter standard errors. This is the point where the function attains an absolute minimum, if they exist, on the given If we do that over a range of R10 values, we get a likelihood profile. This is difficult to visualize in all three parameter dimensions, but as a thought experiment, we can fix the shape parameter, k, we can see how the procedure would work over the two remaining parameters, sigma and mu. 4. The constraint function should return positive values when the constraint is violated. and exploited as entertainment. ... For example, if we take \(n = 365\) then we get annual maxima of daily precipitation of \(m\) years. Now that we Since f(x) is continuous, in thousands of dollars. Then find the y value at that x. For each value of R10, we'll create an anonymous function for the particular value of R10 under consideration. screen similar to the following (remember to put parentheses around the Check Finally, gpd.q returns an updated tailplot which shows the computed estimates and confidence intervals for the estimator. Example 2: Locate We need to find the smallest R10 value, and therefore the objective to be minimized is R10 itself, equal to the inverse CDF evaluated for p=1-1/m. Graph the function to verify your conclusions. For this example, we'll compute a profile likelihood for R10 over the values that were included in the likelihood confidence interval. This 5. This package includes an AutoClaims dataset, containing data on claims experience from a large midwestern (US) P&C insurer for private motor insurance. TV culture, we see the "extreme" of different situations being explored We can try to fit a distribution to the data. A modified version of this example exists on your system. So, we estimate that 99% of claims that occur result in a loss below $13018.15, and in the cases where a claim results in a loss larger than this amount, we estimate that the expected loss will be $21631.43. Modelling Data with the Generalized Extreme Value Distribution, The Generalized Extreme Value Distribution, Fitting the Distribution by Maximum Likelihood, Statistics and Machine Learning Toolbox Documentation, Mastering Machine Learning: A Step-by-Step Guide with MATLAB. Other MathWorks country sites are not optimized for visits from your location. We can examine the QQplot to give an idea of goodness-of-fit. Each red contour line in the contour plot shown earlier represents a fixed value of R10; the profile likelihood optimization consists of stepping along a single R10 contour line to find the highest log-likelihood (blue) contour. Answers to these questions are important in insurance claims modelling to help inform decisions around how to allocate capital, and to comply with regulatory capital requirements. Compare your results from Last Update: January 9, 2010 Leslie Finding an answer using conventional statistical methods based on a representative sample is challenging, as tail events (claims) occur so rarely. In particular, we adopt the peak-over-threshold by GPD (Generalized Pareto Distribution) approach for exceedances (tails), rather than the block maxima approach. on your graphing calculator: using Accelerating the pace of engineering and science. The contours are straight lines because for fixed k, Rm is a linear function of sigma and mu. The function is continuous on [0,2π], and the critcal points are and . The vertical lines are the estimates for the quantiles, and the points where the dashed curves cross the horizontal dashed line are the boundaries of the confidence intervals. What is the larges value, and what is steps 2 and 3. (Note that we will actually work with the negative of the log-likelihood.). visually confirm you results. 2. Finding the lower confidence limit for R10 is an optimization problem with nonlinear inequality constraints, and so we will use the function fmincon from the Optimization Toolbox™. In the limit as k approaches 0, the GEV becomes the type I. Next we can try comparing the loss data to the exponential distribution using a QQplot. That is just the (1-1/m)'th quantile. of this lesson that follows. It also returns an empty value because we're not using any equality constraints here. This is a nonlinear equality constraint. . As with the likelihood-based confidence interval, we can think about what this procedure would be if we fixed k and worked over the two remaining parameters, sigma and mu. qplot in evir creates a QQplot for threshold data against the exponential distribution. minimum values of the given function on the closed interval [ -2, 3]. the endpoints. Find where the profit attains the value(s) where the function attains an absolute maximum and the value(s) Since the function is a polynomial, there won't be your results from steps 2 and 3. have your attention, we make the observation that in today's "reality show" In this case, the estimate for k is positive, so the fitted distribution has zero probability below a lower bound. our results make sense. It also returns an empty value because we're not using any inequality constraints here. and that is exactly what we mean when we use the term in the mathematical The endpoints will have smaller/larger values than Instead, we will use a likelihood-based method to compute confidence limits. it will attain a maximum and a minimum value somewhere on the interval The dependent variable is the amount paid on a closed claim, in $. to produce widgets is 15 thousand widgets . hist(AutoClaims$PAID, breaks=200, xlim=c(0,20000)), fittedGPD <- gpd(AutoClaims$PAID, threshold=1000), fittedGPD$converged # 0 indicates convergence to maximum. # Estimated 99% quantile, with default 95% confidence interval - use argument ci.p to set a custom CI. interval. The bold red contours are the lowest and highest values of R10 that fall within the critical region.