Estimation, Evaluation, and Selection of Actuarial Models

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66 CHAPTER 4. MODEL EVALUATION AND SELECTION but only over those values α and θ for which αθ =1, 200. That means α canbefreetorange over all positive numbers, but θ =1, 200/α. Thus, under the null hypothesis, there is only one free parameter. The likelihood function is maximized at ˆα =0.54955 and ˆθ =2, 183.6. The loglikelihood at this maximum is ln L 0 = −162.466. The test statistic is T =2(−162.293 + 162.466) = 0.346. For a chi-square distribution with one degree of freedom, the critical value is 3.8415. Because 0.346 < 3.8415, the null hypothesis is not rejected. The probability that a chi-square random variable with one degree of freedom exceeds 0.346 is 0.556, a p-value that indicates little support for the alternative hypothesis. ¤ Exercise 73 Using Data Set B, determine if a gamma model is more appropriate than an exponential model. Recall that an exponential model is a gamma model with α =1.Usefulvalueswere obtained in Example 2.32. Exercise 74 Use Data Set C to choose a model for the population that produced those numbers. Choose from the exponential, gamma, and transformed gamma models. Information for the first two distributions was obtained in Example 2.33 and Exercise 28 respectively. 4.3 Representations of the data and model All the approaches to be presented attempt to compare the proposed model to the data or to another model. The proposed model is represented by either its density or distribution function, or perhaps some functional of these quantities such as the limited expected value function or the mean residual life function. The data can be represented by the empirical distribution function or a histogram. The graphs are easy to construct when there is individual, complete data. When there is grouping, or observations have been truncated or censored, difficulties arise. In this Note the only cases to be covered are those where all the data have been truncated at the same value (which could be zero) and are all censored at the same value (which could be infinity). It should be noted that the need for such representations applies only to continuous models. For discrete data, issues of censoring, truncation, and grouping rarely apply. The data can easily be represented by the relative or cumulative frequencies at each possible observation. With regard to representing the data, in this Note the empirical distribution function will be used for individual data and the histogram will be used for grouped data. In order to compare the model to truncated data, we begin by noting that the empirical distribution begins at the lowest truncation point and represents conditional values (that is, they are the distribution and density function given that the observation exceeds the lowest truncation point). In order to make a comparison to the empirical values, the model must also be truncated. Let the common truncation point in the data set be t. The modified functions are ( 0, x < t F ∗ (x) = F (x)−F (t) 1−F (t) , x ≥ t and ( 0, x < t f ∗ (x) = f(x) 1−F (t) , x ≥ t.
4.4. GRAPHICAL COMPARISON OF THE DENSITY AND DISTRIBUTION FUNCTIONS67 For the remainder of this Note, when a distribution function or density function is indicated, a subscript equal to the sample size indicates that it is the empirical model (from Kaplan-Meier, Nelson-Åalen, the ogive, etc.) while no adornment, or the use of an asterisk (*), indicates the estimated parametric model. There is no notation for the true, underlying distribution, because it is unknown and unknowable. 4.4 Graphical comparison of the density and distribution functions The most direct way to see how well the model and data match up is to plot the respective density and distribution functions. Example 4.11 Consider Data Sets B and C. However, for this example and all that follow in this Section, replace the value at 15,743 with 3,476 (this is to allow the graphs to fit comfortably on a page). Truncate Data Set B at 50 and Data Set C at 7,500. Estimate the parameter of an exponential model for each data set. Plot the appropriate functions and comment on the quality of the fit of the model. Repeat this for Data Set B censored at 1,000 (without any truncation). For Data Set B, there are 19 observations (the first observation is removed due to truncation). A typical contribution to the likelihood function is f(82)/[1 − F (50)]. The maximum likelihood estimate of the exponential parameter is ˆθ =802.32. The empirical distribution function starts at 50 and jumps 1/19 at each data point. The distribution function, using a truncation point of 50 is F ∗ (x) = 1 − e−x/802.32 − (1 − e −50/802.32 ) 1 − (1 − e −50/802.32 ) =1− e −(x−50)/802.32 . The following figure presents a plot of these two functions. Exponential Fit F(x) 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 700 1400 2100 2800 3500 x Model Empirical Model vs. data cdf plot for Data Set B truncated at 50
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Estimation, Evaluation, and Selection of Actuarial Models

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