A convenient method for generating inverse Gaussian variates has been developed (Michael, et al., 1976) and is quite useful for simulating the basic travel time model, equation (4.7). The details of the procedure are presented in appendix 3d. A general simulation procedure is to generate N individuals in a cohort and then perform statistical procedures such as parameter estimation or a goodness-of-fit test on the cohort. This is the repeated n times to determine properties associated with the parameter estimation method or the appropriateness of the goodness-of-fit test.
Before continuing discussion of the simulation procedure, though, I should note an artifact of using the X2 test with continuous data. Since the width of the bins (i.e., the values of the ei's) is predetermined by sample size, not by the model or particular parameter estimates, the test statistic X2 takes on discrete values. This is further exacerbated by choosing all the ei's to be the same and is particularly noticeable at small sample sizes. Figure 4.6 demonstrates this effect for N = 20. In this plot I treat the parameters as known, so the test should be unbiased. The problem that arises is that it is difficult to determine if a test is biased because p-values will also be discrete and will be influenced by where they fall in terms of the discrete jumps. To alleviate this, I develop the following procedure to smooth out the jumps. The first step is to generate n1 cohorts of size N, and for each cohort determine a test statistic, X2i (i = 1, 2, ..., n1). These test statistics are then ranked to give X2r's (r = 1, 2, ..., n1). This entire procedure is repeated n2 times, and an average X2r for each of the n1 ranks, 
, is calculated. These 
's are continuous and should more closely follow the theoretical distribution. Figure 4.7 demonstrates the output of this procedure for the same sample size and parameter values in Figure 4.6. Notice that the distribution of the test statistic more closely follows the theoretical distribution.
To determine if a test is biased, one approach is to begin by choosing an 
or several 's (
). If the test is unbiased, then the p-values associated with the (1.0-) · n1th ranked 
should equal (1.0 - ). If this value is greater than (1.0 - ), the test is considered to be liberal; that is, it does not reject the model enough. If the opposite is true, the model is considered to be conservative.
Figure 4.8 contains the results of simulations of the travel time model. I follow the same procedure as above, except the goodness-of-fit test is performed with estimated parameters. I vary the cohort size from 10 to 200 in increments of 10, with n1 = n2 = 1000. In the top graph, = 0.05, and = 0.10 in the bottom graph. In both cases, the test appears to be unbiased for 
.
In this section, I work with the inverse Gaussian distribution (equations (4.9) and (4.10)), the reparameterized version of the basic travel time model. This distribution has two parameters, µ and 
(recall that µ = L/r and = L2/
2), and for these simulations I have chosen µ = 10.0 and = 100.0, roughly corresponding to observed values. The sample size, n, is varied from 25 to 500 in increments of 25. The procedure is as follows. First, create a sample population by selecting n individuals at random from the inverse Gaussian distribution. I use the procedure described by Michael, et al. (1976) to generate variates. These individuals are put into discrete classes (1 day intervals), and model parameters are estimated based on the sample population. This procedure is then repeated 10,000 times so that distributions of parameter estimates can be generated.
In the simulations, I compare the weighted least squares and the maximum likelihood estimation methods. The two methods appear to be very similar in terms of the mean squared errors (MSE) of the parameter estimates (Figure 4.9). In both cases, there is an inverse relationship between MSE and sample size, with MSE increasing substantially at sample sizes below 100. There does appear to be substantial differences in the bias of the two parameter estimation methods. The weighted least squares method gives biased results for estimates for µ, while the maximum likelihood appears to be unbiased, even at low sample sizes (Figure 4.10). Both methods yield biased estimates of for smaller sample sizes. It looks as though the maximum likelihood method is tending toward unbiased estimates as n is gets large, but even at a sample size of 500, the weighted least squares estimates of are still substantially biased. Based on these simulations, the maximum likelihood is a more efficient method of parameter estimation. It should be noted, though, that these simulations were performed with particular parameter values. Additional simulations are needed to show that the results are general.
