4.3. Statistical methods

The parameter estimation methods vary depending on whether the data are discrete or continuous. Also, for both cases, alternative methods are available, so I will present alternatives for each.

• continuous case

The maximum likelihood estimators (mles) for the two parameters r and were first worked out by Shroödinger (1915). They are:

(4.11)

, (4.12)

where t_i is the observed arrival time of the ith individual, is the average arrival time of the group, and N is the number of individuals in the cohort. The maximum likelihood estimators of these two parameters are independent (Chhikara and Folks, 1989), and much of the statistical inference involving the inverse Gaussian distribution parallels that of the normal distribution. Notice that the mle for r is the average migration rate and the mle for involves the difference between the harmonic mean and reciprocal of the arithmetic mean of the travel time. While the mle for r is unbiased, the mle of is biased. An (uniform minimum variance) unbiased estimator for is

(4.13)

(Folks and Chhikara, 1978).

From equations (4.12) and (4.13), the bias of can easily be shown to be

. (4.14)

Plots of this equation for several values of are contained in Figure 4.4. The slopes of these curves are very steep for small sample sizes but flatten out for larger sample sizes.

For the continuous version of the travel time model, theoretical confidence intervals for the parameters r and are available (Tweedie, 1957a, 1957b; Folks and Chhikara, 1978; Chhikara and Folks, 1989).To construct confidence intervals for r, we begin by noting that the statistic

(4.15)

follows a Student's t distribution with N-1 degrees of freedom. Based on this information, we can determine

. (4.16)

Because Student's t distribution is symmetric, b = -a. Thus, a (uniformly most accurate-unbiased) 100(1-) percent confidence interval is

, (4.17)

if , and

otherwise.

For the confidence interval of , we first note that

(4.18)

(Tweedie (1957a)). Equation (4.18) is then used to determine values a and b such that

.

A 100(1-) percent confidence interval for can then be constructed as:

. (4.19)

Notice that the confidence interval for r is determined by the estimates of r and , but the confidence interval for is determined only by the estimate of . Also, as expected, the confidence intervals of r and are dependent on sample size, with the confidence intervals decreasing as N increases.

In Figure 4.5, I use equations (4.17) and (4.19) to construct plots of the length of the 95 percent confidence intervals of r and versus sample size for a variety of parameter values. In these plots, I use the expected values of and , so the confidence intervals can be thought of as "expected" confidence intervals. In all cases, I set L = 100. Since the confidence interval for r is affected by the values of both r and , I made two plots. In the first plot, I set = 10.0 and vary r; in the second plot I set r = 10.0 and vary . Since the confidence interval of is unaffected by r, I made a single plot in which is varied. The behavior of the plots is quite similar in all the cases. For small sample sizes, the slope of the curve is steep and negative. By N = 50 or so, the curve has substantially flattened. This information is useful in determining appropriate sample sizes for the data analysis.

• discrete case

, (4.20)

where the index i refers to the time interval, k is the total number of time intervals, n_i is the number of observed individuals in the ith interval, p_i is taken from equation (4.8), and c is a combinatorial constant unaffected by the choice of parameters. To estimate the parameters, I minimize equation (4.20) with respect to the parameters using a downhill simplex technique (Nelder and Mead, 1965; Press et al. 1988).

Another approach for estimating parameters when the data are discrete is generalized least squares or weighted least squares. Based on the multinomial distribution, the variance of the ith class is

, (4.21)

and to account for unequal variances, the weighting function is w_i = 1/v_i. Thus to estimate the parameters, the following equation is minimized with respect to r and :

. (4.22)

Later in this chapter I compare these two estimation methods using simulations.

When the data are discrete, parametric confidence intervals are not available. In this case, approximate confidence intervals can be constructed using the bootstrap method as described in chapter 3.