1 Example for the t-distribution

Consider the simple linear regression model

yt = α + βxt + σεt, t = 1, 2, . . . 7

where ε ∼ t7 and data as provided in Bedard et al. (2007):

x -3 -2 -1 0 1 2 3

y -2.68 -4.02 -2.91 0.22 0.38 -0.28 0.03

Suppose interest is on the scalar parameter β. The maximum likelihood estimate β̂ is obtained as

0.6504. Table 1 below reports the p-values for testing various values of β using four different meth-

ods. The two first-order methods represent the typical maximum likelihood departure reported in

statistical packages which is denoted as MLE and the log-likelihood ratio in (2) which is denoted

as LR. The two third-order p-values are obtained by using the Lugannani and Rice expression in

(12) and the Barndorff-Nielsen expression given in (13).

Table 1: P-values for the t Distribution

Method β = 0 β = 0.5 β = 1.0 β = 1.5 β = 2.0

MLE 0.0003 0.2179 0.0350 5.33e-06 1.33e-12

LR 0.0035 0.2152 0.0577 0.0015 8.30e-05

Lugannani and Rice 0.0129 0.2349 0.1061 0.0076 0.0010

Barndorff-Nielsen 0.0124 0.2349 0.1052 0.0072 0.0009

Testing a null hypothesis of β = 1 would yield very different conclusions depending on the chosen

significance level of the test. Using the conventional 5% level of significance would lead one to

reject the null using the conventional MLE p-value while not rejecting the null using the LR test

or either of the third-order p-values. If however, a 10% level of significance was chosen, one would

reject the null using either of the two first-order p-value but fail to reject the null using either of

the third-order p-values.

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2 Example for the Pareto distribution

Consider the Pareto distribution given in Section 5 for inference concerning the parameter β.

Using a data set consisting of 50 observations, the maximum likelihood estimate is estimated as

β̂ = 1.256. Table 2 below contains the p-values for testing various values of β using the four

different methods.

Table 2: P-values for the Pareto Distribution

Method β = 0 β = 0.5 β = 1.0 β = 1.5 β = 2.0

MLE 0.0000 0.0000 0.0039 0.0056 4.96e-15

LR 0.0000 0.0000 0.0034 0.0088 2.78e-09

Lugannani and Rice 0.0000 0.0000 0.0362 0.0975 2.11e-08

Barndorff-Nielsen 0.0000 0.0000 0.0308 0.0791 2.05e-08

Once again, it is clear from this table that different conclusions can be reached depending on

which method is employed for inference. For a null of β = 1.5, the first-order methods reject the

hypothesis while the third-order methods do not reject the hypothesis at the conventional 5% level

of significance.

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