1
Example for the t-distribution
Consider the simple linear regression model
yt = α + βxt + σ²t , t = 1, 2, . . . 7
where ² ∼ 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 methods. 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.
1
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.
2