To simplify the notations and calculations that follow we transform M in
order to diagonalise V .Let
1+d 0 -d
W= 010,
-101
where
Eφ [(u4 - 6u2 + 3)(u2 - 1 - 2IF)] _ 1 1 ( 4
Eφ ((u4 - 6u2 + 3)2) = 2 - 12 Eφ u
(8)
(9)
Then
1
WM = —
σ2

- 1 - d(μ4 - 6μ2 + 3)
ʌ O .ʌ
μ3 - 3μ1
μ4 — 6 μ2 + 3
1 / u2 - 1 - 2IF (u; σ; Φ) - d (u4 - 6u2 + 3) ʌ
Wξ ( Y ; θ ) = -2 u3 - 3 u ,
σ u4 - 6u2 +3
and V = WVW' is given by
1
V = σ4diag[A, 6, 24],
where
A = -8 + 4ASV(^) + 2E (u4IF) - 1 [E (u4IF)]2 . (10)
We conclude that the IM test statistic in the normal model can be written
as the sum of three (asymptotically independent) statistics,
A+ (μ2 - 1 - d(μ4 - 6μ2 + 3))2 + 6(μ3 - 3μ1)2
(11)
+ 24(μ4 - 6μ2 + 3)2 .
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