HP 15c User Manual
Page 112
112
Section 4: Using Matrix Operations
112
and
.
)
(
||
ˆ
||
E
2
2
p
n
F
b
X
y
For b ≠ 0. When the simpler model y = r is correct, both of these expectations equal σ
2
.
You can test the hypothesis that the simpler model is correct (against the alternative that the
original model is correct) by calculating the F ratio
)
(
||
ˆ
||
||
ˆ
||
2
2
p
n
p
F
F
F
b
X
y
b
X
F will tend to be larger when the original model is true (b ≠ 0) than when the simpler model
is true (b = 0). You reject the hypothesis when F is sufficiently large.
If the random errors have a normal distribution, the F ratio has a central F distribution with p
and (n − p) degrees of freedom if b = 0, and a non central distribution if b ≠ 0. A statistical
test of the hypothesis (with probability α of incorrectly rejecting the hypothesis) is to reject
the hypothesis if the F ratio is larger than the 100α percentile of the central F distribution
with p and (n – p) degrees of freedom; otherwise, accept the hypothesis.
The following program fits the linear model to a set of n data points x
i1
, x
i2
, …, x
ip
, y
i
by the
method of least-squares. The parameters b
1
, b
2
, …, b
p
are estimated by the solution
bˆ
to the
normal equations X
T
Xb = X
T
y. The program also estimates σ
2
and the parameter covariance
matrix Cov(
bˆ
). The regression and residual sums of squares (Reg SS and Res SS) and the
residuals are also calculated.
The program requires two matrices:
Matrix A: n × p with row i (x
i1
, x
i2
, …, x
ip
) for i = 1, 2, ... , n.
Matrix B: n × 1 with element i (y
i
) for i = 1, 2, ... , n.
The program output is:
Matrix A: unchanged.
Matrix B: n × 1 containing the residuals from the fit (y
i
−
1
bˆ
x
i1
− … −
p
bˆ x
ip
)
for i = 1, 2, ... , n, where
i
bˆ
is the estimate for b
i
.
Matrix C: p × p covariance matrix of the parameter estimates.
Matrix D: p × 1 containing the parameter estimates
1
ˆb
, …,
p
bˆ
.
T-register: contains an estimate of σ
2
.
Y-register: contains the regression sum of squares (Reg SS).
X-register: contains the residual sum of squares (Res SS).