1. Introduction
Many authors have used the likelihood ratio to study the change-point problem (see [1] [2] ). Worsley, K.J. [3] gave exact approximate bounds for the null distributions of likelihood ratio statistics in two case of known and unknown variance. Simulation study results indicated that the approximation of his upper bound is very good for the small sample size, but the study does not support the case of large one. Koul and H.L, Qian. L. [4] studied the change-point by the maximum likelihood and random design. In the case of known variance, Jaruskova, D. [5] derived an asymptotic distribution of log-likelihood type ratio to detect a change-point from a known (or unknown) constant state to a trend state. Aue A., Horvath, L., Huskova, M. and Kokoszka, P. [1] studied the limit distribution of the trimmed version of the likelihood ratio, from which they received the test statistic to detect a change-point for the polynomial regressions. Researchers have used to take simulation studies on the various scenarios of the parameters of alternative hypothesis to find the power of a test. They have found that it depends on the sample size, variance of error and the behavior of the model function under alternative. For two phases’ regression, Lehmann, E.L. and Romano, J.P. [6] gave a formula to calculate the power of change-point through the noncentral F-distribution.
In this paper, the behavior of the model function under alternative is quantified by the roughness that is used to calculate the power of tests. The present paper is organized in the following way. In Section 2, we give a definition of the roughness of the model function and show some its properties; it is possible to take the limit of the roughness when the sequence of designs converges weakly to a limit design as well as designs are random. In Section 3, we present an explicit formula to calculate the noncentrality parameter of F-test in [6] through the roughness, and then the power of change-point test and some of its limits are considered.
2. The Roughness of the Model Function
To approximate the function
by a given system of functions
at the given points
, we consider the model
(1)
. (2)
(3)
(4)
We call this value the roughness of the function
to the system of functions
based on the design
and denote it by
. In the case of a linear trend where ![](//html.scirp.org/file/10-7402476x29.png)
shows the nonlinearity of the curve
based on observations at
.
To study limits cases as well as other purposes, we call a distribution function
whose support belongs to
a (generalized) design on
. A design
is called to be adapted to a system of functions
if its support belongs to
so that the matrix
is invertible. In this paper, the used designs are assumed to be adapted to the system ![]()
To continue, we will establish some assumptions:
(A2) Trend functions
are linearly independent and continuous.
Now suppose that (A1) and (A2) hold, we approximate the function
to
in the equation:
(5)
The estimate for the parameter vector
that minimizes the weighted mean square error
is
(6)
where
. Hence, the estimate for the error of the model (5) is
. (7)
We also call this value the roughness of the function
to the system of trend functions
based on the design
and denote it by
It is easily seen that each discrete design is a generalized design, thus (3), (4) is a special case of (6), (7), respectively.
According to [2] , to evaluate the roughness of the model function based on polynomial trend functions
, by using the linear transformation of independent variables, instead of observing on the arbitrary
interval
one can observe on the standard interval [0, 1]. Then, from now on, the model functions defined on [0, 1] are considered only.
The following theorem in [7] shows the conditions for occurring the convergence of the estimated parameters and the roughness.
1) ![]()
2) ![]()
Now, we consider the model (1) where the observations
are i.i.d. with the distribution function
having support on
The roughness of
is calculated by (4), in which
is replaced by ![]()
(8)
where
![]()
![]()
Theorem 2. Suppose that (A1) and (A2) hold for
and
1)
are independent random variables with the common distribution function
having support on
,
2) ![]()
3) The roughness
is defined by (8).
Then
![]()
![]()
Let
be the least-square estimate of
bases on
observations, we get
![]()
![]()
Because
is a sequence of i.i.d. variables which have finite variance then by the strong law of large numbers,
![]()
Then, according to the assumption 2),
![]()
which follows that elements of the matrix
converge (a.s) to corresponding elements of the matrix ![]()
Similar arguments yield
![]()
Consequently, we obtain the limit
(9)
Note that
is not random and the roughness can be expressed by
![]()
where
![]()
![]()
Because
are i.i.d. and bounded then according to the central limit theorem,
(10)
Inasmuch as
satisfies (9), it can be calculated by (6). Hence, the right side of (10) is ![]()
Again, according to the central limit theorem and (9),
![]()
Combining the above with the fact that
we obtain
![]()
This completes the proof of the theorem. ![]()
3. Applications to the Change-Point Test
Suppose that the model function is defined as:
(11)
where
are known functions,
are unknown parameters. Observations
belong to the closed interval
without the loss of generality, we can assume
some
can be identical. Suppose that a change-point happened at a some time
the model is written:
(12)
where
is a sequence of i.i.d. variables
with the unknown common variance
.
Let
![]()
Using matrix notations, the Equation (12) is written as
(13)
We are interested in testing the hypothesis of structural stability against the alternative of a regime switch at a sometime
that is
(14)
Let
be known as it was studied in Bischoff and Miller [8] . In addition, we assume that the matrices
have full rank:
thence
From that, vector
belongs to a
-dimensional linear subspace
and the null hypothesis
to test that
lies in a
dimensional subspace
of ![]()
The least-squares estimate of
under
and
under
are
,
and
, respectively. Let
are the orthogonal projections of
onto
and
then
and ![]()
We already know that (see Lehmann, E.L. and Romano, J.P. [6] ): Under
the statistics
(15)
will be distributed
Thus, the test rejects the null hypothesis at level
if
(16)
where
is the
-critical value for
a
-distributed random variable with ![]()
and
degrees of freedom. According to [6] , by denoting
where ![]()
are orthogonal projections of
onto
and
respectively, then under
, the statistic
defined by (15) will be noncentral
-distribution with
degrees of freedom and noncentrality parameter
![]()
We note that
and
which implies that
![]()
Now, we call
and
the signal-to-noise of the model (11) based on the design
and
respectively.
Theorem 3. If assumptions (A2), (A3) hold then the power of test (16) is defined by
(17)
Remark. Theorem 3 shows an explicit formula of the power of change-point test. In the case of
and
, if the model function
is continuous segment, the shift of the slope between the first segment and the last one is
by Theorem 1 in [7] , the maximum roughness is obtained if the change-point
is the midpoint of the observations. With the given common variance
of the model, the maximum signal- to-noise
is obtained at this change-point, thence from Theorem 3, the power is maximum. This fits results of simulation studies in [1] .
To increase signal-to-noise ratio, we can decrease the noise or increase the roughness of the model function. When the variance
is small, we can assert that if the model function has a change-point then this test will find it surely. On the other hand, if the variance is large, the test is poorly.
With the sample size
and design
if the variance
decreases to 0 then
increases to
and if
increases to
then
decreases to 0. We have the following corollaries that show the relationship between power and the common variance and the roughness.
Corollary 1. If the assumptions in Theorem 3 are satisfied then the following limits hold:
1) ![]()
2) ![]()
Limits of the powers are obtained by the following corollary.
Corollary 2. 1) With the same conditions as in Theorem 3, assume that
for every
and some
then ![]()
2) Furthermore, if the model function
and a sequence of designs
satisfies the conditions of Theorem 1, then
as long as ![]()
Proof. First of all, it is easy to see that
![]()
where
,
are independent.
Because
then ![]()
Moreover,
and
then the last probability converges to 1 that yields 1).
Now, according to Theorem 1,
![]()
then 2) is implied straight from 1). ![]()