Received 3 September 2014; revised 28 September 2014; accepted 12 October 2014

ABSTRACT

In this paper, the roughness of the model function to the basis functions and its properties have been considered. We also consider some conditions to take the limit of the roughness when the observations are i.i.d. An explicit formula to calculate the power of change-point test for the two phases regression through the roughness was obtained.

Keywords:

Change-Point Test, The Power, The Roughness, Random Design

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 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).

References