Journal of Probability and Statistics
Volume 2017 (2017), Article ID 6219149, 14 pages
https://doi.org/10.1155/2017/6219149

Research Article

Estimation of the Parameters of a Chirp Type Model with Stationary Residuals

K. Perera

Department of Engineering Mathematics, Faculty of Engineering, University of Peradeniya, Peradeniya, Sri Lanka

Correspondence should be addressed to K. Perera; kanthip@pdn.ac.lk

Received 19 July 2016; Accepted 1 December 2016; Published 9 February 2017

Academic Editor: Aera Thavaneswaran

Copyright © 2017 K. Perera. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Let be the observations from a chirp type statistical model , , where is a stationary noise. We consider a method of estimation of parameters, , , , , and , (where is the variance of ’s) which is basically an approximate least-squares method. The main advantage of the proposed approach is that no assumptions are required. We make use of the three theorems which were established associated with the kernel and then use them to prove, under certain conditions, the consistency of the estimators.

1. Introduction

In [1] 1973 Walker considered the problem of estimating the parameters of a sine wave,

where ’s are the observations and ’s are independent, identically distributed random variables with mean zero and finite but unknown variance . The parameters , , , and are assumed unknown and are to be estimated. He showed that, as , the estimators , , , and converge in probability to the actual values , , , and , respectively. That is, he showed that the estimators are consistent. He then showed that the differences between the estimators and the actual values they estimate have a joint normal distribution.

Suppose now that the frequency of the above sine wave changes with time. If is the initial frequency, the frequency at time may be written as

The simplest case is one in which is linear; that is, . Then . This leads to the model

The parameters for this model are , , , , and . This is sometimes called the “chirp” model (see [2, 3]). Several authors have considered the parameter estimation of the chirp model and are found in [4–7]. Different approaches to the estimation of chirp parameters in similar kinds of models are found in [8–12].

But our approach is entirely different from them. We do not make any assumptions. The method used to estimate the parameters and to prove the consistency of the estimators is similar to that by Walker in [13]. We consider not only estimates of the parameters of , , , and but also an estimate of . Although this leads to an interesting problem in estimation, the model is somewhat unrealistic. Over the course of observations, the frequency changes from to . As goes to , the sine wave oscillates faster and faster (unless ). Its frequency becomes infinite, and its period approaches . We therefore change the model by assuming that the change in frequency over the course of the observations is a number independent of . We also assume, as in (3), that the change in frequency is linear. This leads to the model or, more precisely, sequence of models

We assume, as Walker does in his 1971 paper, that ’s are independent and identically distributed with mean zero and variance . Thus the parameters are , , , , and and the estimators of the parameters are , , , , and . Our objective is to show that these estimators are consistent.

We make use of the following three theorems which we have established.

Theorem 1 (see [14]). Let , where is a nonnegative real number. Then , for not a rational multiple of , uniformly in .

Theorem 2 (see [15]). Let be a sequence of independent random variables such that , , and . Then

Theorem 3 (see [16]). For any sufficiently small and

2. Estimation of the Parameters

If ’s are normally distributed, the likelihood function of the observations is

Then the log likelihood function of the observations is

where

Now consider :

By using the identities , , and we obtain

From Theorem 1, the last two terms in (11) are of order . Therefore if we let

then is an approximation of . More precisely, . Thus

is an approximation of the log likelihood function . Now fix in (13). Then we maximize to obtain the estimates for , , , and . That is, we minimize over the region . Now fix and . If or then . Since is continuous in , the minimum is achieved at a point . The partial derivatives and must vanish at . Thus the estimators of and are solutions of the equations

The solution to these equations is given by

To obtain estimates for and we substitute and for and , respectively, in (12) and minimize as a function of and . Since this last expression is equal to

with respect to and ; this is same as maximizing

Since varies over a compact set, there is a point which maximizes this expression. We then estimate and by

The minimum value of is assumed at .

Now we obtain an estimator of . If is zero then, since and , almost surely. Thus there is no randomness in the model. We assume, then, that . To obtain we substitute , , , and in (13) and maximize . From (13),

We will show that , as a function of , achieves its positive maximum provided , which we show is positive with high probability for large values of . From (4) and (9) we have

Using (11) and (12) we have

Using the mean value theorem,

where is a point on the line segment connecting and .

Thus

In Section 3 we prove that and . Therefore we have

Also in Section 3 we show that

Thus (25) implies that

Similarly

In Section 3 we also show that and . Thus (22) implies that

Using (21) in the above equation we obtain

By the weak law of large numbers

Using this in (30) we obtain that with probability arbitrarily close to 1 for sufficiently large values of . Suppose, then, that . Since

and , as or . Thus since is a continuous function in the maximum is achieved when . The partial derivative vanishes at . Therefore is the solution to equation

So

Note that exits only with high probability for large . Using (12) and substituting for and we obtain

Now, using (18) in the above equation, we obtain

Therefore, substituting in (34) we have

3. The Consistency of the Estimators

Now we will establish under certain conditions the consistency of the estimators , , , , and

Theorem 4. Let be a sequence of independent random variables with , , and . Let be any real numbers and . Let and let . For each , let

The estimators , , , , and of , , , , and given in (12), (13), (15), and (17), respectively, are consistent estimators of , , , , and , respectively. Furthermore and .

Proof. From (15)

But

Thus

If we use complex exponentials instead of sines and cosines we have

where and , and then

Let

Then

Expanding the third term on the right, we get

Lemma 5.

Proof. From (46) we have

By virtue of Theorem 2

Hence

Thus we obtain the following estimates for the first and second terms in (48):

Consider the term in (48). Using (44) we have

Now we can write

where .
Also

It follows by virtue of Lemma [16] that there is for which

Thus we have

Let . Then we obtain

In the proof of Theorem 3 [16, p. 65, eq 1.9], we showed that if at least one of or tends to then

Therefore, since , we have

This together with (57) implies that

Thus using this equation (51), and (60) in (48) we obtain

Thus

This completes the proof of Lemma 5.

Lemma 6. Let be defined by

where and . Then

Proof. Throughout the proof it is assumed that and . From (46) we have

From Theorem 2

This implies that

Thus we obtain the following estimates for the first and second terms in (65):

Consider the term . Using (44) we have

We can write

where .
Then

It follows by virtue of Lemma [9, p. 1] that there is for which

Similarly we can show that

Thus we have

Let in (74). Then we have

Let . We will show that

Suppose

Then there is a sequence and for which

for each . Then we can find for which

for each . Let , . Since , , ; hence . It follows from equation 1.9 [16, p. 65] in the proof of Theorem 3 that

Thus we have a contradiction. Therefore

Hence by (76)

This implies that

Substituting (68) and (84) in (65) we obtain

Thus

This completes the proof of Lemma 6.

Now combining Lemmas 5 and 6 we obtain

Thus

where

and .

From Theorem 3 we have

Substituting and in the above inequality we obtain

Therefore since and , we have

Therefore .

Let . Since , there exits such that

Since (<0), there exits such that

Now let . If , then

Hence

Now, from (88),

Therefore

Since is arbitrary,

In other words,

where

In the last statement the maximum is taken over the region shown in Figure 1.

Figure 1

Suppose belongs to the shaded region of the figure. Then does not exceed . If this implies that is not the point which maximizes which in turn implies that lies in the unshaded rectangle, namely, the region . Thus (100) implies that, for large , the probability is high that the maximum value lies in that region, that is,

In other words

Since and are arbitrary positive numbers, we have

Next we prove the consistency of and . That is, we show that

Recall from (15) and (16)

Thus

But

so

Letting

where and , we have

Since we obtain, using (44),

Using the triangle inequality we get

Now we estimate each of the terms on the right hand side of the above inequality.

Let .

By the mean value theorem

where is a point on the open line segment connecting and .

Now

and similarly

Thus

Using the triangle inequality and the fact that , we obtain

Substituting and for and , respectively, we get

so that

It follows from (104) that

Similarly

Thus

Now consider the first term on the right in inequality (113). Again by the mean value theorem,

where is a point on the open line segment connecting and . Using the above expressions for and , we have as before

Thus

Therefore

Substituting and for and , respectively, we get

so that

Using (60) and (104) we obtain

Similarly

Thus

Now consider the last term in inequality (113). As a consequence of Theorem 2, we have

By using (123), (132), and (133) in inequality (113), we obtain

Therefore

That is,

Now we prove the consistency of . That is, we show that .

Recall from (37) that

From the definition of , we have

But from (15) and (16) we have

Hence

Therefore

Since

we have

Using the identities , , and we obtain

From (60) we have

Thus

Therefore

Now we estimate each term on the right in (147). First consider the term . ’s are independent and identically distributed with . Therefore by the Weak Law of Large Numbers

Next consider the second and third terms in the right hand side of (147).

Using Theorem 2 we have

Now consider the last term in (136). From (132) we have

Thus

Therefore

By using (148), (149), and (152) in (147) we obtain

This completes the proof of the theorem.

Competing Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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