Chapter 2 Minimum Variance Unbiased estimation

1. Chapter 2Minimum Variance Unbiased estimation

2. Introduction • In this chapter we will begin our search for good estimators of unknown deterministic parameters. • We will restrict our attention to estimators which on the average yield the true parameter value. • Then, within this class of estimators the goal will be to find the one that exhibits the least variability. • The estimator thus obtained will produce values close to the true value most of the time. • The notion of a minimum variance unbiased estimator is examined within this chapter.

3. Unbiased Estimators • For an estimator to be unbiased we mean that on the average the estimator will yield the true value of the unknown parameter. • Since the parameter value may in general be anywhere in the interval , unbiasedness asserts that no matter what the true value of θ, our estimator will yield it on the average. (2.1)

4. Example 2.1 (1/2) • Consider the observations where A is the parameter to be estimated and w[n] is WGN. The parameter A can take on any value in the interval . • The reasonable estimator for the average value of x[n] is or the sample mean. (2.2)

5. Example 2.1 (2/2) • Due to the linearity properties of the expectation operator for all A. The sample mean estimator is unbiased.

6. Unbiased Estimators • The restriction that for allθ is an important one. • It is possible that may hold for some values of θ and not others.

7. Example 2.2 • Consider again Example 2.1 but with the modified sample mean estimator • Then, • It is seen that (2.3) holds for the modified estimator only for A = 0. • Clearly, it is a biased estimator.

8. Unbiased Estimators • That an estimator is unbiased does not necessarily mean that it is a good estimator. • It only guarantees that on the average it will attain the true value. • A persistent bias will always result in a poor estimator. • As an example, the unbiased property has an important implication when several estimators are combined. A reasonable procedure is to combine these estimates into a better one by averaging them to form

9. Unbiased Estimators • Assuming the estimators are unbiased, with the same variance, and uncorrelated with each other, and so that as more estimates are averaged, the variance will decrease.

10. Unbiased Estimators • However, if the estimators are biased or , then and no mater how many estimators are averaged, willnot converge to the true value. • Note that, in general, is defined as the bias of the estimator.

12. Minimum Variance Criterion • In searching for optimal estimators we need to adopt some optimality criterion. • A natural one is the mean square error (MSE), defined as • Unfortunately, adoption of this natural criterion leads to unrealizable estimators, ones that cannot be written solely as a function of the data.

13. Minimum Variance Criterion • To understand the problem which arises we first rewrite the MSE as which shows that the MSE is composed of errors due to the variance of the estimator as well as the bias. (2.6)

14. Minimum Variance Criterion • As an example, for the problem in Example 2.1 consider the modified estimator for come constant a. • We will attempt to find the a which results in the minimum MSE. • Since and , we have

15. Minimum Variance Criterion • Differentiating the MSE with respect to a yields which upon setting to zero and solving yields the optimum value • It is seen that the optimal value of a depends upon the unknown parameter A. The estimator is therefore not realizable.

16. Minimum Variance Criterion • In retrospect the estimator depends upon A since the bias term in (2.6) is a function of A. • It would seem that any criterion which depends on the bias will lead to an unrealizable estimator. • From a practical view point the minimum MSE estimator needs to be abandoned.

17. Minimum Variance Criterion • An alternative approach is to constrain the bias to be zero and find the estimator which minimizes the variance. • Such an estimator is termed the minimum variance unbiased(MVU) estimator. • Note that from (2.6) that the MSE of an unbiased estimator is just the variance. • Minimizing the variance of an unbiased estimator also has the effect of concentrating the PDF of the estimation error about zero. • The estimation error will therefore be less likely to be large.

18. Existence of the Minimum Variance Unbiased Estimator • The question arises as to whether a MVU estimator exists, i.e., an unbiased estimator with minimum variance for all θ.

19. Example 2.3 (1/3) • Assume that we have two independent observations x[0] and x[1] with PDF The two estimators can easy be shown to be unbiased.

20. Example 2.3 (2/3) • To compute the variances we have that so that and

21. Example 2.3 (3/3) • Clearly, between these two estimators no MVU estimator exists. • No single estimator can have a variance uniformly less than or equal the minima.

22. Finding the Minimum Variance Unbiased Estimator • Even if a MV estimator exists, we may not be able to find it. • In the next few chapters we shall discuss several possible approaches. • They are: • Determine the Cramer-Rao lower bound (CRLB) and check to see if some estimator satisfies it (Chapters 3 and 4). • Apply the Rao-Blackwell-Lehmann-Scheffe (RBLS) theorem (Chapter 5). • Further restrict the class of estimators to be not only unbiased but also linear. Ten, find the minimum variance estimator within this restricted class (Chapter 6).

23. Finding the Minimum Variance Unbiased Estimator • The CRLB allow us to determine that for any unbiased estimator the variance must be greater than or equal to a given value. • If an estimator exists whose variance equals the CRLB for each value of θ, then it must be the MVU estimator.

24. Extension to a Vector Parameter • If is a vector of unknown parameters, then we say that an estimator is unbiased if for i = 1, 2, …, p. • By defining (2.7)

25. Extension to a Vector Parameter • We can equivalently define an unbiased estimator to have the property for every θ contained within the space defined in (2.7). • A MVU estimator has the additional property that for i = 1, 2, …, p is minimum among all unbiased estimators.

26. Chapter 3Cramer-Rao Lower Bound

27. Introduction • Place a lower bound on the variance of any unbiased estimator and assert that an estimator is the MVU estimator. • Although many such variance bounds exist [McAulay and Hofstetter 1971, Kendall and Stuart 1979, Seidman 1970, Ziv and Zakai 1969], the Cramer-Rao lower bound (CRLB) is the easiest to determine.

28. 3.3 Estimator Accuracy Considerations • Consider the hidden factors that determine how well we can estimate a parameter. • The more the PDF is influenced by the unknown parameter, the better we should be able to estimate it. • Example 3.1 - PDF dependence on unknown parameterIf a single sample is observed aswhere , and it is desired to estimate A

29. 3.3 Estimator Accuracy Considerations • Example 3.1(cont.)A good unbiased estimator isThe variance isThe estimator accuracy improves as decreases. If and

30. 3.3 Estimator Accuracy Considerations • Example 3.1(cont.)the latter is a much weaker dependence on A.

31. 3.3 Estimator Accuracy Considerations • The “sharpness” of the likelihood functions determines how accurately we can estimate the unknown parameter.

32. 3.3 Estimator Accuracy Considerations • For this examplethe second derivative does not depend on • In general ,a more appropriate measure of curvature is

33. 3.3 Estimator Accuracy Considerations • Which measures the average curvature of the log-likelihood function. • The expectation is taken with respect to ,resulting in a function of A only. • The larger the quantity, the smaller the variance of the estimator.

34. 3.4 Cramer-Rao Lower Bound • Theorem 3.1 (CRLB – Scalar Parameter)It is assumed that the PDF satisfies the “regularity” condition for allthen , the variance of any unbiased estimator must satisfy

35. 3.4 Cramer-Rao Lower Bound • Theorem 3.1(cont.)furthermore, an unbiased estimator attains the bound if and only ifand min variance

36. 3.4 Cramer-Rao Lower Bound • Prove when the CRLB is attained, thenproof：Because CRLB is attained and

37. 3.4 Cramer-Rao Lower Bound • Proof：(cont.)so we getand thenfinally,

38. 3.4 Cramer-Rao Lower Bound • Regularity

39. 3.4 Cramer-Rao Lower Bound • Example 3.2 – CRLB for Example 3.1

40. 3.4 Cramer-Rao Lower Bound • Example 3.3 – DC level in white Gaussian Noiseconsider the multiple observationsPDF

41. 3.4 Cramer-Rao Lower Bound • Example 3.3(cont.)

42. 3.4 Cramer-Rao Lower Bound • Example 3.3(cont.)we see that the sample mean estimator attains the bound and must therefore be the MVU estimator.

43. 3.4 Cramer-Rao Lower Bound • Example 3.4 – Phase EstimatorA and f0are assumed known, and we wish to estimate the phase

44. 3.4 Cramer-Rao Lower Bound • Example 3.4(cont.) So we get

45. 3.4 Cramer-Rao Lower Bound • Example 3.4(cont.)In this example the condition for the bound to hold is not satisfied.Hence, a phase estimator does not exist which unbiased and attains the CRLB. • But, a MVU estimator may exist

46. 3.4 Cramer-Rao Lower Bound • Efficiency vs min variance

47. 3.4 Cramer-Rao Lower Bound • Fisher information properties: • Nonnegative • Additive for independent observations

48. 3.4 Cramer-Rao Lower Bound • The latter property leads to the result that the CRLB for N IID observations is 1/N times that for one observation. • For completely dependent samples,

49. 3.5 General CRLB for Signals in White Gaussian Noise • Consider

50. 3.5 General CRLB for Signals in White Gaussian Noise • finally,