Chapter 2 Minimum Variance Unbiased estimation - PowerPoint PPT Presentation

1 / 64
About This Presentation

Chapter 2 Minimum Variance Unbiased estimation


Chapter 2 Minimum Variance Unbiased estimation Wireless Information Transmission System Lab. National Sun Yat-sen University Institute of Communications Engineering ... – PowerPoint PPT presentation

Number of Views:40
Avg rating:3.0/5.0
Slides: 65
Provided by: 4518


Transcript and Presenter's Notes

Title: Chapter 2 Minimum Variance Unbiased estimation

Chapter 2Minimum Variance Unbiased estimation
  • 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
  • Then, within this class of estimators the goal
    will be to find the one that exhibits the least
  • 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.

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 ?, our estimator will yield it on the

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

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

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

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.

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
  • 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

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.

Unbiased Estimators
  • However, if the estimators are biased or
  • , then
  • and no mater how many estimators are averaged,
    will not converge to the true value.
  • Note that, in general,
  • is defined as the bias of the estimator.

(No Transcript)
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

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.

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

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.

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.

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.

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

Example 2.3 (1/3)
  • Assume that we have two independent observations
    x0 and x1 with PDF
  • The two estimators
  • can easy be shown to be unbiased.

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

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.

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).

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 ?, then it must be the MVU

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

Extension to a Vector Parameter
  • We can equivalently define an unbiased estimator
    to have the property
  • for every ? 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.

Chapter 3Cramer-Rao Lower Bound
  • 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.

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

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

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

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

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

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.

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

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

3.4 Cramer-Rao Lower Bound
  • Prove when the CRLB is attained,
    thenproofBecause CRLB is attained and

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

3.4 Cramer-Rao Lower Bound
  • Regularity

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

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

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

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.

3.4 Cramer-Rao Lower Bound
  • Example 3.4 Phase EstimatorA and f0 are
    assumed known, and we wish to estimate the phase

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

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

3.4 Cramer-Rao Lower Bound
  • Efficiency vs min

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

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,

3.5 General CRLB for Signals in White Gaussian
  • Consider

3.5 General CRLB for Signals in White Gaussian
  • finally,

3.5 General CRLB for Signals in White Gaussian
  • Example 3.5 Sinusoidal Frequency
    EstimationAssume where A and phase are known.
    So we get the CRLB
  • If , the CRLB goes to infinity.

3.5 General CRLB for Signals in White Gaussian
  • Example 3.5 (cont.)

3.6 Transformation of Parameters
  • Usually, the parameter we wish to estimate is a
    function of some more fundamental parameter.
  • In Example 3.3, we wish to estimate A2. Knowing
    the CRLB for A, we can easily obtain it for A2.
  • As shown in Appendix 3A, if it is desired to
  • , then the CRLB is

3.6 Transformation of Parameters
  • For the present example this becomes
  • In Example 3.3, the sample mean estimator was
    efficient for A. It might be supposed that
    is efficient for A2.
  • But actually, is not even an unbiased
  • proof Since

3.6 Transformation of Parameters
  • The efficiency of an estimator is destroyed by a
    nonlinear transformation.
  • But the efficiency is maintained for linear
  • Proof Assume that an efficient estimator for
    exists and is given by . It is desired to
    estimate .
  • We choose .
  • So that is unbiased.

3.6 Transformation of Parameters
  • The CRLB for
  • But
    , so that the CRLB is achieved.
  • So, the efficiency is maintained for linear

3.6 Transformation of Parameters
  • The efficiency is approximately maintained over
    nonlinear transformations if the data record is
    large enough.
  • Ex The example of estimating .
    Although is biased, we note that
    is asymptotically
    unbiased or unbiased as .
  • Since , we can evaluate the

3.6 Transformation of Parameters
  • Using the result that if
  • therefore
  • For our problem, we have
  • Hence, as , is an asymptotically
    efficient estimator of A2.

3.6 Transformation of Parameters
  • This situation occurs due to the statistical
    linearity of the transformation, as illustrated
    in figure. As N increased, the PDF of
    becomes more concentrated about the mean A.

3.6 Transformation of Parameters
  • If we linearize g about A, we have the
  • Within this approximation,
  • the estimator is unbiased (asymptotically).
  • so the estimator achieves the CRLB

3.7 Extension to a Vector Parameter
  • Now we extend the results to a vector parameter
  • . As derived in Appendix 3B,
    the CRLB is found as the i, i element of the
    inverse of a matrix
  • where is the Fisher
    information matrix.
  • for .

3.7 Extension to a Vector Parameter
  • Example 3.6 DC Level in White Gaussian Noise
  • We now extend example 3.3
  • to the case where in addition to A the noise
  • is also unknown.
  • The parameter vector is , hence
    p 2. The 2x2 Fisher information matrix is

3.7 Extension to a Vector Parameter
  • The log-likelihood function is
  • The derivatives are easily found as

3.7 Extension to a Vector Parameter
  • Taking the negative expectation, the Fisher
    information matrix becomes
  • Although not true in general, for this example
    the Fisher information matrix is diagonal and
    hence easily inverted to yield
Write a Comment
User Comments (0)