Page 90 - Classification Parameter Estimation & State Estimation An Engg Approach Using MATLAB

EXERCISES 79

A robust estimator is one which can cope with outliers in the measure-
ments. Such an estimator can be achieved by application of a robust
error norm.

3.5 SELECTED BIBLIOGRAPHY

Kay, S.M., Fundamentals of Statistical Signal Processing – Estimation Theory, Prentice
Hall, New Jersey, 1993.
Papoulis, A., Probability, Random Variables and Stochastic Processes, Third Edition,
McGraw-Hill, New York, 1991.
Sorenson, H.W., Parameter Estimation: Principles and Problems, Marcel Dekker, New
York, 1980.
Tarantola, A., Inverse Problem Theory: Methods for Data Fitting and Model Parameter
Estimation, Elsevier, New York, 1987.
Van Trees, H.L., Detection, Estimation and Modulation Theory, Vol. 1, Wiley, New
York, 1968.

3.6 EXERCISES

1. Prove that the linear MMSE estimator, whose form is ^ x lMMSE (z) ¼ Kz, is found as:
x
def
K ¼ M xz M 1 with M xz ¼E xz T and M z ¼E zz T ð Þ
def
z
2. In the Gaussian case, in Section 3.1.3, we silently assumed that the covariance matrices
C x and C v are invertible. What can be said about the elements of x if C x is singular?
And what about the elements of v if C v is singular? What must be done to avoid such
a situation? ( )
3. Prove that, in Section 3.1.3, the posterior density is Gaussian, and prove equation
(3.20). ( ) Hint: use equation (3.2), and expand the argument of the exponential.
4. Prove equation (3.32). (0)
5. Use the matrix inversion lemma (b.10) to prove that the form given in (3.20):

1 T 1 T 1 1 1
^ x x MMSE ðzÞ¼ C e C m þ H C z with C e ¼ H C H þ C x
v
v
x
x
is equivalent to the Kalman form given in (3.33). ( )
R
6. Explain why (3.42) cannot be replaced by C e ¼ C e (x)p(x)dx.( )
7. Prove that the unbiased linear MMSE estimator is indeed unbiased. (0)
8. Given that the random variable z is binominally distributed (Appendix C.1.3) with
parameters (x,M). x is the probability of success of a single trial. M is the number of
independent trials. z is the number of successes in M trials. The parameter x must be
estimated from an observed value of z.
. Develop the ML estimator for x. (0)
. What is the bias and the variance of this estimator? (0)