Detection and Estimation, Fall 2015Problem Set 1 (Due October 18, 2015)1. Consider N i.i.d samples from a Gaussian distribution of mean and variance both unknown. Find maximum likelihood (ML) estimates for the mean and variance. Examine whether the estimators are unbiased and consistent. If the estimator for the variance is biased, suggest an unbiased estimator.

2. Consider an estimation problem in which the observations are real-valued random variables given by

where are i.i.d. . Find a sufficient statistic for .3. Consider an observation given by

where is an unknown fixed parameter to be estimated and Z is uniformly distributed over the interval . Find the ML estimate of given

4. Let

for x > 0. Show that there exist no unbiased estimators for x. Note that becauseonly x > 0 are possible values, an unbiased estimator need only be unbiased for x > 0.

5. Consider a point x on a unit circle in . Given N i.i.d observations from of the form

where , and is zero-mean Gaussian with covariance matrix with Find the ML estimator for x.

6. Consider N i.i.d samples , where , from a Bernoulli distribution with parameter p, i.e., with probability p and with probability . Find the ML estimate of p. Find the Cramer-Rao lower bound on the variance of the estimator. Is the estimator efficient?

7. Consider N samples from a noisy sinusoidal signal with known frequency and amplitude A

where is zero-mean Gaussian with variance equal to , and is the sampling time such that is equal to one period of the sinusoidal signal. Parameter is much smaller than the period. Angle is an unknown phase to be estimated. Find the ML estimate for . Sketch a block diagram of estimator operation.

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