SOLVED CH5115: Parameter and State Estimation Assignment 3

1. Suppose a random sample of N observations drawn from the following PDF are given: f(y) = ( e −(y−θ) , x > θ, −∞ < θ < ∞ 0 otherwise (1) (a) Consider the statistic TN = 2 min(yN ). Derive the PDF of TN . (b) If TN is used as an estimator of θ, verify theoretically if it is unbiased. If TN is found to be biased, apply a correction factor to make it unbiased. Denote the corrected statistic as T 0 N . (c) Prove that T 0 N converges to θ in probability using your answer in (1a). 2. Given an N-length realization of a random process, v[k], compute the DFT coefficients and an estimator of the PSD as: V [n] , V (fn) = N X−1 k=0 v[k] exp (−j2πfnk), fn = n N , n = 0, 1, · · · , N − 1 (2) P[n] , P(fn) = |V (fn)| 2 N = a 2 n + b 2 n N a = <(V [n]), b = =(V [n]) (3) For analysis, take up the zero-mean, unit variance GWN process. Recall that the true PSD is γvv(f) = X∞ l=−∞ σvv[l] exp (−j2πfl) (4) where σvv[l] is the ACVF. For MC simulations, wherever required, use N = 1000 and R = 300 (no. of realizations). Choose a frequency fn corresponding to n = last two digits of your Roll No. + 1. (a) Show (theoretically) that the DFT coefficients, an and bn are Gaussian distributed. Verify your answer through MC simulations. (b) Determine the correlation between an and bn. (c) Next, determine the distribution of ζn = 2Pvv(fn) γvv(fn) . From this distribution, determine the mean and variance of ζn. (d) Finally, verify if P(fn) is a consistent estimator of γvv(fn) in the mean-square sense both theoretically and by MC simulations. 2 3. (a) In the frequency detection example discussed in the lecture, assume both amplitude and frequency are unknown. Given N observations, determine the CRLB for amplitude and frequency estimates. Compare the variance bounds with those obtained when each of the parameters is estimated assuming the other is known. (b) Design a BLUE that estimates variance of a zero-mean GWN process y[k] = e[k] ∼ N (0, σ2 ) from N observations of y[k]. If such an estimator does not exist, design a BLUE for σ 2 using transformed data. 4. (a) A random collection of 100 samples of a polymer manufactured by a leading chemical industry revealed a mean molecular weight of 14578 with a standard deviation of 1845. With what degree of confidence can we assert that the average molecular weight of that polymer is in between 12000 and 16000? (b) Suppose that we want to investigate whether on the average, students from an elite institution A perform better in an exam than students from another elite institution B. If 60 students of A demonstrated on the average x¯1 = 85.2 marks (out of 100) with s1 = 6.8 marks, while 55 students from institute B produced an average x¯1 = 87.2 marks (out of 100) with s1 = 8.8 marks, what can we conclude? 5. (a) Prove that for Q.1, TN is a sufficient statistic for θ. Design the MVUE for θ. (b) Compare the efficiency of estimator 9using sample MC simulations) obtained in (5a) with that of ˆθ1 = ¯y − 1 and ˆθ2 = ˜y − ln 2, where y¯ and y˜ are the sample mean and sample median, respectively