Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

PROBLEMS

Sufficient Statistics and UMVU Estimation

9.1 For N iid Poisson samples, use the definition of sufficiency to find a sufficient statistic for parameter α.

9.2 For N iid uniform samples with pdf , use the factorization theorem to find a sufficient statistic for .

9.3 Repeat Problem 9.2 for N iid gamma samples to find a sufficient statistic for .

9.4 Derive the expectations in (9.30) for the minimum and maximum.

9.5 Find the minimal sufficient statistic from M iid samples of the chi-square distribution with parameter N.

9.6 Use the ratio approach in (9.34) to find the minimal sufficient statistic from N iid samples of the Poisson random variable with parameter α.

9.7 For the exponential random variable with location parameter c:

(9.398)

show that the minimal sufficient statistic for c (with fixed λ) is X(1) from N iid samples.

9.8 For N iid Laplace samples with parameters , find a minimal sufficient statistic for (a) α (with fixed) and then (b) μ (with α = 1).

9.9 For N iid uniform ...