problem 1:

a) State the prove Lehmann-Pearson Lemma. Give its significance in testing of hypothesis.

b) Construct a LR test to test H_o: µ = µ_o against H₁: µ ≠ µ_o in sampling from N (µ, σ²), where both µ and σ² are unknown.

problem 2:

a) describe Kolomogolor-Sminnor one sample and two sample tests.

b) Describe the term Mann-Whitney U-test.

problem 3:

a) Show that SPRT terminates finally with certainty.

b) Define the term OC and ASN functions of SPRT. Derive them for testing the proportion of a binomial distribution.

problem 4:

a) State and prove the Wald’s fundamental identify.

b) describe the Wald’s SPRT. Derive OC and ASN functions for testing the mean of a normal distribution with unit variance.

problem 5: prepare brief notes on any two of the given:

a) Factorization theorem.
b) CAN estimators.
c) Monotone likelihood ratio and UMP tests.
d) Median test.
e) SPRT for testing Poisson parameter.