STA3702 Assignment 2 2024 - DUE 7 June 2024

STA3702 Assignment 2 2024 (Unique Nr. 199414) - DUE 7 June 2024 ;100 % TRUSTED workings, explanations and solutions. For assistance call or W.h.a.t.s.a.p.p us on ...(.+.2.5.4.7.7.9.5.4.0.1.3.2)........... Question 1 [44] The number of cars, X , passing through a certain intersection every week has Poisson distribution with mean λ. Let X1 , X2 , ..., Xn be independent variables representing the number of cars passing through the intersection onn randomly chosen weeks of the year. Furthermore, let S2 = 1 n − 1 nX i=1 Xi − X 2 S⋆2 = n − 1 n S2 and X = 1 n nX i=1 Xi be three competing estimators of λ. Given: V (S 2 ) = E [X i − λ] 4 n − λ 2 (n − 3) n(n − 1) . (a) Determine: (i) E X ; and (ii) prove or disprove thatE (S 2 ) = λ. (10) Hint: nX i=1 Xi − X 2 = nX i=1 X 2 i − nX 2 . (b) Prove or disprove thatS⋆2 is a method of moments estimator of λ. Hint: nX i=1 Xi − X 2 = nX i=1 X 2 i − nX 2 . (6) (c) Determine the bias of S⋆2 in estimating λ. (4) (d) Determine the variances of: (i) X ; and (ii) S⋆2. (8) (e) Determine the mean square errors of: (i) X ; (ii) S2 ; and (iii) S⋆2. (4) (f) Which of X , S2 and S⋆2 is the least accurate estimator of λ. Justify your answer. (3) (g) Prove or disprove thatX , S2 and S⋆2 are consistent estimators of λ. (9) Question 2 [42] The lifetime (X in years) of an electronic component manufactured by certain company has a distribution with probability density function: f (x|θ) =    1 θ exp − (x − µ) θ if θ > 0 and x > µ , 0 otherwise; where θ is unknown andµ is known. The company has hired you to estimateθ.Suppose that X1 , X2 , ..., Xn are the lifetimes ofn randomly chosen electronic components manufactured by the company. 2 STA3702/012/0/2024 Given: E(X) = θ − µ and V (X) = θ 2 . (a) What are the simplified likelihood and log-likelihood functions ofθ? (6) (b) Show that nX i=1 Xi is a sufficient statistic for θ. (4) (c) Show that nX i=1 Xi is a minimal sufficient statistic for θ. (5) (d) Prove or disprove that the maximum likelihood estimate ofθ is ˆθ = x − µ. (5) (e) What is the maximum likelihood estimate of θ 2 ? Justify your answer. (2) (f) Show that the method of moments estimate of θ 2 is ˜θ 2 = (x − µ) 2. (3) (g) Calculate the Fisher information about θ in the sample. (6) (h) What is the observed information about θ in the sample? (2) (i) Determine the Crammer-Rao lower bound for the variance of an unbiased estimator of θ. (2) (j) Prove or disprove that the maximum likelihood estimator of θ is also the minimum variance unbiased estimator of θ. (7) Question 3 [12] Refer to Question 2. Given: A distribution belongs to the regular 1-parameter exponential family if among other regularity conditions its probability density or mass function has the form: f (x|β) = g (x) exp {βt (x) − ψ (β)} , β ∈ Ω ⊂ (∞, ∞). Furthermore, for this distribution E[t(X)] = ψ ′ (β) and V ar[t(X)] = ψ ′′ (β). (a) Show that f (x|θ) belongs to the 1-parameter exponential family by expressing it in the form f (x|β) above. Do not forget to identify β, t(x) and ψ(β). (6) (b) What is the complete sufficient statistic for θ? Justify your answer. (2) (c) Find the mean and the variance of the complete sufficient statistic forθ. (4) Total: [98]