Term Test October 3, 003 Name Math 56 Studet Number Directio: This test is worth 50 poits. You are required to complete this test withi 50 miutes. I order to receive full credit, aswer each problem completely ad must show all work. Please make sure that you have all the 7 pages. GOOD LUCK!
. 4 poits a 0 poits. Defie a ubiased estimator for a populatio parameter θ. Aswer: A estimator θ of θ is said to be a ubiased estimator of θ if ad oly if θ = θ. b 3 poits. Let X, X,..., X be a radom sample from a ormal distributio with mea µ ad variace. Show that the sample variace, S = Xi X, is a ubiased estimator of the populatio variace. Aswer: We wat to show that the estimator is a ubiased estimator of. Cosider [ S ] = S = [ S ] Xi X = [ χ ] sice S χ = =. Hece S is a ubiased estimator of.
. 4 poits a 0 poits. Defie a sufficiet estimator for a populatio parameter θ. Aswer: A estimator θ of the parameter θ is said to be a sufficiet estimator of θ if the coditioal distributio of the sample give the estimator θ does ot deped o the parameter θ. b 3 poits. Let X, X,..., X be a radom sample from a populatio X with desity fuctio θ for 0 x < +x fx; θ = θ+ 0 otherwise, where θ > 0 is a ukow parameter. What is a sufficiet statistic for the parameter θ? Aswer: The Pitma-Koopma Theorem says that if the probability desity fuctio ca be expressed i the form of fx; θ = e {KxAθ+Sx+Bθ} the KX i is a sufficiet statistic for θ. The give populatio desity ca be writte as Hece by Pitma-Koopma Theorem, is a sufficiet statistic for θ. θ fx; θ = + x [ θ+ = e l θ +x θ+ ] = e {l θ θ+ l+x}. KX i = l + X i = l + X i 3
3. 4 poits a 0 poits. Defie a efficiet estimator for a populatio parameter θ. Aswer: A ubiased estimator θ is called a efficiet estimator if it satisfies Cramér-Rao lower boud, that is V ar θ = [ ]. l Lθ θ b 3 poits. Let X, X,..., X be a radom sample from a ormal populatio with kow mea µ ad ukow variace > 0. The maximum likelihood estimator of is θ = X i µ. Is this maximum likelihood estimator a uiform miimum variace ubiased estimator UMVU of? Aswer: First, we show that this estimator is ubiased. Let θ =. Sice θ = X i µ = Xi µ = θ χ = θ = θ =, hece θ is a ubiased estimator of. The variace of θ ca be obtaied as follows: V ar θ = V ar X i µ = 4 V ar Xi µ = θ V ar χ = θ = θ = 4. Fially we determie the Cramér-Rao lower boud for the variace of θ. The secod derivative of l Lθ with respect to θ is Hece d l Lθ dθ Thus Therefore d l Lθ dθ = θ θ 3 x i µ. = θ θ 3 X i µ = θ θ θ 3 χ = d l Lθ V ar θ dθ = = θ = 4. d l Lθ dθ θ θ = θ. ad hece θ is a efficiet estimator of θ. Sice every efficiet estimator is a uiform miimum variace ubiased estimator, therefore X i µ is a uiform miimum variace ubiased estimator of. 4
4. 4 poits a 0 poits. Defie a pivotal quatity for a parameter θ. Aswer: A pivotal quatity Q is a fuctio of the sample X, X,..., X ad the parameter θ whose probability distributio is idepedet of the parameter θ. b 3 poits. Let X, X,..., X be a radom sample from a ormal populatio X with kow mea µ ad ukow variace. What the pivotal quatity for the parameter? Fid the distributio of this pivotal quatity. Usig this pivotal quatity, costruct a 00 α% cofidece iterval for the variace. Aswer: The pivotal quatity QX, X,..., X, for is give by QX, X,..., X, = Xi µ. Sice X i N µ,, Xi µ N0,, Xi µ χ. Xi µ χ. Hece Q has a chi-square distributio with degrees of freedom. Hece α a Q b Xi µ a b a X i µ b X i µ X i µ a b X i µ X i µ b a X i µ χ X i µ χ α α Therefore, the α% cofidece iterval for whe mea is kow is give by [ X i µ χ, α ] X i µ χ. α 5
5. 4 poits a 0 poits. Let X, X,..., X be a radom sample from a cotiuous populatio X with a distributio fuctio F x; θ. Suppose that F x; θ is mootoe i θ. What is the distributio of the radom variable Q = l F X i ; θ o proof required? Aswer: The distributio of the ststistics Q = l F X i ; θ is chi-squared with degrees of freedom. b 3 poits. If X, X,..., X is a radom sample from a populatio with desity θx θ if 0 < x < fx; θ = 0 otherwise, where θ > 0 is a ukow parameter, what is a 00 α% cofidece iterval for θ? Aswer: To costruct a cofidece iterval for θ, we eed a pivotal quatity. That is, we eed a radom variable which is a fuctio of the sample ad the parameter, ad whose probability distributio is kow but does ot ivolve θ. We use the radom variable Q = θ l X i χ as the pivotal quatity. The 00 α% cofidece iterval for θ ca be costructed from α χ α Q χ α χ α θ l X i χ α χ α θ l X i χ α. l X i Hece, 00 α% cofidece iterval for θ is give by χ α, l X i χ α l X i. 6
6. 4 poits If X, X,..., X is a radom sample from a populatio with desity θ x θ if 0 < x < fx; θ = 0 otherwise, where θ > 0 is a ukow parameter, what is a α00% approximate cofidece iterval for θ if the sample size is large? Aswer: The likelihood fuctio Lθ of the sample is Lθ = l Lθ = l θ + θ l x i. The first derivative of the logarithm of the likelihood fuctio is d dθ l Lθ = θ + l x i. Settig this derivative to zero ad solvig for θ, we obtai θ = likelihood estimator of θ is give by θ = l X. i θ xθ i. Hece l xi. Hece, the maximum Fidig the variace of this estimator is difficult. We compute its variace by computig the Cramér-Rao boud for this estimator. The secod derivative of the logarithm of the likelihood fuctio is give by Hece Therefore d dθ l Lθ = d dθ θ + l x i = θ. d l Lθ = dθ θ. θ V θ. Thus we take θ V θ. θ Sice V has θ i its expressio, we replace the ukow θ by its estimate θ so that V θ θ. The 00 α% approximate cofidece iterval for θ is give by [ ] θ θ θ z α, θ + z α, which is [ l X i + z α l X, i l X i z α ] l X. i 7