exact confidence interval requires detailed knowledge of the sampling

# Exact confidence interval requires detailed knowledge

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exact confidence interval requires detailed knowledge of the sampling distribution as well as some cleverness. An alternative method of constructing confidence intervals is based on the large sample theory of the previous section. According to the large sample theory result, the distribution of p nI ( θ 0 )( ˆ θ - θ 0 ) is approx- imately the standard normal distribution. Since the true value of θ , θ 0 , is unknown, we will use the estimated value ˆ θ to estimate I ( θ 0 ). It can be further argued that the distribu- tion of q nI ( ˆ θ )( ˆ θ - θ 0 ) is also approximately standard normal. Since the standard normal distribution is symmetric about 0, P - z (1 - α/ 2) q nI ( ˆ θ )( ˆ θ - θ 0 ) z (1 - α/ 2) 1 - α

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MAXIMUM LIKELIHOOD ESTIMATION and CRAM ´ ER-RAO INEQUALITY Fourth IIA - Penn State Astrostatistics School July 22-29, 2013. VBO, Kavalur Notes by Donald Richards,Penn State Univ. notes revised and lecture delivered by B. V. Rao;Chennai Math. Inst.
The ML Method for Testing Hypotheses Suppose that X N ( μ, σ 2 ), so that f ( x ; μ, σ 2 ) = 1 2 πσ 2 exp - ( x - μ ) 2 2 σ 2 . Using a random sample X 1 , . . . , X n , we wish to test H 0 : μ = 3 vs. H a : μ 6 = 3. Here the parameter space is the space of all permissible values of the parameters Ω = { ( μ, σ ) : -∞ < μ < , σ > 0 } . The hypotheses H 0 and H a represent restrictions on the parameters, so we are led to parameter subspaces ω 0 = { ( μ, σ ) : μ = 3 , σ > 0 } ; ω a = { ( μ, σ ) : μ 6 = 3 , σ > 0 } .

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L ( μ, σ 2 ) = f ( x 1 ; μ, σ 2 ) · · · f ( x n ; μ, σ 2 ) = 1 (2 πσ 2 ) n / 2 exp " - 1 2 σ 2 n X i =1 ( x i - μ ) 2 # Maximize L ( μ, σ 2 ) over ω 0 and ω a . The likelihood ratio test statistic (LRT) is λ = max ω 0 L ( μ, σ 2 ) max ω a L ( μ, σ 2 ) = max σ> 0 L (3 , σ 2 ) max μ 6 =3 ,σ> 0 L ( μ, σ 2 ) .

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