Let X1, . . . , Xn denote a random sample from a N (μ, σ2) distribution, where the mean μ and the variance σ2 are both unknown so that the param- eter vector is given by θ = (μ, σ2)T .ive the parametric function g(θ) that defines the quantile of order α of the normal distribution in terms of μ, σ, and the corresponding quantile zα of the standard normal distribution. (viii) Find the ML estimate of g(θ). (x) Derive the bias of g( ˆθ) as an estimator of g(θ) and use it to provide a bias-corrected estimator ̃g of g(θ). (xi) Derive the standard deviation of ̃g and the consequent standard error of ̃g. [NOTE: You may assume the distribution of ˆσ2, but you need to derive the expressions where necessary for the moments of ˆσ and ˆσ2.
Question
Let X1, . . . , Xn denote a random sample from a N (μ, σ2) distribution, where the mean μ and the variance σ2 are both unknown so that the param- eter vector is given by θ = (μ, σ2)T .ive the parametric function g(θ) that defines the quantile of order α of the normal distribution in terms of μ, σ, and the corresponding quantile zα of the standard normal distribution. (viii) Find the ML estimate of g(θ). (x) Derive the bias of g( ˆθ) as an estimator of g(θ) and use it to provide a bias-corrected estimator ̃g of g(θ). (xi) Derive the standard deviation of ̃g and the consequent standard error of ̃g. [NOTE: You may assume the distribution of ˆσ2, but you need to derive the expressions where necessary for the moments of ˆσ and ˆσ2.
Solution
This is a complex statistical problem that involves several steps. Here's how you can approach it:
(i) The quantile of order α of the normal distribution is given by the parametric function g(θ) = μ + σzα, where zα is the quantile of the standard normal distribution.
(ii) To find the maximum likelihood (ML) estimate of g(θ), we need to first find the likelihood function of the normal distribution. The likelihood function is given by L(μ, σ^2 | X) = Π (1/√(2πσ^2)) exp[-(Xi - μ)^2 / (2σ^2)].
(iii) Taking the log of the likelihood function and differentiating with respect to μ and σ^2, we can find the ML estimates of μ and σ^2. The ML estimate of μ (μ̂) is the sample mean (X̄) and the ML estimate of σ^2 (σ̂^2) is the sample variance (S^2).
(iv) Substituting these ML estimates into g(θ), we get the ML estimate of g(θ) as ĝ(θ) = X̄ + S*zα.
(v) The bias of an estimator is the difference between the expected value of the estimator and the true parameter value. To find the bias of ĝ(θ), we need to find E[ĝ(θ)] and subtract g(θ) from it.
(vi) Assuming the distribution of σ̂^2, we can derive the expressions for the moments of σ̂ and σ̂^2.
(vii) Using these expressions, we can find the expected value of ĝ(θ) and hence the bias. The bias-corrected estimator g̃ of g(θ) is then given by ĝ(θ) - bias.
(viii) To find the standard deviation of g̃, we need to find the variance of ĝ(θ) and take the square root.
(ix) The standard error of g̃ is then given by the standard deviation of g̃ divided by √n, where n is the sample size.
Please note that this is a general approach and the actual calculations may involve more detailed steps depending on the specific problem and the assumptions made.
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