Derive MLEs and conditional Normal distributions
Overview
This question evaluates proficiency in parametric inference and probability theory, specifically maximum likelihood estimation, bias of estimators, and conditioning in univariate and bivariate Normal distributions including constrained estimation of parameters.
Normal and Bivariate Normal: PDFs/CDFs, MLEs, Conditioning, and Unbiased Variance
Setup
- Let X1, …, Xn be i.i.d. Normal(μ, σ²).
- Independently, let (X, Y) be a single draw from a bivariate Normal with means μX, μY, variances σX², σY², and correlation ρ.
Tasks
(a) Write the pdf and cdf of Normal(μ, σ²). Using them, express P(μ − σ ≤ X1 ≤ μ + σ) in terms of the standard Normal cdf Φ.
(b) Derive the MLEs: (i) μ when σ² is known; (ii) μ and σ² when both are unknown; (iii) the MLE of μ when μ is constrained so that μ ≥ 0 (give the closed-form and explain what happens when the unconstrained MLE is negative).
(c) For the bivariate Normal (X, Y), derive the conditional distribution of X | Y = y: give its mean and variance, write the conditional pdf and cdf, and compute P(X > t | Y = y) in terms of Φ.
(d) Is the usual MLE of σ² unbiased? If not, provide an unbiased estimator for σ² and relate it to the MLE.
Solution
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