MLE for θ in N(θ, θ) | Step-by-Step Visual Guide

Maximum Likelihood Estimation (MLE) for θ in N(θ, θ)

We consider a sample X₁, …, Xₙ from a normal distribution with mean and variance both equal to θ (Xᵢ ∼ N(θ, θ)), where θ > 0. This post shows a step-by-step visual derivation of the MLE and its approximate variance.

If you’re a graduate student trying to finish homework or a project and the method isn’t clicking, you’re not alone. Online tutoring can help you connect definitions, steps, and interpretation so the workflow becomes repeatable.

This article provides a graduate-level explanation that prioritizes clarity and correctness. It focuses on definitions, assumptions, and a step-by-step approach that you can apply to real problems. The goal is a self-contained discussion that supports coursework, exams, and applied projects.

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Step 1: Likelihood Function

The PDF of each Xᵢ is:

f(xᵢ; θ) = 1 / √(2πθ) · exp(-(xᵢ - θ)² / (2θ))

The likelihood function is:

L(θ) = ∏ f(xᵢ; θ) = (1 / √(2πθ))ⁿ · exp(- (1/(2θ)) ∑ (xᵢ - θ)²)

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Step 2: Log-Likelihood

Take the log of the likelihood:

ℓ(θ) = -n/2 · log(2πθ) - (1/(2θ)) ∑ (xᵢ - θ)²

Expand the squared term:

(xᵢ - θ)² = xᵢ² - 2θxᵢ + θ²

Sum over i:

∑(xᵢ - θ)² = ∑xᵢ² - 2θ∑xᵢ + nθ²

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Step 3: Solve for MLE

Substitute back into log-likelihood:

ℓ(θ) = -n/2 · log(2πθ) - (1/(2θ))(∑xᵢ² - 2θ∑xᵢ + nθ²)

Differentiating and setting to zero gives a quadratic equation:

θ² + θ - W = 0, where W = (1/n) ∑ xᵢ²

Solution of the quadratic:

θ = (-1 ± √(1 + 4W)) / 2

Since θ > 0, the MLE is:

θ̂ = (-1 + √(1 + 4W)) / 2

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Step 4: Approximate Variance of θ̂

The approximate variance of the MLE uses the observed Fisher information:

Var(θ̂) ≈ [ - d²ℓ(θ) / dθ² ]⁻¹ evaluated at θ = θ̂

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Many students first encounter this concept in graduate coursework or applied assignments. Additional explanation can help connect theory to practice. Online tutoring support is available for graduate-level quantitative topics.

Related Topics

• Learn more about Statistics and Probability concepts for combinations and estimation.
• Check out our Stata, R, and SPSS tutorials for implementing MLE in software.
• Explore Research Troubleshooting for step-by-step guidance on statistical estimation.
• See more examples on our Statistics Blog Hub.

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