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Questions › GATE › GATE 2024 › Q38
1 markStatistics › Point and interval estimation › Maximum likelihood estimationHardabout 2 min

GATE 2024 Question 38: Point and interval estimation

Let x1,x2,⋯ ,xnx_1,x_2,\cdots,x_nx1​,x2​,⋯,xn​ be an independently, and identically distributed (iid) random sample drawn from a population that follows the Normal Distribution N(μ,σ2)N(\mu,\sigma^2)N(μ,σ2), where both the mean (μ\muμ) and variance (σ2\sigma^2σ2) are unknown. Let xˉ\bar xxˉ be the sample mean. The maximum likelihood estimator (MLE) of the variance (σ^MLE2\hat\sigma^2_{MLE}σ^MLE2​) is/are then characterized by

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Let x₁,x₂,⋯,xₙ be an independently, and identically distributed (*iid*) random sample drawn from a population that follows the Normal Distribution N(μ,σ²), where both the mean (μ) and variance (σ²) are unknown. Let x̄ be the sample mean. The maximum likelihood estimator (MLE) of the variance (σ̂²_(MLE)) is/are then characterized by (A) σ̂²_(MLE) = 1/nΣᵢ₌₁ⁿ(xᵢ-x̄)² which is a biased estimator of σ² (B) σ̂²_(MLE) = 1/nΣᵢ₌₁ⁿ(xᵢ²-x̄)² which is a consistent estimator of σ² (C) σ̂²_(MLE) = 1/(n-1)Σᵢ₌₁ⁿ(xᵢ-x̄)² which is an unbiased estimator of σ² (D) σ̂²_(MLE) = 1/(n-1)Σᵢ₌₁ⁿ⁻¹(xᵢ-x̄)² which is an unbiased and consistent estimator of σ²

Also written as: Let x_1,x_2,...,x_n be an independently, and identically distributed (*iid*) random sample drawn from a population that follows the Normal Distribution N(mu,sigma^2), where both the mean (mu) and variance (sigma^2) are unknown. Let x_bar be the sample mean. The maximum likelihood estimator (MLE) of the variance (sigma_hat^2_(MLE)) is/are then characterized by

Point and interval estimation has come up 28 times in 6 exams since 2008.

What this question tests

This question is on Maximum likelihood estimation, part of Point and interval estimation in Statistics.

Choosing the parameter value that makes the data most likely.

TestStop has 5 past-paper questions on this subtopic; the nearest in year are below.

  • IIT JAM 2021 · Q16Let X be uniform on [a, b]. A sample of size five is {1.3, 0.8, 9.5, 20.2, 8.2}. The Maximum Likelihood…
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