IES ISS 2025 Paper-II Statistics Question Paper (21-June-2025)

Question 1

Consider the following statements :I. MLEs are unbiased but not necessarily unique.II. Unbiased estimator is always unique.III. If U and W are consistent estimators of θ₁ and θ₂, then UW is also consistent for θ₁θ₂.Which of the statements given above is/are correct?

Accepted Answer

III only

Answer

I and II

Answer

I and III

Answer

II and III

Question 2

With the increase in sample size if the estimator becomes closer and closer to the parameter, then it is called :

Accepted Answer

Consistent

Answer

Completeness

Answer

Sufficient

Answer

Unbiasedness

Question 3

Let X₁, X₂, X₃, ... , Xₙ ~ N(μ, σ²) where mean μ and variance σ² are unknown. Let s² = (1/n)Σ(Xᵢ – X̄)² and S² = (1/(n-1))Σ(Xᵢ – X̄)².
Which one of the following is correct according to the efficiency criterion ?

Accepted Answer

s² is more efficient than S².

Answer

s² is less efficient than S².

Answer

s² and S² cannot be compared.

Answer

Efficiency of s² is equal to efficiency of S².

Question 4

Suppose X₁, X₂, X₃, ... , Xₙ be a random sample of size n from Poisson distribution with parameter λ. Then uniformly minimum variance unbiased estimator (UMVUE) of λ is:

Accepted Answer

(Σᵢ₌₁ⁿ Xᵢ)/n

Answer

(Σᵢ₌₁ⁿ Xᵢ²)/n

Answer

(Σᵢ₌₁ⁿ Xᵢ)/(n+1)

Answer

(Σᵢ₌₁ⁿ Xᵢ²)/(n+1)

Question 5

Let S be the set of all unbiased estimators T of θ ∈ Θ such that E₀T²<∞ for all θ ∈ Θ. An estimator T₀ ∈ S is called a uniformly minimum variance unbiased estimator (UMVUE) of θ for T ∈ S if :

Accepted Answer

E₀(T₀ – θ)² ≤ E₀(T – θ)²

Answer

E₀(T₀ - θ) ≤ E₀(T – θ)

Answer

E₀(T₀ – θ)² > E₀(T – θ)²

Answer

E₀(T₀ – θ) > E₀(T – θ)