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GATE 2025 ST – Question 22

Non-parametric Statistics · Empirical Distribution and Goodness of Fit · 1 mark · Multiple choice

For iid X_n with CDF F, let F_n be the empirical CDF. At a fixed real x, which holds?

  1. √n(F_n(x)−F(x))→0 in probability
  2. n(F_n(x)−F(x))/√[F(x)(1−F(x))]⇒N(0,1)
  3. F_n(x)→F(x) almost surely
  4. lim n Var(F_n(x))=0

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Correct answer: (C) F_n(x)→F(x) almost surely

Explanation

F_n(x) is the average of iid indicators 1_(X_i≤x), so the strong law gives almost-sure convergence to F(x), proving **C**. For 0<F(x)<1, the CLT has √n scaling and nonzero limiting variance F(x)(1−F(x)). Also nVar(F_n)=F(x)(1−F(x)), generally nonzero. Endpoint CDF values do not invalidate C.

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