2023 Statistics Paper - II

The radius of a circle is measured with an error of measurement, which is N(0, σ2). Let x1, x2, ... xn be n measurements of the radius. Let \( \bar{x} = \frac{\sum_{i=1}^n x_i}{n} \) be the sample mean and \( S² = \frac{\sum_{i=1}^n (x_i - \bar{x})²}{n-1} \) be the sample variance. What is the unbiased estimate of the area of the circle?

Let 2·5, −2·0, 1·5, 3·5, 0·5 be the observations of a random sample of size 5 from the continuous distributions with pdf \( f(x) = 1/8 e^{-|x-2|} + 3/4√{2\pi} e^{-1/2(x-\theta)²} \); x, θ ∈ R and θ is unknown. Then the method of moment estimators of θ belongs to the interval:

Consider the following statements : Statement-I : The method of moments provides consistent estimators of population moments. Statement-II : From the Weak Law of Large Numbers, it follows that \( 1/n \sum_{i=1}^n x_i^j \xrightarrow{P} E(X_j) \) provided E(X_j) exists. Which one of the following is correct in respect of the above Statements ?

Let X be a random variable which assumes only two values 1 and 0, respectively representing occurrence of success or failure in an experiment with only two outcomes. The probability of getting success in the experiment is p(θ) defined as : p(θ) = \left\{ \begin{array}{ll} \theta, & if \theta is rational \\ 1-\theta, & if \theta is algebraic irrational \end{array} \right. Define an estimator, T = \frac{\sum_{i=1}^n x_i}{n} = Sample proportion of success. Which one of the following is correct ? (a) T is MLE as well as consistent for θ. (b) T is unbiased and consistent for θ. (c) T is MLE but not consistent for θ. (d) T is consistent but not MLE tending to θ.

Let {6, 11, 4, 13, 5} be a random sample from the exponential distribution f(x, θ) = e^{-(x-θ)}, x ≥ θ, θ ∈ (-∞, 3] is unknown. Then the Maximum Likelihood Estimate of e^θ is :

Let {1, 0, 0, 0, 1, 1} be a random sample from a binomial distribution b(1, θ), 0 < θ < 1. Then UMVUE of θ(1 + θ) is :