List of top Statistics Questions on Estimation asked in GATE ST

Let \(X_1,X_2\) be a random sample from the following probability density function
\[ f_\alpha(x)= \begin{cases} \alpha x^{\alpha-1}e^{-x^\alpha} & \text{if } x>0 \\ 0 & \text{otherwise,} \end{cases} \]
where \(\alpha\in(0,\infty)\) is an unknown parameter. Let
\[ X_{(1)}=\min\{X_1,X_2\} \quad\text{and}\quad X_{(2)}=\max\{X_1,X_2\}. \]
Then which of the following statements is correct?

Let \(X_1\) and \(X_2\) be independent and identically distributed random variables following normal distribution with mean \(\theta\in(-\infty,\infty)\) and variance \(1\). Then which of the following estimators of their expected values attains the Cramer-Rao lower bound?

For a random sample \(X_1,\ldots,X_n\), \(n\geq2\), from a population with distribution function \(F_X\), let \(Z_n(x)\) be the proportion of sample values less than or equal to \(x\), \(x\in\mathbb{R}\). Which of the following statements is/are true?

Let \(x_1,x_2,\ldots,x_n\) (\(n\ge2\)) be the observed values of a random sample from the following probability density function
\[f(x)=\begin{cases}\dfrac{\lambda^{\alpha}}{\Gamma(\alpha)}x^{\alpha-1}e^{-\lambda x} & \text{if } x>0\\0 & \text{otherwise,}\end{cases}\]
where \(\alpha\in(0,\infty)\) and \(\lambda\in(0,\infty)\) are unknown parameters. If
\[\frac{x_1+x_2+\cdots+x_n}{n}=2 \quad\text{and}\quad \frac{x_1^2+x_2^2+\cdots+x_n^2}{n}=5,\]
then the method of moments estimate of \(\alpha\) equals ______ (answer in integer).

Let \(X_1,X_2,\ldots,X_{10}\) be a random sample from the following probability density function
\[f(x)=\begin{cases}2(x-\mu)e^{-(x-\mu)^2} & \text{if } x>\mu\\0 & \text{otherwise,}\end{cases}\]
where \(\mu\in(-\infty,\infty)\) is an unknown parameter. It is given that the observed value of \(\min\{X_1,X_2,\ldots,X_{10}\}\) is \(1\). Using the pivot \(\min\{X_1,X_2,\ldots,X_{10}\}-\mu\), suppose a 95% confidence interval of \(\mu\) is of the form \((c,1)\), then \(c\) equals ______ (rounded off to two decimal places).

Let \(X_1,X_2,X_3\) be a random sample from a distribution having probability mass function
\[ f_{\theta}(x)=\begin{cases}\theta & \text{if } x=1\\ 1-\theta & \text{if } x=2\\ 0, & \text{otherwise},\end{cases} \]
where \(\theta\in\Theta=(0,1)\). Let \(\underline{X}=(X_1,X_2,X_3)\). Then which of the following is NOT a sufficient statistic for \(\theta\)?

Let \(X_1, X_2, \ldots, X_n\) \((n \geq 2)\) be a random sample from the following probability density function
\[ f(x) = \frac{1}{2} e^{-|x-\mu|}, \quad -\infty < x < \infty, \] where \(\mu \in (-\infty, \infty)\) is an unknown parameter. Let \(\bar{X} = \frac{1}{n}\sum_{i=1}^{n} X_i\) and \(\hat{\mu}\) denote the maximum likelihood estimator of \(\mu\), whenever it exists. Then which of the following statements is/are correct?

Let \(X_1,X_2\) be a random sample from a distribution having the population density function
\[ f(x)=\begin{cases}\dfrac{1}{\theta}&\text{if }0<x<\theta\\0&\text{otherwise,}\end{cases} \]
where \(\theta\in(0,\infty)\). Let \(X_{(2)}=\max\{X_1,X_2\}\) and
\[ \psi(\theta)=P_\theta(X_1+X_2<1),\quad\theta>0. \]
Let \(\delta\big(X_{(2)}\big)\) be an unbiased estimator of \(\psi(\theta)\) that depends on observations \(X_1\) and \(X_2\) only through \(X_{(2)}\). If \(\delta(t)\) is a continuous function on \((0,\infty)\), then the value of \(18\,\delta\!\left(\dfrac{3}{4}\right)\) equals ______ (answer in integer).