List of top Estimation Questions

Let $0,1,1,2,0$ be five observations of a random variable $X$ which follows a Poisson distribution with parameter $\theta>0$. Let the minimum variance unbiased estimate of $P(X\le 1)$, based on this data, be $\alpha$. Then $5^{4}\alpha$ (in integer) is equal to $\;__________$

A random sample of size $5$ is taken from the distribution with density \[ f(x;\theta)= \begin{cases} \dfrac{3x^2}{\theta^3}, & 0[6pt] 0, & \text{elsewhere}, \end{cases} \] where $\theta$ is unknown. If the observations are $3,6,4,7,5$, then the maximum likelihood estimate of the $1/8$ quantile of the distribution (rounded off to one decimal place) is __________.

Consider a gamma distribution with pdf \[ f(x;\beta)= \begin{cases} \dfrac{1}{24\beta^{5}}x^{4}e^{-x/\beta}, & x>0,\\ 0, & \text{elsewhere}, \end{cases} \] with $\beta>0$. For $\beta=2$, find the Cramér–Rao lower bound (rounded to one decimal place) for the variance of any unbiased estimator of $\beta^{2}$ based on a random sample of size 8.
A random sample of size 4 is taken from the distribution with the probability density function \[ f(x; \theta) = \begin{cases} \frac{2(\theta - x)}{\theta^2}, & 0<x<\theta, \\ 0, & \text{elsewhere}. \end{cases} \] If the observed sample values are 6, 5, 3, 6, then the method of moments estimate (in integer) of the parameter \( \theta \), based on these observations, is ________
Let \( X_1, X_2, \dots, X_{18} \) be a random sample from the distribution \[ f(x; \theta) = \begin{cases} \frac{2x}{\theta} e^{-x^2/\theta}, & x>0,\\ 0, & x \leq 0. \end{cases} \] What is the maximum likelihood estimator (MLE) of \( \theta \)?
Let \( X_1, X_2, \dots, X_n \) be a random sample from a population \( f(x; \theta) \), where \( \theta \) is a parameter. Then which one of the following statements is NOT true?
Let \( S^2 \) be the variance of a random sample of size \( n>1 \) from a normal population with an unknown mean \( \mu \) and an unknown finite variance \( \sigma^2>0 \). Consider the following statements:
(I) \( S^2 \) is an unbiased estimator of \( \sigma^2 \), and \( S \) is an unbiased estimator of \( \sigma \).
(II) \( \frac{n-1}{n} S^2 \) is a maximum likelihood estimator of \( \sigma^2 \), and \( \sqrt{\frac{n-1}{n}} S \) is a maximum likelihood estimator of \( \sigma \).
Which of the above statements is/are true?
Let \( \{0.90, 0.50, 0.01, 0.95\} \) be a realization of a random sample of size 4 from the probability density function

\[ f(x) = \begin{cases} \frac{\theta}{(1-\theta)} x^{(2\theta-1)/(1-\theta)}, & 0<x<1 \\ 0, & \text{otherwise} \end{cases} \]

where \( 0.5 \leq \theta<1 \). Then the maximum likelihood estimate of \( \theta \) based on the observed sample equals

\[ \underline{\hspace{2cm}} \]

(round off to 2 decimal places).

Let \( Y_i = \alpha + \beta x_i + \epsilon_i \), where \( x_i \)'s are fixed covariates, \( \alpha \) and \( \beta \) are unknown parameters, and \( \epsilon_i \)'s are independent and identically distributed random variables with mean zero and finite variance. Let \( \hat{\alpha} \) and \( \hat{\beta} \) be the ordinary least squares estimators of \( \alpha \) and \( \beta \), respectively. Given the following observations: 

The value of \( \hat{\alpha} + \hat{\beta} \) equals _________ (round off to 2 decimal places).