MCQs Estimation 1

1. If $E(\hat{\theta})=\theta$, then $\hat{\theta}$ is said to be

2. If $Var(\hat{\theta})\rightarrow 0$ as $n \rightarrow 0$, then $\hat{\theta}$ is said to be

3. Let $X_1,X_2,\cdots,X_n$ be a random sample from the density $f(x;\theta)$, where $\theta$ may be vector. If the conditional distribution of $X_1,X_2,\cdots,X_n$ given $S=s$ does not depend on $\theta$ for any value of $s$ of $S$, then statistic is called.

4. If $X_1,X_2,\cdots, X_n$ is the joint density of n random variables, say, $f(X_1, X_2,\cdots, X_n;\theta)$ which is considered to be a function of $\theta$. Then $L(\theta; X_1,X_2,\cdots, X_n)$ is called

5. If the conditional distribution of $X_1, X_2,\cdots,X_n$ given $S=s$, does not depend on $\theta$, for any value of $S=s$, the statistics $S=s(X_1,X_2,\cdots,X_n)$ is called

6. Let $L(\theta;X_1,X_2,\cdots,X_n)$ be the likelihood function for a sample $X_1,X_2,\cdots, X_n$ having joint density $f(x_1,x_2,\cdots,x_n;\theta)$ where ? belong to parameter space. Then a test defined as $\lambda=\lambda_n=\lambda(x_1,x_2,\cdots,x_n)=\frac{Sup_{\theta\varepsilon \Theta_0}L(\theta;x_1,x_2,\cdots,x_n)}{Sup_{\theta\varepsilon \Theta}L(\theta;x_1,x_2,\cdots,x_n)}$

7. A test is said to be the most powerful test of size $\alpha$, if

8. $Var_\theta (T) \geq \frac{[\tau'(\theta)]^2}{nE[{\frac{\partial}{\partial \theta}log f((X;\theta)}^2]}$, where $T=t(X_1,X_2,\cdots, X_n)$ is an unbiased estimator of $\tau(\theta)$. Then above inequality is called

9. Which of the following assumptions are required to show the consistency, unbiasedness, and efficiency of the OLS estimator?

1. $E(\mu_t)=0$
2. $Var(\mu_t)=\sigma^2$
3. $Cov(\mu_t,\mu_{t-j})=0;t\neq t-j$
4. $\mu_t \sim N(0,\sigma^2)$

10. For two estimators $T_1=t_1(X_1,X_2,\cdots,X_n)$ and $T_2=t_2(X_1,X_2,\cdots,X_n)$ then estimator $t_1$ is defined to be $R_{{t_1}(\theta)}\leq R_{{t_2}(\theta)}$ for all $\theta$ in $\Theta$

11. A set of jointly sufficient statistics is defined to be minimal sufficient if and only if

12. In statistical inference, the best asymptotically normal estimator is denoted by

13. Let $X_1,X_2,\cdots,X_n$ be a random sample from a density $f(x|\theta)$, where $\theta$ is a value of the random variable $\Theta$ with known density $g_\Theta(\theta)$. Then the estimator $\tau(\theta)$ with respect to the prior $g_\Theta(\theta)$ is defined as $E[\tau(\theta)|X_1,X_2,\cdots,X_n]$ is called

14. If $Var(T_2) < Var(T_1)$, then $T_2$ is

15. If $f(x_1,x_2,\cdots,x_n;\theta)=g(\hat{\theta};\theta)h(x_1,x_2,\cdots,x_n)$, then $\hat{\theta}$ is

16. For a biased estimator $\hat{\theta}$ of $\theta$, which one is correct

17. Let $Z_1,Z_2,\cdots,Z_n$ be independently and identically distributed random variables, satisfying $E[|Z_t|]<\infty$. Let N be an integer-valued random variable whose value n depends only on the values of the first n $Z_i$s. Suppose $E(N)<\infty$, then $E(Z_1+Z_2+\cdots+Z_n)=E( N)E(Z_i)$ is called