True or false. with explaination. 1. In case of poisson distribution with parame

Question

True or false. with explaination.
1. In case of poisson distribution with parameter lemda, x bar is sufficient for lemda.

2. If (X1, X2, ..., Xn) be a sample of independent observations from the uniform distribution on (theta, theta+1) then the maximum likelihood estimator of theta is unique.

3. unbiased estimator is necessarily consistent.

4. A consistent estimator is also unbiased.

5. An unbiased estimator whose variance tends to 0 as sample size increases is consistent.

6. If t is a sufficient statistic for theta, then f(t) is a sufficient statistic for f(theta).

7. If t1 and t2 are two independent estimators of theta, then (t1+t2) is less efficient then both t1 and t2.

8. If T is consistent estimator of parameter theta, then aT+b is a consistent estimator of (a*theta+b), where a and b are constants.

9. If X is the num of success in n independent trials with a consistent probability p of success in each trial, then x/n is a consistent estimator of p.

10. The power function of a test is always important when evaluating a hypothesis test.

11. If we take negative of expected values of Hessian matrix, it will provide us Fisher Information matrix.

12. In 1000 tosses of a coin, 560 heads and 440 tails appear and it is tested that coin is fair.

13. What is the appropriate test statistics for testing several correlation coefficient?

Explanation / Answer

1. In case of poisson distribution with parameter lemda, x bar is sufficient for lemda. - yes its true for finding equation it should be there

2. If (X1, X2, ..., Xn) be a sample of independent observations from the uniform distribution on (theta, theta+1) then the maximum likelihood estimator of theta is unique. 0-- because estimation of unbiased variance tends to 0 so its true

3. unbiased estimator is necessarily consistent. because - An estimator can be unbiased but not consistent. so its false

4) A consistent estimator is also unbiased. because An estimator can be unbiased but not consistent. so its true

5) . An unbiased estimator whose variance tends to 0 as sample size increases is consistent.--becauseif an estimator is unbiased and the variance tends to 0, the estimator is consistent. so its true

6. If t is a sufficient statistic for theta, then f(t) is a sufficient statistic for f(theta). -- obviously its true because to caluclate f(t) we should use f(teta)

7. If t1 and t2 are two independent estimators of theta, then (t1+t2) is less efficient then both t1 and t2.

its false because it will be more efficent not less efficient

8) If T is consistent estimator of parameter theta, then aT+b is a consistent estimator of (a*theta+b), where a and b are constants. - yes its true beacuse its basic representation of that equation

9) If X is the num of success in n independent trials with a consistent probability p of success in each trial, then x/n is a consistent estimator of p. -- yes its true

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