1a. All maximum likelihood estimators are asymptotically normal.
True
False
b. )
Let X1…Xn be i.i.d. Bernoulli random variables with some unknown parameter . Then which of the following is/are valid confidence interval(s) for with nonasymptotic confidence level 95%?
(Choose all that apply.)
• [[Xn]-1.96(p(1-p)/n)^1/2/n , Xn+1.96(p(1-p)/n)]
•(0,1)
•[0,Xn+.83/n]
•None of above
c.) Which of the following is/are valid statistical model(s)?
•(R,{N(θ,1) θ>10)
•(θ,inf) {t->e^θ-t*1(t>θ)}
•(0,inf), {N(σ, μ)
•(0,inf), {x->e^θx*1(x>0) θ>0}
1. True
2. 1
3. a,b
a) False. Not all maximum likelihood estimators are asymptotically normal. It depends on the specific case and conditions of the estimator.
b) The valid confidence interval(s) for p with nonasymptotic confidence level 95% are:
• [[Xn - 1.96(p(1-p)/n)^(1/2)/n, Xn + 1.96(p(1-p)/n)^(1/2)/n]
• [0, 1]
c) The valid statistical model(s) are:
• (0, inf), {N(σ, μ)}
• (0, inf), {x -> e^(θx)*1(x > 0), θ > 0}
a. False. Not all maximum likelihood estimators are asymptotically normal. The asymptotic normality of an estimator depends on several conditions, such as the regularity of the likelihood function and the asymptotic properties of the estimator itself. While many maximum likelihood estimators are asymptotically normal under certain conditions, it is not a universal property.
b. To determine the valid confidence intervals for the unknown parameter in the given scenario, we need to consider the properties of the Bernoulli distribution and the concept of confidence intervals.
A confidence interval is a range of values within which we can be confident that the true parameter lies with a certain level of confidence. In this case, we want nonasymptotic confidence level 95%, meaning we want to construct a confidence interval that holds true for any sample size.
Here are the valid confidence intervals:
• [[Xn] - 1.96(p(1-p)/n)^1/2/n, Xn + 1.96(p(1-p)/n)]:
This interval is valid because it is based on the Central Limit Theorem, which guarantees that the sample mean (Xn) approximately follows a normal distribution for large sample sizes. The formula takes into account the sample proportion (p) and the sample size (n) to construct the confidence interval.
• [0, Xn + 0.83/n]:
This interval is valid because it is a conservative interval based on the ffding's inequality. It guarantees that the true parameter lies within the interval with a nonasymptotic confidence level of at least 95%.
Therefore, the valid confidence intervals for the unknown parameter with nonasymptotic confidence level 95% are:
[[Xn] - 1.96(p(1-p)/n)^1/2/n, Xn + 1.96(p(1-p)/n)] and [0, Xn + 0.83/n].
c. To determine the valid statistical models, we need to consider the properties and requirements of each model:
• (R, {N(θ,1) θ>10}):
This is a valid statistical model because it represents a random variable (R) that follows a normal distribution with mean (θ) greater than 10. The restriction on θ being greater than 10 defines the parameter space for this model.
• (θ, inf) {t->e^θ-t*1(t>θ)}:
This is a valid statistical model because it represents a random variable (t) that follows an exponential distribution with parameter (θ). The parameter space for this model is θ being any positive real number.
• (0, inf), {N(σ, μ)}:
This is a valid statistical model because it represents a random variable following a normal distribution with parameters (σ) and (μ). The restriction on the parameter space is σ being positive and μ being any real number.
• (0, inf), {x->e^θx*1(x>0) θ>0}:
This is a valid statistical model because it represents a random variable (x) that follows an exponential distribution with parameter (θ). The parameter space for this model is θ being any positive real number.
Therefore, the valid statistical models are:
(R, {N(θ,1) θ>10}),
(θ, inf) {t->e^θ-t*1(t>θ)},
(0, inf), {N(σ, μ)},
(0, inf), {x->e^θx*1(x>0) θ>0}.