[3:58 p. m., 10/7/2021] Ejh: As on the previous page, let X1,…,Xn be i.i.d. with pdf
F_θ(x)= θx^θ-1*1(0<x<1)
Assume we do not actually get to observe X1,…,Xn. Instead let Y1,…,Yn be our observations where Yi=1(Xi≤0.5). Our goal is to estimate θ based on this new data.
What distribution does Yi follow?
First, choose the type of the distribution:
• Bernoulli
• Poisson
• Normal
• Exponential
Second, enter the parameter of this distribution in terms of θ. Denote this parameter by mθ. (If the distribution is normal, enter only 1 parameter, the mean).
mθ=
[4:03 p. m., 10/7/2021] Ejh: 3 a. As on the previous page, let X1,…,Xn be i.i.d. with pdf
F_θ(x)= θx^θ-1*1(0<x<1)
Assume we do not actually get to observe X1,…,Xn. Instead let Y1,…,Yn be our observations where Yi=1(Xi≤0.5). Our goal is to estimate θ based on this new data.
What distribution does Yi follow?
First, choose the type of the distribution:
• Bernoulli
• Poisson
• Normal
• Exponential
Second, enter the parameter of this distribution in terms of θ. Denote this parameter by mθ. (If the distribution is normal, enter only 1 parameter, the mean).
mθ=
b. Write down a statistical model associated to this experiment. Is the parameter θ identifiable?
Yes
No
c. Compute the Fisher information I(θ).
(To answer this question correctly, your answer to part (a) needs to be correct.)
I(θ)=
d. Compute the maximum likelihood estimator ˆθ for θ in terms of ¯Yn.
(Enter barY_n for ¯Yn.)
ˆθ=
e. Compute the method of moments estimator ˜θ for θ.
(Enter barY_n for ¯Yn.)
˜θ=
f. What is the asymptotic variance V(˜θ) of the method of moments estimator ˜θ?
V(˜θ)=
g. Give a formula for the p-value for the test of
H0:θ≤1vs.H1:θ>1
based on the asymptotic distribution of ˆθ.
To avoid double jeopardy, you may use V for the asymptotic variance V(θ0), I for the Fisher information I(θ0), hattheta for ˆθ, or enter your answer directy without using V or I or hattheta.
(Enter barY_n for ¯Yn, hattheta for ˆθ. If applicable, enter Phi(z) for the cdf Φ(z) of a normal variable Z, q(alpha) for the quantile qα for any numerical value α. Recall the convention in this course that P(Z≤qα)=1−α for Z∼N(0,1).)
p-value:
Assume n=50, and ¯Yn=0.46. Will you reject the null hypothesis at level α=5%?
Yes, reject the null hypothesis at level α=5%.
No, cannot reject the null hypothesis at level α=5%.
To answer these questions, we need to understand the distribution followed by Yi, the observations obtained from Xi by comparing them to 0.5.
a. The distribution followed by Yi is a Bernoulli distribution.
Explanation: In this scenario, Yi takes on the value 1 if Xi ≤ 0.5 and 0 otherwise. Since Yi can only take two possible values (1 or 0), it follows a Bernoulli distribution.
b. The parameter of this Bernoulli distribution, denoted as mθ, is the probability of Yi being 1 (success) given the parameter θ.
Explanation: In a Bernoulli distribution, the parameter represents the probability of success. In this case, Yi takes on the value 1 when Xi ≤ 0.5, so mθ represents the probability of Xi ≤ 0.5 given the parameter θ.
c. To compute the Fisher information I(θ), we need to differentiate the log-likelihood function with respect to θ and take the expected value.
Explanation: The Fisher information measures the amount of information that the data provides about the parameter. It is computed by taking the expected value of the second derivative of the log-likelihood function with respect to the parameter.
d. The maximum likelihood estimator ˆθ for θ can be obtained by maximizing the log-likelihood function with respect to θ.
Explanation: The maximum likelihood estimator is obtained by finding the value of θ that maximizes the likelihood function, which is the probability of observing the given data.
e. The method of moments estimator ˜θ for θ can be obtained by setting the sample mean ¯Yn equal to the population mean.
Explanation: The method of moments estimator is obtained by equating the sample moments (in this case, the sample mean) to their corresponding population moments (in this case, the population mean).
f. The asymptotic variance V(˜θ) of the method of moments estimator ˜θ can be computed using the formula for the asymptotic variance.
Explanation: The asymptotic variance measures the variability of the estimator as the sample size increases indefinitely. It can be computed using certain theoretical properties and formulas.
g. The p-value for the test of H0: θ ≤ 1 vs. H1: θ > 1 can be calculated based on the asymptotic distribution of the maximum likelihood estimator ˆθ.
Explanation: The p-value is a measure of the evidence against the null hypothesis. It is calculated based on the estimated distribution of the test statistic, which in this case is the maximum likelihood estimator.
Given the provided information about n = 50 and ¯Yn = 0.46, we can use the estimators and formulas described above to determine if we will reject the null hypothesis at a significance level of α = 5%.
a. The distribution that Yi follows is Bernoulli.
b. The parameter of this distribution, denoted by mθ, is the probability of success in a Bernoulli trial. In this case, mθ represents the probability that Xi ≤ 0.5, which is equal to the cumulative distribution function Fθ(0.5).
c. To compute the Fisher information I(θ), we need the probability mass function (pmf) of Yi, which is given by the Bernoulli distribution. The Fisher information is calculated as the expected value of the square of the derivative of the log-likelihood function with respect to θ, which can be expressed as:
I(θ) = n * mθ * (1 - mθ)
d. The maximum likelihood estimator (MLE) for θ in terms of ¯Yn is given by:
ˆθ = -log(1 - ¯Yn)
e. The method of moments estimator (MME) for θ is equal to the sample mean ¯Yn.
˜θ = ¯Yn
f. The asymptotic variance V(˜θ) of the method of moments estimator ˜θ is given by:
V(˜θ) = 1 / I(θ)
g. The formula for the p-value for the test of H0:θ≤1 vs. H1:θ>1 based on the asymptotic distribution of ˆθ can be calculated using the standard normal distribution. The p-value is given by:
p-value = 1 - Φ((ˆθ - θ0) / sqrt(V(ˆθ)))
where Φ(z) represents the cumulative distribution function of a standard normal variable and θ0 is the specific value being tested against.
Given n=50 and ¯Yn=0.46, you can use these values to calculate the p-value and determine whether to reject the null hypothesis at level α=5%.