Consider an i.i.d. sample X1,…,Xn∼Poiss(λ) for λ>0 .
Starting from the Central Limit Theorem, find a confidence interval I=[A,B] with asymptotic level 1−α that is centered about Xn using the plug-in method.
now consider the following hypothesis with a fixed number λ0>0 :
H0:λ=λ0vsH1:λ≠λ0.
Define a test for the above hypotheses with asymptotic level α , and rewrite it in the form
ψ=1{λ0∉J},
for some interval J=[C,D] .
( Write barX_n for Xn . If applicable, type abs(x) for |x| , Phi(x) for Φ(x)=P(Z≤x) where Z∼N(0,1) , and q(alpha) for qα , the 1−α quantile of a standard normal variable. )
C=?
D=?
To find a confidence interval using the plug-in method, we can start from the Central Limit Theorem. The Central Limit Theorem states that for a large enough sample size, the distribution of the sample mean approaches a normal distribution.
Let's define Y_n as the sample mean, which is equal to X_n/n. Using the Central Limit Theorem, we know that Y_n follows a normal distribution with mean λ and variance λ/n.
To construct a confidence interval, we need to calculate the standard error of Y_n. The standard error is the standard deviation of the sample mean, which can be calculated as the square root of the variance. So the standard error is sqrt(λ/n).
Since we are using the plug-in method, we replace λ with its estimate, which is the sample mean X_n. So the standard error is sqrt(X_n/n).
Next, we need to find the appropriate z-value for our desired confidence level. Let's denote this value as Z_alpha/2, where alpha is the desired significance level and Z_alpha/2 is the 1-alpha/2 quantile of the standard normal distribution.
Now, we can construct the confidence interval by using the formula:
I = [X_n - Z_alpha/2 * sqrt(X_n/n), X_n + Z_alpha/2 * sqrt(X_n/n)]
So the confidence interval is given by I = [X_n - Z_alpha/2 * sqrt(X_n/n), X_n + Z_alpha/2 * sqrt(X_n/n)].
Now, let's move on to the hypothesis testing.
To test the hypothesis H0: λ = λ0 vs H1: λ ≠ λ0, we can use the likelihood ratio test. The likelihood ratio test compares the likelihood of the data under the null hypothesis to the likelihood under the alternative hypothesis.
Under the null hypothesis H0: λ = λ0, the likelihood of the data is given by L(λ0) = (e^(-λ0) * λ0^X_n)/X_n!
Under the alternative hypothesis H1: λ ≠ λ0, the likelihood of the data is given by L(λ) = (e^(-λ) * λ^X_n)/X_n!
To find the test statistic, we take the ratio of the likelihoods:
T = L(λ0)/L(λ) = (e^(-λ0) * λ0^X_n)/(e^(-λ) * λ^X_n)
Taking the logarithm of the test statistic simplifies the calculation:
log(T) = -X_n * (λ0 - λ) + X_n * log(λ0/λ)
To decide whether to reject or not reject the null hypothesis H0, we compare log(T) to a threshold value c. If log(T) <= c, we reject H0. If log(T) > c, we fail to reject H0.
We need to find the values C and D that define the interval J = [C, D] such that ψ = 1{λ0 ∉ J}.
Since we compare the test statistic log(T) to a threshold value c, we have:
ψ = 1{λ0 ∉ J} = 1{log(T) > c}
Therefore, the test can be rewritten as:
ψ = 1{log(T) > c}
So, C is negative infinity, and D is equal to c.