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11. Chi-Squared Test for a Family of Discrete Distributions A Bookmark this page In the problems...
Degrees of Freedom of a Known Test 2 points possible graded) Let us consider a statistical...
Degrees of Freedom of a Known Test 2 points possible graded) Let us consider a statistical model with parameter ER". Let O be the parameter that generates the n lid samples X1,..., X, Let I ) be the Fisher information and assume that the MLE is asymptotically normal. Assume that I(C) is a diagonal matrix with positive entries 1/t1,...,1/td. We wish to perform a test for the hypotheses H : 8 - and H:8 + . Let the test statistic...
5. A confidence interval for Poisson variables Bookmark this page (a) 2 points possible (graded) Let...
5. A confidence interval for Poisson variables Bookmark this page (a) 2 points possible (graded) Let X1,..., Xn bei.i.d. Poisson random variables with parameter 1 > 0 and denote by Xn their empirical average, Xn=1 İx; and (bn), such that an (Xn-bn) converges in distribution to a standard Gaussian random variable Find two sequences (an) Z~ N(0,1). an =
Relating M-estimation and Maximum Likelihood Estimation 1 point possible (graded) Let (E,{Pθ}θ∈Θ) denote a discrete statistical...
Relating M-estimation and Maximum Likelihood Estimation 1 point possible (graded) Let (E,{Pθ}θ∈Θ) denote a discrete statistical model and let X1,…,Xn∼iidPθ∗ denote the associated statistical experiment, where θ∗ is the true, unknown parameter. Suppose that Pθ has a probability mass function given by pθ. Let θˆMLEn denote the maximum likelihood estimator for θ∗. The maximum likelihood estimator can be expressed as an M-estimator– that is, θˆMLEn=argminθ∈Θ1n∑i=1nρ(Xi,θ) for some function ρ. Which of the following represents the correct choice of the function...
Concept Question: Maximum Likelihood Estimator for the Laplace distribution 1 point possible (graded) As in the...
Concept Question: Maximum Likelihood Estimator for the Laplace distribution 1 point possible (graded) As in the previous problem, let mn MLE denote the MLE for an unknown parameter m* of a Laplace distribution. MLE Can we apply the theorem for the asymptotic normality of the MLE to mn? (You must choose the correct answer that also has the correct explanation.) No, because the log-likelihood is not concave. No, because the log-likelihood is not twice-differentiable, so the Fisher information does not...
This question uses a discrete probability distribution known as the Poisson distribution. A discrete random variable...
This question uses a discrete probability distribution known as the Poisson distribution. A discrete random variable X follows a Poisson distribution with parameter λ if Pr(X = k) = Ake-A ke(0, 1,2, ) k! You are a warrior in Peter Jackson's The Hobbit: Battle of the Five Armies. Because Peter decided to make his battle scenes as legendary as possible, he's decided that the number of orcs that will die with one swing of your sword is Poisson distributed (lid)...
Negative binomial probability function: is the mean is the dispersion parameter Let there be two groups...
Negative binomial probability function:
is the mean
is the dispersion
parameter
Let there be two groups with numbers and means of
1) Write down the log-likelihood for the full model
2) Calculate the likelihood equations and find the general form
of the MLE for and
3) Write down the likelihood function in the reduced model (ie.
assuming )
and derive the MLE for in general
terms
4) Using the above answers only, give the MLE and standard error
for where...
Let X1,X2,X3..Xn be iid of f(x)= theta. x^(theta-1), with x(0,1) and theta being a positive numbe...
Let X1,X2,X3..Xn be iid of f(x)= theta. x^(theta-1), with x(0,1) and theta being a positive number. Is the parameter identifiable?.Compute the maximum likelihood estimate. If instead of X1,X2,,, We observe, Y1,Y2,...Yn, where Yi=1(Xi<=0.5).What distribution does Yi follow? What is the parameter of this distribution? Compute MLE and the method of moments and Fisher information.
Let X1,X2,X3..Xn be iid of f(x)= theta. x^(theta-1), with x(0,1) and theta being a positive number....
Let X1,X2,X3..Xn be iid of f(x)= theta. x^(theta-1), with x(0,1) and theta being a positive number. Is the parameter identifiable?.Compute the maximum likelihood estimate. If instead of X1,X2,,, We observe, Y1,Y2,...Yn, where Yi=1(Xi<=0.5).What distribution does Yi follow? What is the parameter of this distribution? Compute MLE and the method of moments and Fisher information.
1 Bookmark this page Setup: For all problems on this page, suppose you have data X],...,x...
1 Bookmark this page Setup: For all problems on this page, suppose you have data X],...,x . N (0,1) that is a random sample of identically and independently distributed standard normal random variables. Useful facts: The following facts might be useful: For a standard normal random variable X1, we have: E[X] =0, E[X{1=1, E(X) = 3. Sample mean 1.5 points possible (graded, results hidden) Consider the sample mean: X = x + X2+...+X,). What are the mean E [Xn] and...
8. A Union-Intersection Test Bookmark this page Let X1,…,Xn be i.i.d. Bernoulli random variables with unknown...
8. A Union-Intersection Test Bookmark this page Let X1,…,Xn be i.i.d. Bernoulli random variables with unknown parameter p∈(0,1). Suppose we want to test H0:p∈[0.48,0.51]vsH1:p∉[0.48,0.51] We want to construct an asymptotic test ψ for these hypotheses using X¯¯¯¯n. For this problem, we specifically consider the family of tests ψc1,c2 where we reject the null hypothesis if either X¯¯¯¯n<c1≤0.48 or X¯¯¯¯n>c2≥0.51 for some c1 and c2 that may depend on n, i.e. ψc1,c2=1((X¯¯¯¯n<c1)∪(X¯¯¯¯n>c2))where c1<0.48<0.51<c2. Throughout this problem, we will discuss possible choices...
Degrees of Freedom of a Known Test 2 points possible graded) Let us consider a statistical model with parameter ER". Let O be the parameter that generates the n lid samples X1,..., X, Let I ) be the Fisher information and assume that the MLE is asymptotically normal. Assume that I(C) is a diagonal matrix with positive entries 1/t1,...,1/td. We wish to perform a test for the hypotheses H : 8 - and H:8 + . Let the test statistic...
5. A confidence interval for Poisson variables Bookmark this page (a) 2 points possible (graded) Let X1,..., Xn bei.i.d. Poisson random variables with parameter 1 > 0 and denote by Xn their empirical average, Xn=1 İx; and (bn), such that an (Xn-bn) converges in distribution to a standard Gaussian random variable Find two sequences (an) Z~ N(0,1). an =
Concept Question: Maximum Likelihood Estimator for the Laplace distribution 1 point possible (graded) As in the previous problem, let mn MLE denote the MLE for an unknown parameter m* of a Laplace distribution. MLE Can we apply the theorem for the asymptotic normality of the MLE to mn? (You must choose the correct answer that also has the correct explanation.) No, because the log-likelihood is not concave. No, because the log-likelihood is not twice-differentiable, so the Fisher information does not...
This question uses a discrete probability distribution known as the Poisson distribution. A discrete random variable X follows a Poisson distribution with parameter λ if Pr(X = k) = Ake-A ke(0, 1,2, ) k! You are a warrior in Peter Jackson's The Hobbit: Battle of the Five Armies. Because Peter decided to make his battle scenes as legendary as possible, he's decided that the number of orcs that will die with one swing of your sword is Poisson distributed (lid)...
Negative binomial probability function:
is the mean
is the dispersion
parameter
Let there be two groups with numbers and means of
1) Write down the log-likelihood for the full model
2) Calculate the likelihood equations and find the general form
of the MLE for and
3) Write down the likelihood function in the reduced model (ie.
assuming )
and derive the MLE for in general
terms
4) Using the above answers only, give the MLE and standard error
for where...
1 Bookmark this page Setup: For all problems on this page, suppose you have data X],...,x . N (0,1) that is a random sample of identically and independently distributed standard normal random variables. Useful facts: The following facts might be useful: For a standard normal random variable X1, we have: E[X] =0, E[X{1=1, E(X) = 3. Sample mean 1.5 points possible (graded, results hidden) Consider the sample mean: X = x + X2+...+X,). What are the mean E [Xn] and...
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