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Likelihood for a Categorical Distribution 3 points possible graded) Suppose that K = 3, and let...
Practice: Compute Likelihood of a Poisson Statistical Model 0/3 points (graded) Let X1,…,Xn∼iidPoiss(λ∗) for some unknown...
Practice: Compute Likelihood of a Poisson Statistical Model 0/3 points (graded) Let X1,…,Xn∼iidPoiss(λ∗) for some unknown λ∗∈(0,∞). You construct the associated statistical model (E,{Poiss(λ)}λ∈Θ) where E and Θ are defined as in the answers to the previous question. Suppose you observe two samples X1=1,X2=2. What is L2(1,2,λ)? Express your answer in terms of λ. L2(1,2,λ)= Next, you observe a third sample X3=3 that follows X1=1 and X2=2. What is L3(1,2,3,λ)? L3(1,2,3,λ)= Suppose your data arrives in a different order: X1=2,X2=3,X3=1....
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...
Practice: Compute the Kolmogorov-Smirnov Test Statistic 1 point possible (graded) Let X1, ..., Xn be iid...
Practice: Compute the Kolmogorov-Smirnov Test Statistic 1 point possible (graded) Let X1, ..., Xn be iid samples with cdf F, and let F° denote the cdf of Unif(0,1). Recall that F° (t) = t. 1(t € [0,1]) +1.1(t > 1). We want to use goodness of fit testing to determine whether or not X1,...,x, iid Unif(0,1). To do so, we will test between the hypotheses = 70 H : F(t) H :F(t) + F. To make computation of the test...
Do I get the right answers? If not, can someone please explain? (a) 2 points possible (graded, results hidden) Conside...
Do I get the right answers? If not, can someone please
explain?
(a) 2 points possible (graded, results hidden) Consider a Gaussian linear model Y = aX + e in a Bayesian view. Consider the prior (a) = 1 for all a eR. Determine whether each of the following statements is true or false. (a) is a uniform prior. O True C False n(a) is a Jeffreys prior when we consider the likelihood L (Y = y|A = a, X...
Concept Check: Terminology 0/3 points (graded) Suppose you observe iid samples X1,…,Xn∼P from some unknown distribution...
Concept Check: Terminology
0/3 points (graded)
Suppose you observe iid samples X1,…,Xn∼P from some
unknown distribution P. Let F denote a parametric
family of probability distributions (for example, F could be the
family of normal distributions {N(μ,σ2)}μ∈R,σ2>0).
In the topic of goodness of fit testing, our
goal is to answer the question "Does P
belong to the family F, or is P
any distribution outside of F
?"
Parametric hypothesis testing is a particular case of goodness
of fit testing...
As on the previous page, let Xi,...,Xn be i.i.d. with pdf where >0 2 points possible...
As on the previous page, let Xi,...,Xn be i.i.d. with pdf where >0 2 points possible (graded, results hidden) Assume we do not actually get to observe X, . . . , Xn. to estimate based on this new data. Instead let Yİ , . . . , Y, be our observations where Yi-l (X·S 0.5) . our goals What distribution does Yi follow? First, choose the type of the distribution: Bernoulli Poisson Norma Exponential Second, enter the parameter of...
As on the previous page, let X1,... ,Xn be iid with pdf where θ > 0. (to) 2 Possible points (qual...
As on the previous page, let X1,... ,Xn be iid with pdf where θ > 0. (to) 2 Possible points (qualifiable, hidden results) Assume we do not actually get to observe Xı , . . . , X. . Instead let Yı , . . . , Y, be our observations where Yi = 1 (Xi 0.5) . Our goal is to estimate 0 based on this new data. What distribution does Y follow? First, choose the type of distribution:...
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...
Practice: Compute the Kolmogorov-Smirnov Test Statistic 1 point possible (graded) Let X1, ..., Xn be iid samples with cdf F, and let F° denote the cdf of Unif(0,1). Recall that F° (t) = t. 1(t € [0,1]) +1.1(t > 1). We want to use goodness of fit testing to determine whether or not X1,...,x, iid Unif(0,1). To do so, we will test between the hypotheses = 70 H : F(t) H :F(t) + F. To make computation of the test...
Do I get the right answers? If not, can someone please
explain?
(a) 2 points possible (graded, results hidden) Consider a Gaussian linear model Y = aX + e in a Bayesian view. Consider the prior (a) = 1 for all a eR. Determine whether each of the following statements is true or false. (a) is a uniform prior. O True C False n(a) is a Jeffreys prior when we consider the likelihood L (Y = y|A = a, X...
Concept Check: Terminology
0/3 points (graded)
Suppose you observe iid samples X1,…,Xn∼P from some
unknown distribution P. Let F denote a parametric
family of probability distributions (for example, F could be the
family of normal distributions {N(μ,σ2)}μ∈R,σ2>0).
In the topic of goodness of fit testing, our
goal is to answer the question "Does P
belong to the family F, or is P
any distribution outside of F
?"
Parametric hypothesis testing is a particular case of goodness
of fit testing...
As on the previous page, let Xi,...,Xn be i.i.d. with pdf where >0 2 points possible (graded, results hidden) Assume we do not actually get to observe X, . . . , Xn. to estimate based on this new data. Instead let Yİ , . . . , Y, be our observations where Yi-l (X·S 0.5) . our goals What distribution does Yi follow? First, choose the type of the distribution: Bernoulli Poisson Norma Exponential Second, enter the parameter of...
As on the previous page, let X1,... ,Xn be iid with pdf where θ > 0. (to) 2 Possible points (qualifiable, hidden results) Assume we do not actually get to observe Xı , . . . , X. . Instead let Yı , . . . , Y, be our observations where Yi = 1 (Xi 0.5) . Our goal is to estimate 0 based on this new data. What distribution does Y follow? First, choose the type of distribution:...
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