Homework Answers
Invariance Property of
MLE:
If T is the MLE of p, then g(T) is the mle of g(p)
where g(p) is any function of p
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1. Suppose Yi,½, , Yn is an iid sample from a Bernoulli(p) population distribution, where 0< p<1 is unknown. The population pmf is py(ulp) otherwise 0, (a) Prove that Y is the maximum likelihoo...
Let Yı, Y2, ...,Yn be an iid sample from a population distribution described by the pdf...
Let Yı, Y2, ...,Yn be an iid sample from a population distribution described by the pdf fy(y|0) = (@+ 1) yº, o<y<1 for 0> - -1. (a) Find the MOM estimator of 0. (b) Find the maximum likelihood estimator (MLE) of 0. (c) Find the MLE of the population mean E(Y) = 0 +1 0 + 2 You do not need to prove that the above is true. Just find its MLE.
Suppose Y1, Y2, ..., Yn is an iid sample from a Pareto population distribution described by...
Suppose Y1, Y2, ..., Yn is an iid sample from a Pareto population distribution described by the pdf fy(y|0) = 4ao y -0-1 y > 20, 2 where the parameter do is known. The unknown parameter is 0 > 0. (a) Find the MOM estimator of 0. (b) Find the MLE of 0.
Let Sı, S2,... , Sn be a random sample from a Bernoulli distribution with pmf (a) Calculate E(S) and find the MOM estim...
Let Sı, S2,... , Sn be a random sample from a Bernoulli distribution with pmf (a) Calculate E(S) and find the MOM estimator of p. (b) Construct the log-likelihood function and use this to find the max- imum likeligood (ML) estimator of p.
Let Sı, S2,... , Sn be a random sample from a Bernoulli distribution with pmf (a) Calculate E(S) and find the MOM estimator of p. (b) Construct the log-likelihood function and use this to find the max-...
Suppose that X1, X2,., Xn is an iid sample from the probability mass function (pmf) given...
Suppose that X1, X2,., Xn is an iid sample from the probability mass function (pmf) given by (1 - 0)0r, 0,1,2, 0, otherwise, where 001 (a) Find the maximum likelihood estimator of θ. (b) Find the Cramer-Rao Lower Bound (CRLB) on the variance of unbiased estimators of Eo(X). Can this lower bound be attained? (c) Find the method of moments estimator of θ. (d) Put a beta(2,3) prior distribution on θ. Find the posterior mean. Treating this as a fre-...
Suppose Y1, Y2, ..., Yn are such that Y; ~ Bernoulli(p) and let X = 2h+Yi....
Suppose Y1, Y2, ..., Yn are such that Y; ~ Bernoulli(p) and let X = 2h+Yi. (a) [1 point] Use the distribution of X to show that the method of moments estimator of p is ÔMM = Lizzi. (Work that is unclear or that cannot be followed from step to step will not recieve full credit.) (b) [2 points] Show that the method of moments estimator PMM is a consistent estimator of p. Please show your work to support your...
7.5.12 Suppose that X.., X, are iid Bernoulli(p) where 0
7.5.12 Suppose that X.., X, are iid Bernoulli(p) where 0<p s an unknown parameter. Consider the parametric function T(p)-p + qe with q p. (i) Find a suitable unbiased estimator T for (p); (ii) Since the complete sufficient statistic is = Ση!Xi, use the Lehmann-Scheffé theorems and evaluate the conditional expec tation, E [I, I u-11]; (iii) Hence, derive the UMVUE for T(p) Hint: Try and use the mgf of the Xs appropriately.)
7.5.12 Suppose that X.., X, are iid...
2. Suppose Y1,...,Yn are IID discrete random variables with P(Y; = 0) = 60 P(Y; =...
2. Suppose Y1,...,Yn are IID discrete random variables with P(Y; = 0) = 60 P(Y; = 1) = 01, P(Y; = 2) = 62, where the parameter vector 6 = (60,61,62) satisfies: 0; > 0 and 200; = 1. (a) Calculate E[Y] and EY?), and use the results to derive a method of moments estimator for the parameters (01,02). (b) Show that the maximum likelihood estimator for 6 = 60, 61, 62) is - Ôno = ôz = = 1(Y;=0),...
B1. A random sample of n observations, Yi, ., Yn, is selected from a pop- ulation...
B1. A random sample of n observations, Yi, ., Yn, is selected from a pop- ulation in which Yi, for i = 1, 2, ,n, possesses a common distribution the same as that of the population distribution Y (a) Suppose that we know Y has a Geometric distribution with parameter p, p unknown. Find the estimator using the method of moments. (b) Suppose that we know that Y has an exponential distribution with parameter λ, λ unknown. Find the estimator...
Exercice 6. Let be (Xi,..., Xn) an iid sample from the Bernoulli distribution with parameter θ,...
Exercice 6. Let be (Xi,..., Xn) an iid sample from the Bernoulli distribution with parameter θ, ie. I. What is the Maximum Likelihood estimate θ of θ? 2. Show that the maximum likelihood estimator of θ is unbiased. 3. We're looking to cstimate the variance θ (1-9) of Xi . x being the empirical average 2(1-2). Check that T is not unli ator propose an unbiased estimator of θ(1-0).
3. Let Yi,... , Y be a random sample from a distribution with probability mass function...
3. Let Yi,... , Y be a random sample from a distribution with probability mass function f(a; ?)-|(1-0)20" a--1 a=0,1,2, where 0 01 a. [6 pts] Show that the maximum likelihood estimator of ? is Hint: With the use of indicator functions, a Bernoulli distribution can be written as f(a; ?)-8111}(a) + (1-0)1101 (a) or, equivalently, One of these will simplify the likelihood equation for this problem.
Let Yı, Y2, ...,Yn be an iid sample from a population distribution described by the pdf fy(y|0) = (@+ 1) yº, o<y<1 for 0> - -1. (a) Find the MOM estimator of 0. (b) Find the maximum likelihood estimator (MLE) of 0. (c) Find the MLE of the population mean E(Y) = 0 +1 0 + 2 You do not need to prove that the above is true. Just find its MLE.
Suppose Y1, Y2, ..., Yn is an iid sample from a Pareto population distribution described by the pdf fy(y|0) = 4ao y -0-1 y > 20, 2 where the parameter do is known. The unknown parameter is 0 > 0. (a) Find the MOM estimator of 0. (b) Find the MLE of 0.
Let Sı, S2,... , Sn be a random sample from a Bernoulli distribution with pmf (a) Calculate E(S) and find the MOM estimator of p. (b) Construct the log-likelihood function and use this to find the max- imum likeligood (ML) estimator of p.
Let Sı, S2,... , Sn be a random sample from a Bernoulli distribution with pmf (a) Calculate E(S) and find the MOM estimator of p. (b) Construct the log-likelihood function and use this to find the max-...
Suppose that X1, X2,., Xn is an iid sample from the probability mass function (pmf) given by (1 - 0)0r, 0,1,2, 0, otherwise, where 001 (a) Find the maximum likelihood estimator of θ. (b) Find the Cramer-Rao Lower Bound (CRLB) on the variance of unbiased estimators of Eo(X). Can this lower bound be attained? (c) Find the method of moments estimator of θ. (d) Put a beta(2,3) prior distribution on θ. Find the posterior mean. Treating this as a fre-...
Suppose Y1, Y2, ..., Yn are such that Y; ~ Bernoulli(p) and let X = 2h+Yi. (a) [1 point] Use the distribution of X to show that the method of moments estimator of p is ÔMM = Lizzi. (Work that is unclear or that cannot be followed from step to step will not recieve full credit.) (b) [2 points] Show that the method of moments estimator PMM is a consistent estimator of p. Please show your work to support your...
7.5.12 Suppose that X.., X, are iid Bernoulli(p) where 0<p s an unknown parameter. Consider the parametric function T(p)-p + qe with q p. (i) Find a suitable unbiased estimator T for (p); (ii) Since the complete sufficient statistic is = Ση!Xi, use the Lehmann-Scheffé theorems and evaluate the conditional expec tation, E [I, I u-11]; (iii) Hence, derive the UMVUE for T(p) Hint: Try and use the mgf of the Xs appropriately.)
7.5.12 Suppose that X.., X, are iid...
2. Suppose Y1,...,Yn are IID discrete random variables with P(Y; = 0) = 60 P(Y; = 1) = 01, P(Y; = 2) = 62, where the parameter vector 6 = (60,61,62) satisfies: 0; > 0 and 200; = 1. (a) Calculate E[Y] and EY?), and use the results to derive a method of moments estimator for the parameters (01,02). (b) Show that the maximum likelihood estimator for 6 = 60, 61, 62) is - Ôno = ôz = = 1(Y;=0),...
B1. A random sample of n observations, Yi, ., Yn, is selected from a pop- ulation in which Yi, for i = 1, 2, ,n, possesses a common distribution the same as that of the population distribution Y (a) Suppose that we know Y has a Geometric distribution with parameter p, p unknown. Find the estimator using the method of moments. (b) Suppose that we know that Y has an exponential distribution with parameter λ, λ unknown. Find the estimator...
Exercice 6. Let be (Xi,..., Xn) an iid sample from the Bernoulli distribution with parameter θ, ie. I. What is the Maximum Likelihood estimate θ of θ? 2. Show that the maximum likelihood estimator of θ is unbiased. 3. We're looking to cstimate the variance θ (1-9) of Xi . x being the empirical average 2(1-2). Check that T is not unli ator propose an unbiased estimator of θ(1-0).
3. Let Yi,... , Y be a random sample from a distribution with probability mass function f(a; ?)-|(1-0)20" a--1 a=0,1,2, where 0 01 a. [6 pts] Show that the maximum likelihood estimator of ? is Hint: With the use of indicator functions, a Bernoulli distribution can be written as f(a; ?)-8111}(a) + (1-0)1101 (a) or, equivalently, One of these will simplify the likelihood equation for this problem.
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