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Q1. Consider a random variable Y having probability density function otherwise. Given Yi, . . ....
6. Let Y be a continuous random variable with probability density function Oyo-1, for 0< y<...
6. Let Y be a continuous random variable with probability density function Oyo-1, for 0< y< k; f(y) 0, otherwise, where 0 > 1 and k > 0. (a) Show that k = 1. (b) Find E(Y) and Var(Y) in terms of 0. (c) Derive 6, the moment estimator of 0 based on a random sample Y1,...,Y. (d) Derive ô, the maximum likelihood estimator of 0 based on a random sample Y1,..., Yn. (e) A random sample of n =...
Lct Yi, Y2.. . Yn denote a random sample from the probability density function Show that...
Lct Yi, Y2.. . Yn denote a random sample from the probability density function Show that Y is a consistent estimator of 1
1. Suppose that xi,..., xn are a random sample having probability density function f(x; δ)-¡0 otherwise...
1. Suppose that xi,..., xn are a random sample having probability density function f(x; δ)-¡0 otherwise (a) Determine the method of moments estimator of δ based on the first moment. (b) Determine the MLE of δ.
1. Consider a GLM (generalised linear model) for a Poisson random sample Y1,. .. , Y, with \Vi each Yi having a pdf or...
1. Consider a GLM (generalised linear model) for a Poisson random sample Y1,. .. , Y, with \Vi each Yi having a pdf or pmf f(y; A;) = i= 1, . .. ,n. Yi = 0, 1,2, -..; ^; > 0; Y;! Note that the pdf from an exponential family has the following general form b(0) + c(y, a(o) y0 exp f(y; 0, 6) = Suppose the linear predictor of the GLM is n = a+Bxi, with (a,B) being the...
I. Consider a variable y = θ + where θ is an unknown parameter and e is a random variable with me...
I. Consider a variable y = θ + where θ is an unknown parameter and e is a random variable with mean zero. (a) What is the expected value of y? (b) Suppose you draw a sample of yi yn. Derive the least squares estimator for θ. For full credit you must check the 2nd order condition c) Can this estimator (0) be described as a method of moments estimator? (d) Now suppose є is independent normally distributed with mean...
1. Suppose that xi, ,Zn are a random sample having probability density function f(x,6) =(0 otherwise...
1. Suppose that xi, ,Zn are a random sample having probability density function f(x,6) =(0 otherwise (a) Determine the method of moments estimator of δ based on the first moment. (b) Determine the MLE of δ.
Let with Y, Y, ..., Yn be i id random variables the following probability density function,...
Let with Y, Y, ..., Yn be i id random variables the following probability density function, 1 x)/x fyly) = f I y ocyc1 o otherwise a) b) where x>0 is an unknown parameter. Find the maximum likelihood estimator , ã of x. Show this is an unbaised estimator for a. Hint : make use of the fact that in y follows an exponential distribution with mean a. Toe., -lny ~ Exp(x) c) Find the MSE of the manimum likelihood...
1. Suppose that xi,... ,n are a random sample having probability density function otherwise (a) Determine...
1. Suppose that xi,... ,n are a random sample having probability density function otherwise (a) Determine the method of moments estimator of 6 based on the first moment. (b) Determine the MLE of o
In class, we showed that the MLE of the mean of a random sample {yi, y2,...
In class, we showed that the MLE of the mean of a random sample {yi, y2, ..., Yn} from an N(u, 6-) population is exactly the sample average (i.e., û = y). Show that the MLE for the standard deviation is: 1 n @ = – Ëly;-)) ? V ni=1 Hint: Find the partial derivative of the likelihood function with respect to o, then set this equal to zero and solve for o. The result will be a function of...
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.
6. Let Y be a continuous random variable with probability density function Oyo-1, for 0< y< k; f(y) 0, otherwise, where 0 > 1 and k > 0. (a) Show that k = 1. (b) Find E(Y) and Var(Y) in terms of 0. (c) Derive 6, the moment estimator of 0 based on a random sample Y1,...,Y. (d) Derive ô, the maximum likelihood estimator of 0 based on a random sample Y1,..., Yn. (e) A random sample of n =...
Lct Yi, Y2.. . Yn denote a random sample from the probability density function Show that Y is a consistent estimator of 1
1. Suppose that xi,..., xn are a random sample having probability density function f(x; δ)-¡0 otherwise (a) Determine the method of moments estimator of δ based on the first moment. (b) Determine the MLE of δ.
1. Consider a GLM (generalised linear model) for a Poisson random sample Y1,. .. , Y, with \Vi each Yi having a pdf or pmf f(y; A;) = i= 1, . .. ,n. Yi = 0, 1,2, -..; ^; > 0; Y;! Note that the pdf from an exponential family has the following general form b(0) + c(y, a(o) y0 exp f(y; 0, 6) = Suppose the linear predictor of the GLM is n = a+Bxi, with (a,B) being the...
I. Consider a variable y = θ + where θ is an unknown parameter and e is a random variable with mean zero. (a) What is the expected value of y? (b) Suppose you draw a sample of yi yn. Derive the least squares estimator for θ. For full credit you must check the 2nd order condition c) Can this estimator (0) be described as a method of moments estimator? (d) Now suppose є is independent normally distributed with mean...
1. Suppose that xi, ,Zn are a random sample having probability density function f(x,6) =(0 otherwise (a) Determine the method of moments estimator of δ based on the first moment. (b) Determine the MLE of δ.
Let with Y, Y, ..., Yn be i id random variables the following probability density function, 1 x)/x fyly) = f I y ocyc1 o otherwise a) b) where x>0 is an unknown parameter. Find the maximum likelihood estimator , ã of x. Show this is an unbaised estimator for a. Hint : make use of the fact that in y follows an exponential distribution with mean a. Toe., -lny ~ Exp(x) c) Find the MSE of the manimum likelihood...
1. Suppose that xi,... ,n are a random sample having probability density function otherwise (a) Determine the method of moments estimator of 6 based on the first moment. (b) Determine the MLE of o
In class, we showed that the MLE of the mean of a random sample {yi, y2, ..., Yn} from an N(u, 6-) population is exactly the sample average (i.e., û = y). Show that the MLE for the standard deviation is: 1 n @ = – Ëly;-)) ? V ni=1 Hint: Find the partial derivative of the likelihood function with respect to o, then set this equal to zero and solve for o. The result will be a function of...
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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