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iid Let Yı, Y2, ..., Yn N(u,), where the population mean y and population variance o...
Let Yı, Y2, ..., Yn iid N (4,02), where the population mean and population variance o2...
Let Yı, Y2, ..., Yn iid N (4,02), where the population mean and population variance o2 are both unknown. Show that the Method of Moments (MOM) estimators of u and o2 are given by n û =Ý ΣΥ, n i=1 п n - - 1 ô2 - = s2 Ü(Y; – 7) n п i=1 Note: In this case, (Y, S2) is a sufficient statistic for (u, 02). The MOM estimators of u and o2 are therefore functions of a...
Let Yı,Y2, ..., Yn be iid from a population following the shifted exponential distribution with scale...
Let Yı,Y2, ..., Yn be iid from a population following the shifted exponential distribution with scale parameter B = 1. The pdf of the population distribution is given by fy(y\0) = y-0) = e x I(y > 0). The "shift" @ > 0 is the only unknown parameter. (a) Find L(@ly), the likelihood function of 0. (b) Find a sufficient statistic for 0 using the Factorization Theorem. (Hint: O is bounded above by y(1) min{Y1, 42, ..., .., Yn}.) (c)...
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.
1. Let Yı,Y2,..., Yn denote a random sample from a population with mean E (-0,) and...
1. Let Yı,Y2,..., Yn denote a random sample from a population with mean E (-0,) and variance o2 € (0,0). Let Yn = n- Y. Recall that, by the law of large numbers, Yn is a consistent estimator of . (a) (10 points) Prove that Un="in is a consistent estimator of . (b) (5 points) Prove that Vn = Yn-n is not a consistent estimator of (c) (5 points) Suppose that, for each i, P(Y, - of ? Prove what...
7. (12 points) Let Yı,Y2, ..., Yn be a random sample from Gamma(a,b), where a =...
7. (12 points) Let Yı,Y2, ..., Yn be a random sample from Gamma(a,b), where a = 2 and 3 is an unknown parameter. 2 (a) Find the method of moments (MOM) estimator of B. (b) Find the maximum likelihood estimator (MLE) of B. (€) Are the estimators in parts (a) and (b) MVUEs for B? Justify your answer.
Question 4 Let Yı: Y2, .... Yn denote a random sample and let E(Y) = u...
Question 4 Let Yı: Y2, .... Yn denote a random sample and let E(Y) = u and Var(Y) = o-y, i = 1, 2, ..., n. (b) Prove that the standard error of the sample mean Y SEⓇ) =
Suppose Yı, Y2, ..., Yn|7 vid N(10, 7-2). The population mean Mo is known. The un-...
Suppose Yı, Y2, ..., Yn|7 vid N(10, 7-2). The population mean Mo is known. The un- known parameter T > 0, which is the inverse of the population variance, is called the precision. The pdf of N(Mo, T-1) is given by Syl-(wl=) = Vb exp (-5(v – wo)"] Let's now derive the posterior distribution of t from the Bayesian perspective. (a) Define U = (Y; – Mo)? i=1 Show that U is a sufficient statistic for t using the Factorization...
Let Y, Y2, ..., Yn be n i.i.d random variables drawn from the population distribution of...
Let Y, Y2, ..., Yn be n i.i.d random variables drawn from the population distribution of Y-(My, oy). Suppose we want to estimate My and we are asked to choose between two possible estimators of Wy: (1)Y, and (2) Y = (x + 3) (a) Show both estimators are unbiased (2 points) (b) Derive the variance of both estimators and discuss which estimator is more efficient (3 points)
Let yı, y2,-. ., yn be a sample drawn from a normal population with unknown mean...
Let yı, y2,-. ., yn be a sample drawn from a normal population with unknown mean μ an model d unknown variance σ2. One way to estimate μ is to fit the linear (2.61) and use the least squares (LS), that is, to minimize the sum of squares, Σ (Vi-A)2. Another way is to use the least absolute value (L AV), that is, to minimize the sum of absolute value of the vertical distances, Σ bi-μ| (a) Show that the...
. Suppose the Y1, Y2, · · · , Yn denote a random sample from a...
. Suppose the Y1, Y2, · · · , Yn denote a random sample from a
population with Rayleigh distribution (Weibull distribution with
parameters 2, θ) with density function f(y|θ) = 2y θ e −y 2/θ, θ
> 0, y > 0
Consider the estimators ˆθ1 = Y(1) = min{Y1, Y2, · · · , Yn},
and ˆθ2 = 1 n Xn i=1 Y 2 i .
ii) (10 points) Determine if ˆθ1 and ˆθ2 are unbiased
estimators, and in...
Let Yı, Y2, ..., Yn iid N (4,02), where the population mean and population variance o2 are both unknown. Show that the Method of Moments (MOM) estimators of u and o2 are given by n û =Ý ΣΥ, n i=1 п n - - 1 ô2 - = s2 Ü(Y; – 7) n п i=1 Note: In this case, (Y, S2) is a sufficient statistic for (u, 02). The MOM estimators of u and o2 are therefore functions of a...
Let Yı,Y2, ..., Yn be iid from a population following the shifted exponential distribution with scale parameter B = 1. The pdf of the population distribution is given by fy(y\0) = y-0) = e x I(y > 0). The "shift" @ > 0 is the only unknown parameter. (a) Find L(@ly), the likelihood function of 0. (b) Find a sufficient statistic for 0 using the Factorization Theorem. (Hint: O is bounded above by y(1) min{Y1, 42, ..., .., Yn}.) (c)...
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.
1. Let Yı,Y2,..., Yn denote a random sample from a population with mean E (-0,) and variance o2 € (0,0). Let Yn = n- Y. Recall that, by the law of large numbers, Yn is a consistent estimator of . (a) (10 points) Prove that Un="in is a consistent estimator of . (b) (5 points) Prove that Vn = Yn-n is not a consistent estimator of (c) (5 points) Suppose that, for each i, P(Y, - of ? Prove what...
7. (12 points) Let Yı,Y2, ..., Yn be a random sample from Gamma(a,b), where a = 2 and 3 is an unknown parameter. 2 (a) Find the method of moments (MOM) estimator of B. (b) Find the maximum likelihood estimator (MLE) of B. (€) Are the estimators in parts (a) and (b) MVUEs for B? Justify your answer.
Question 4 Let Yı: Y2, .... Yn denote a random sample and let E(Y) = u and Var(Y) = o-y, i = 1, 2, ..., n. (b) Prove that the standard error of the sample mean Y SEⓇ) =
Suppose Yı, Y2, ..., Yn|7 vid N(10, 7-2). The population mean Mo is known. The un- known parameter T > 0, which is the inverse of the population variance, is called the precision. The pdf of N(Mo, T-1) is given by Syl-(wl=) = Vb exp (-5(v – wo)"] Let's now derive the posterior distribution of t from the Bayesian perspective. (a) Define U = (Y; – Mo)? i=1 Show that U is a sufficient statistic for t using the Factorization...
Let Y, Y2, ..., Yn be n i.i.d random variables drawn from the population distribution of Y-(My, oy). Suppose we want to estimate My and we are asked to choose between two possible estimators of Wy: (1)Y, and (2) Y = (x + 3) (a) Show both estimators are unbiased (2 points) (b) Derive the variance of both estimators and discuss which estimator is more efficient (3 points)
Let yı, y2,-. ., yn be a sample drawn from a normal population with unknown mean μ an model d unknown variance σ2. One way to estimate μ is to fit the linear (2.61) and use the least squares (LS), that is, to minimize the sum of squares, Σ (Vi-A)2. Another way is to use the least absolute value (L AV), that is, to minimize the sum of absolute value of the vertical distances, Σ bi-μ| (a) Show that the...
. Suppose the Y1, Y2, · · · , Yn denote a random sample from a
population with Rayleigh distribution (Weibull distribution with
parameters 2, θ) with density function f(y|θ) = 2y θ e −y 2/θ, θ
> 0, y > 0
Consider the estimators ˆθ1 = Y(1) = min{Y1, Y2, · · · , Yn},
and ˆθ2 = 1 n Xn i=1 Y 2 i .
ii) (10 points) Determine if ˆθ1 and ˆθ2 are unbiased
estimators, and in...
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