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2. (a) Suppose that x1,... , Vn are a random sample from a gamma distribution with...
2. (a) Suppose that xi,...,In are a random sample from a gamma distribution with shape parameter...
2. (a) Suppose that xi,...,In are a random sample from a gamma distribution with shape parameter and rate parameter λ, Γ(a) Here α > 0 and λ > 0. Let θ sufficient statistic for the data (α, β). Determine the log-likelihood, I(0), and a 2-dimensional b) Suppose that xi,...,In are a random sample from a U(-0,) distribution, 1/(20) if- otherwise x-θ f(x;0)-' 0, Here θ > 0, Determine the likelihood, L(0), and a one-dimensional sufficient statistic. Note that the likelihood...
. Suppose that x1, . . . , xn are a random sample having probability density...
. Suppose that x1, . . . , xn are a random sample having probability density function f(x; θ) = (θ + 1)x^θ , 0 < x < (1) Here the parameter θ > 0. (a) Determine the log-likelihood, l(θ), and a 1-dimensional sufficient statistic. (b) Show that P(Xi ≤ b; θ) = b θ+1 for f(x; θ) given in (1). (c) Suppose now that because of a recurring computer glitch in storing the observations, only a random subset of...
Suppose that x1, . . . , xn are a random sample having probability density function...
Suppose that x1, . . . , xn are a random sample having probability density function f(x; θ) = (θ + 1)x^θ , 0 < x < 1. (1) Here the parameter θ > 0. (a) Show that P(Xi ≤ b; θ) = b^(θ+1) for f(x; θ) given in (1). (b) Suppose now that because of a recurring computer glitch in storing the observations, only a random subset of the xi are observed. For the rest of the observations, it...
1. Suppose that xi, ,xn are a random sample having probability density function Here the parameter...
1. Suppose that xi, ,xn are a random sample having probability density function Here the parameter θ > 0. (a) Determine the log likelibood, 10), and a 1- dimensional (a) Determine the log-likelihood, l(0), and a 1-dimensional sufficient statistic. (b) Show that P(XiS b;0) = +1 for f(x:0) given in (1). (c) Suppose now that because of a recurring computer glitch in storing the observations, only a random subset of the r, are observed. For the rest of the observations,...
Let X1 ,……, Xn be a random sample from a Gamma(α,β) distribution, α> 0; β> 0....
Let X1 ,……, Xn be a random sample from a Gamma(α,β) distribution, α> 0; β> 0. Show that T = (∑n i=1 Xi, ∏ n i=1 Xi) is complete and sufficient for (α, β).
Let X1, . . . , Xn be a sample taken from the Gamma distribution Γ(2,...
Let X1, . . . , Xn be a sample taken from the Gamma distribution Γ(2, θ−1) with pdf f(x,θ)= θ^2xexp(−θx) if x ≥ 0, θ ∈ (0,∞), and 0 otherwise, (A) Show that Y = ∑ni=1 Xi is a complete and sufficient statistic. (B) Find E(1/Y) . Hint: If W ∼ χ2(k) then E(W^m) = 2mΓ(k/2+m) for m > −k/2. Note also that Y Γ(k/2) Γ(n) = (n − 1)!, n ∈ N∗ . Facts from 1(C) are useful:...
Let Xi,, X be a random sample from a gamma(a, B) distribution a. Identify a two-dimensional...
Let Xi,, X be a random sample from a gamma(a, B) distribution a. Identify a two-dimensional sufficient statistics for (α, β). b. Is the two-dimensional sufficient statistic in part (a) minimal sufficient?
Let Xi,...,Xn be a random sample from a two parameter exponential distribution with pa- rameter θ...
Let Xi,...,Xn be a random sample from a two parameter exponential distribution with pa- rameter θ (λ, μ), (a) Show that the distribution of Ti = log(X(n)-X) +log λ is free of θ. Îs an ancillary statistics (b) show that 72- Xu is ancillary X-X
Let Xi,...,Xn be a random sample from a two parameter exponential distribution with pa- rameter θ (λ, μ), (a) Show that the distribution of Ti = log(X(n)-X) +log λ is free of θ. Îs an...
1. Suppose that ri,...,In are a random sample having probability density function Here the parameter 0...
1. Suppose that ri,...,In are a random sample having probability density function Here the parameter 0 >0 (a) Determine the log-likelihood, l(0), and a 1-dimensional sufficient statistic. (b) Show that P(Xi b;0)-μ+1 for f(x;0) given in (1). (c) Suppose now that because of a recurring computer glitch in storing the observations, only a random subset of the ai are observed. For the rest of the observations, it is only known that z; < 1/2. Let δί-1 or 0 according to...
7. Suppose X1, X2, ..., Xn is a random sample from an exponential distribution with parameter...
7. Suppose X1, X2, ..., Xn is a random sample from an exponential distribution with parameter K. (Remember f(x;2) = 2e-Ax is the pdf for the exponential dist”.) a) Find the likelihood function, L(X1, X2, ..., Xn). b) Find the log-likelihood function, b = log L. c) Find dl/d, set the result = 0 and solve for 2.
2. (a) Suppose that xi,...,In are a random sample from a gamma distribution with shape parameter and rate parameter λ, Γ(a) Here α > 0 and λ > 0. Let θ sufficient statistic for the data (α, β). Determine the log-likelihood, I(0), and a 2-dimensional b) Suppose that xi,...,In are a random sample from a U(-0,) distribution, 1/(20) if- otherwise x-θ f(x;0)-' 0, Here θ > 0, Determine the likelihood, L(0), and a one-dimensional sufficient statistic. Note that the likelihood...
1. Suppose that xi, ,xn are a random sample having probability density function Here the parameter θ > 0. (a) Determine the log likelibood, 10), and a 1- dimensional (a) Determine the log-likelihood, l(0), and a 1-dimensional sufficient statistic. (b) Show that P(XiS b;0) = +1 for f(x:0) given in (1). (c) Suppose now that because of a recurring computer glitch in storing the observations, only a random subset of the r, are observed. For the rest of the observations,...
Let Xi,, X be a random sample from a gamma(a, B) distribution a. Identify a two-dimensional sufficient statistics for (α, β). b. Is the two-dimensional sufficient statistic in part (a) minimal sufficient?
Let Xi,...,Xn be a random sample from a two parameter exponential distribution with pa- rameter θ (λ, μ), (a) Show that the distribution of Ti = log(X(n)-X) +log λ is free of θ. Îs an ancillary statistics (b) show that 72- Xu is ancillary X-X
Let Xi,...,Xn be a random sample from a two parameter exponential distribution with pa- rameter θ (λ, μ), (a) Show that the distribution of Ti = log(X(n)-X) +log λ is free of θ. Îs an...
1. Suppose that ri,...,In are a random sample having probability density function Here the parameter 0 >0 (a) Determine the log-likelihood, l(0), and a 1-dimensional sufficient statistic. (b) Show that P(Xi b;0)-μ+1 for f(x;0) given in (1). (c) Suppose now that because of a recurring computer glitch in storing the observations, only a random subset of the ai are observed. For the rest of the observations, it is only known that z; < 1/2. Let δί-1 or 0 according to...
7. Suppose X1, X2, ..., Xn is a random sample from an exponential distribution with parameter K. (Remember f(x;2) = 2e-Ax is the pdf for the exponential dist”.) a) Find the likelihood function, L(X1, X2, ..., Xn). b) Find the log-likelihood function, b = log L. c) Find dl/d, set the result = 0 and solve for 2.
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