Let X1, ..., X50 denote a random sample of size 50 from the geometric distribution f(x; θ) = θ(1 − θ) x−1 for x = 1, 2, ... and 0 < θ < 1. Suppose that after taking the observations we find that ¯x = 5.
a) Find E[X¯] and var(X¯).
b) Use part (a) above together with the CLT and delta method to find the limiting distribution of √ n( ˆθ − θ)
Let X1, ..., X50 denote a random sample of size 50 from the geometric distribution f(x;...
Let X1, ..., X50 denote a random sample of size 50 from the geometric distribution f(x; θ) = θ(1 − θ) x−1 for x = 1, 2, ... and 0 < θ < 1. Suppose that after taking the observations we find that ¯x = 5. 8. a) Find the maximum likelihood estimator ˆθ of θ. b) Find E[X¯] and var(X¯). c) Use part (b) above together with the CLT and delta method to find the limiting distribution of √...
Let X1, X2, .. , Xn be a random sample of size n from a geometric distribution with pmf =0.75 . 0.25z-1, f(x) X-1.2.3. ) Let Zn 3 n n-2ућ. Find Mz, (t), the mgf of Žn. Then find the limiting mgf limn→oo MZm (t). What is the limiting distribution of Z,'? Let X1, X2, .. , Xn be a random sample of size n from a geometric distribution with pmf =0.75 . 0.25z-1, f(x) X-1.2.3. ) Let Zn 3...
Let X1 Xn be a random sample from a distribution with the pdf f(x(9) = θ(1 +0)-r(0-1) (1-2), 0 < x < 1, θ > 0. the estimator T-4 is a method of moments estimator for θ. It can be shown that the asymptotic distribution of T is Normal with ETT θ and Var(T) 0042)2 Apply the integral transform method (provide an equation that should be solved to obtain random observations from the distribution) to generate a sam ple of...
Let X1,X2,...,Xn denote a random sample from the Rayleigh distribution given by f(x) = (2x θ)e−x2 θ x > 0; 0, elsewhere with unknown parameter θ > 0. (A) Find the maximum likelihood estimator ˆ θ of θ. (B) If we observer the values x1 = 0.5, x2 = 1.3, and x3 = 1.7, find the maximum likelihood estimate of θ.
Let X1, X2, ...,Xn denote a random sample of size n from a Pareto distribution. X(1) = min(X1, X2, ..., Xn) has the cumulative distribution function given by: αη 1 - ( r> B X F(x) = . x <B 0 Show that X(1) is a consistent estimator of ß.
Let X1, X2, ... , Xn be a random sample of size n from the exponential distribution whose pdf is f(x; θ) = (1/θ)e^(−x/θ) , 0 < x < ∞, 0 <θ< ∞. Find the MVUE for θ. Let X1, X2, ... , Xn be a random sample of size n from the exponential distribution whose pdf is f(x; θ) = θe^(−θx) , 0 < x < ∞, 0 <θ< ∞. Find the MVUE for θ.
4. Let X1, . . . , Xn be a random sample from a normal random variable X with probability density function f(x; θ) = (1/2θ 3 )x 2 e −x/θ , 0 < x < ∞, 0 < θ < ∞. (a) Find the likelihood function, L(θ), and the log-likelihood function, `(θ). (b) Find the maximum likelihood estimator of θ, ˆθ. (c) Is ˆθ unbiased? (d) What is the distribution of X? Find the moment estimator of θ, ˜θ.
Let X1, X2, ..., X48 denote a random sample of size n = 48 from the uniform distribution U(?1,1) with pdf f(x) = 1/2, ?1 < x < 1. E(X) = 0, Var(X) = 1/3 Let Y = (Summation)48, i=1 Xi and X= 1/48 (Summation)48, i=1 Xi. Use the Central Limit Theorem to approximate the following probability. 1. P(1.2<Y<4) 2. P(X< 1/12)
8. Let X1,...,Xn denote a random sample of size n from an exponential distribution with density function given by, 1 -x/0 -e fx(x) MSE(1). Hint: What is the (a) Show that distribution of Y/1)? nY1 is an unbiased estimator for 0 and find (b) Show that 02 = Yn is an unbiased estimator for 0 and find MSE(O2). (c) Find the efficiency of 01 relative to 02. Which estimate is "better" (i.e. more efficient)? 8. Let X1,...,Xn denote a random...
Let X be a random variable with p.d.f. f(x) = θx^(θ−1) , for 0 < x < 1. Let X1, ..., Xn denote a random sample of size n from this distribution. (a) Find E(X) [2] (b) Find the method of moment estimator of θ [2] (c) Find the maximum likelihood estimator of θ [3] (d) Use the following set of observations to obtain estimates of the method of moment and maximum likelihood estimators of θ. [1 each] 0.0256, 0.3051,...