) Show that = ₁₁X₁ is the maximum likelihood estimator of the parameter 0. Zi=1
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- The time X of a radioactive isotope decays with an expected value of 10 years, and is modeled as an exponential random variable. a) What is the parameter λ of the exponential RV X? b) Compute P(X < 10).I need the answer as soon as possibleConsider independent observations y₁, ..., yn from the model Y;~ Poisson(μ). Using likelihood L(μ) and log-likelihood (μ) as appropriate, compute the following items. 3. Give an expression of the approximated asymptotic standard error of û by plugging in the estimate μ. To this end, estimate the Fisher Information Matrix by and then s. e. () = √V V _2_-1(P) | ₁-² 1
- Consider the maximum likelihood estimation of a parameter 0 and a test of the hypothesis Ho: c(0) = 0. Describe the 3 basic approaches used for testing the hypothesis.Assume the data are generated from the GN-SLR setting, Y = B1Xin + ti i = 1, (a) Show that the likelihood function is where €; ~ N (0,0²). We only have the observed versions of (Y₁, X₁1), which are (x₁1,₁),..., (Xn1, Yn). We want to estimate B₁ and o² with these data using the Maximum Likelihood Method. n i=1 1 L(Bo, B₁,0²) = (2ño²)−¹/² exp{-22 (Yi - B₁x₁)²} (b) Compute the negative log-likelihood function. (c) Show that the MLE estimator for ₁ is ., n 9.9 1(Bo, B₁,0²) = -log L(Bo, B₁, 0²) ÂMLE _ Σi=1 X₁Yi = An 2 (d) Show that the MLE estimator for o2 is i=1 - (62) MLE _ Σi=1(Yi – ÎMLEƑ¡1)² n(4) Consider n i.i.d. samples of X ~ N(µ,0²). Find the maximum likelihood estimate of o?.
- Let X1,..., X, a random sample with distribution f(x; 0) = (0 + 1)æº, 0 -1. (a) Find the method of moments estimator of 0. (b) Show that the method of moments estimator is consistent. (c) Find the maximum likelihood estimator of 0. (d) takes value 0 = 1.2. Use the parametric bootstrap method to obtain a 95% revised bootstrap percentile confidence interval using the maximum likelihood estimate. Make sure to include a plot of the bootstrapped values and an interpretation of your confidence interval. Given a sample of size n = 500, suppose that the maximum likelihood estimate of 07 Find the expected value of the function g (X) = X², where x is a random variable defined by the đensity, fx ) = a. eax, u u (x), where 'a' is a constant.B we regression Given the data (Xi) and (1₁) will assume that a model Y₁ =B₁Xi + Ei is with normally distributed independent Error and í 12 3 4 Xi 7 12 25 30 Yi 14 17 30 42 a) State the likelihood function for the four y observations. b) Evaluate the likelihood function for By and Bo and B₁ = 1 and B₁ = 2- For which of these is the likelihood • function the largest.
- Suppose that X₁ = 1; X₂ = 1; X3 = 0; X₁ = 1; X5 = 1, X6 = 1; X₂ = 0; X = 1; X9 = 0, X10 = 0, represents a random sample. Each of these X's comes from the same population and has a density of fx₁ = 0¹-* (1 - 0)*; x = 0,1 First determine the form of the maximum likelihood estimator for 0, then use the MLE formula and the data provided to find an estimate for 0. Round your answer to 3 decimal places. (Answer is 0.4)Let x₁ = 6, x2 = 4, x3 = 5, x4 = 15 be a random sample from a population with probability density function a 10a (x + 10)α+1 Compute â, the maximum likelihood estimate of a. f(x) = You do not have to verify that your estimate is a maximum. 2 x > 0.
