Consider independent observations y₁, ..., yn from the model Y;~ Poisson(μ). Using likelihood L(μ) and log-likelihood (μ) as appropriate, compute the following items. 1. Derive the maximum likelihood estimate û.
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- The following table provides values of the function f(x,y). However, because of potential; errors in measurement, the functional values may be slightly inaccurately. Using the statistical package included with a graphical calculator or spreadsheet and critical thinking skills, find the function f(x,y)=a+bx+cy that best estimate the table where a, b and c are integers. Hint: Do a linear regression on each column with the value of y fixed and then use these four regression equations to determine the coefficient c. x y 0 1 2 3 0 4.02 7.04 9.98 13.00 1 6.01 9.06 11.98 14.96 2 7.99 10.95 14.02 17.09 3 9.99 13.01 16.01 19.02Find the minimum mean square error forecast Y(1), forecast error e, (1) and Varfe, (1)1 for the following modes. Y, = 0.8Y, +e,. Y, = 3+21+e,.Consider independent observations (rı, y1), ... (rns yn), from the model Y, Bin(ri, p) for i = 1, ... , n, where the r, are fixed constants. Using likelihood L(p) and log-likelihood I(p) as appropriate, compute the following items. N 1. Derive the maximum likelihood estimate p. 2. Write the second derivative of log-likelihood /(p). == 3. Give an expression of the approximated asymptotic standard error of p by plugging in the estimate p. To this end, estimate the Fisher Information Matrix by : and then s. e. ap² 4. Consider data (15,11), (20,14), (15,9), (10,7), (25,17), (15,12), (10,8). Using your formulæ, compute and write numerical estimates p, s. e. (p) and give a 95% confidence interval for p using the normal approximation. e. (p) = √v¹¹ p=p Note: To answer this question you will work by hand. Do not write in the textbox but upload a single page pdf image of your workings and results. For theoretical computations (a) to (c) you are expected to show your equations and developments…
- Arm circumferences (cm) and heights (cm) are measured from randomly selected adult females. The 139 pairs of measurements yield x = 31.99 cm, y = 163.33 cm, r= 0.032, P-value = 0.708, and y = 158 + 0.1703x. Find the best predicted value of y (height) given an adult female with an arm circumference of 35.0 cm. Let the predictor variable x be arm circumference and the response variable y be height. Use a 0.05 significance level. %3D ..... The best predicted value is cm. (Round to two decimal places as needed.)Find the maximum likelihood estimator by getting the derivative and equating it to 0Simplify the likelihood ratio e (µ1,..., Hr, o2) max (µ1,...,Hr,o²)EO0 e (41,...,Hr; o2) ' max (41,...,Hr,0²)EO SSTR/(r-1) SSE/(nt-r) and show that A is a decreasing function of F =
- ...., n, Consider independent observations (ri, y₁), ... (rn, yn), from the model Y₁ Bin(ri, p) for i = 1, ... where ther, are fixed constants. Using likelihood L(p) and log-likelihood /(p) as appropriate, compute the following items. 1. Derive the maximum likelihood estimate p. 2. Write the second derivative of log-likelihood /(p). 3. Give an expression of the approximated asymptotic standard error of p by plugging in the estimate p. To this end, estimate the Fisher Information Matrix by = -1(P) and then s. e. (p) = ») = √v p=p 4. Consider data (15,11), (20,14), (15,9), (10,7), (25,17), (15,12), (10,8). Using your formulæ, compute and write numerical estimates p, s. e. (p) and give a 95% confidence interval for p using the normal approximation.Let Y1, . . . , YN be a random sample from the Normal distribution Yi ∼ N(ln β, s2) where s2is known. Find the maximum likelihood estimator of b from first principles.Find the Score function, the estimating equation and the information matrix.where is the graphical estimation method (by hand)