Suppose a researcher collects x1,...,n i.i.d. measurements of the background radiation in Boston. Suppose also that these observations follow a Rayleigh distribution with parameter T, with pdf given by XTе¯¯x² f(x) = = Find the maximum likelihood estimate for T.
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- Show that the mean of a random sample of size n from an exponential population is a minimum variance unbi-ased estimator of the parameter θ.Suppose X1,...,Xn are measurements of the mass of an object. The measurements are normally distributed, with mean equal to mu , the mass of the object, and standard deviation equal to 1 microgram. Find (a) The likelihood function. (b) The log-likelihood function (c) the maximum likelihood estimator for the unknown parameter, andConsider 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…
- Suppose that n observations are chosen at random from a continuous pdf fY(y). What is the probability that the last observation recorded will be the smallest number in the sample?...., 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.Suppose that Y is a uniform continuous random variable on the interval (1,11). Calculate the expected value of the random variable (Y− 1)². Use one decimal place accuracy.
- Q4. Let X1,X 2,.X , be a random sample from an exponential distribution with parameter 0. (a) Find the Cramer-Rao lower bound for the variance of any unbiased estimator of e. (b) Find the maximum likelihood estimator (m.l.e.) of 0. (c) Is this estimator unbiased and does its variance attain the Cramer-Rao lower bound? (d) What is the method of moments estimator of 0?The p.d.f. of a random variable X' is as shown in the figure. The pdf is zero for X 5. Calculate (i) the maximum value of p.d.f. (ii) expectation of X, E(X) (iii) variance of X. fx (x) k