*9.93 Let Y₁, Y₂...., Y, be a random sample from a population with density function20²y30 C20110EXAMPLE 9.16 Let Y₁...... Y, be a random sample of observations from a uniform distributionwith probability density function (18)= 1/8,for0 ≤ y ≤ and i = 1, 2, ..., n.Find the MLE of 8.Solution In this case, the likelihood is given byL(B) = f(y-2--- % ]®) = S(x]8) × ƒ(y₂]6) × --- × ƒ(8)110 ≤ y ≤0.1=1,2,.....otherwise.Obviously, L() is not maximized when L(8)= 0. You will notice that 1/8" is amonotonically decreasing function of 6. Hence, nowhere in the interval 0 <<is d[1/8"1/49 equal to zero. However, 1/6 increases as 8 decreases, and 1/8" ismaximized by selecting to be as small as possible, subject to the constraint that allof the y values are between zero and 8. The smallest value of that satisfies thisconstraint is the maximum observation in the set y. y. y That is, 8=Y=max(₁. Y..... Y) is the MLE for 8. This MLE for is not an unbiased estimatorof 8, but it can be adjusted to be unbiased, as shown in Example 9.1.We have seen that sufficient statistics that best summarize the data have desirableproperties and often can be used to find an MVUE for parameters of interest. If Uis any sufficient statistic for the estimation of a parameter 8, including the sufficientstatistic obtained from the optimal use of the factorization criterion, the MLE isalways some function of U. That is, the MLE depends on the sample observationsonly through the value of a sufficient statistic. To show this, we need only observe480 Chapter 9 Properties of Point Estimators and Methods of Estimationw okkapuenjiž.that if U is a sufficient statistic for 6, the factorization criterion (Theorem 9.4) impliesthat the likelihood can be factored asL(8)=L(... y) = g(u,0)h(y)where g(u,) is a function of only u and 6 and k(y₁. 2. y) does not dependon 8. Therefore, it follows thatIn[L(9)] = Ing(u,9)]+[(y)].Notice that In[h(y. y... y)] does not depend on and therefore maximizingIn[L(9)] relative to is equivalent to maximizing In[g(u, 8)] relative to 6. BecauseIng(4,8)] depends on the data only through the value of the sufficient statistic U, theMLE for 8 is always some function of U. Consequently, if an MLE for a parametercan be found and then adjusted to be unbiased, the resulting estimator often is anMVUE of the parameter in question.MLEs have some additional properties that make this method of estimation par-ticularly attractive. In Example 9.9, we considered estimation of 8², a function of theparameter 8. Functions of other parameters may also be of interest. For example, thevariance of a binomial random variable is mp(1-p). a function of the parameter p.If Y has a Poisson distribution with mean A, it follows that P(Y=0)=we maywish to estimate this function of à Generally, if is the parameter associated witha distribution, we are sometimes interested in estimating some function of 8-say1(6) rather than itself. In Exercise 9.94, you will prove that if 1(8) is a one-to-onefunction of and it is the MLE for e, then the MLE of 1() is given by1(0)=1(0).This result, sometimes referred to as the invariance property of MLEs, also holds forany function of a parameter of interest (not just one-to-one functions). See Casellaand Berger (2002) for details.

Answered: *9.93 Let Y₁, ₂,..., Y, be a random…

*9.93 Let Y₁, ₂,..., Y, be a random sample from a population with density function 202 f(y 10) = y3 0, 0

6th Edition

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Author:Amos Gilat

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Chapter1: Starting With Matlab

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