Fundamentals of Probability and Statistics for Engineers

Following similar procedures as given above, we can show that where k is the kth moment of population X. 9.1.4 Order Statistics ...

Before proceeding, a remark is in order regarding the notation to be used. As seen in Equation (9.2), our objective in parameter ...

9.2.2 M inimum Variance It seems natural that, if h(X 1 ,X 2 ,...,Xn) is to qualify as a good estimator for , not only its mean ...

Theorem 9. 2: the Crame ́r– R a o ineq ua lit y.LetX 1 ,X 2 ,...,Xn denote a sample of size n from a population X with pdf f(x; ...

Let us define a new random variable Y by Equation (9.30) shows that Moreover, since Y is a sum of n independent random variables ...

variance of any unbiased estimator and it expresses a fundamental limitation on the accuracy with which a parameter can be estim ...

which is a one-to-one transformation and differentiable with respect to ; then, CR LB for var where is an unbiased estimator for ...

Answer: let us denote^2 by. Then, and Hence, according to Equation (9.36), the CRLB for the variance of any unbiased estimator f ...

Example 9.4.Problem: determine the CRLB for the variance of any unbiased estimator for in the lognormal distribution Answer: we ...

whereX is the sample mean based on a sample of size n. The choice of 1 is intuitively obvious since , and the choice of 2 is bas ...

a valid reason for choosing 2 as a better estimator, compared with 1 ,for , in certain cases. 9.2.3 Consistency An estimator is ...

9.2.4 Sufficiency Let X 1 ,X 2 ,...,Xn be a sample of a population X the distribution of which depends on unknown parameter. If ...

The foregoing results can be extended to the multiple parameter case. Let be the parameter vector. Then Y 1 h 1 (X 1 ,...,Xn),.. ...

where is the unknown parameter. We have which can be factorized in the form of Equation (9.50) by letting and It is seen that is ...

9.3.1.1 Method of Moments The oldest systematic method of point estimation was proposed by Pearson (1894) and was extensively us ...

Let us remark that it is not necessary to consider m consecutive moment equations as indicated by Equations (9.58); any convenie ...

and and hence is consistent. Example 9.9.Problem: let us select the normal distribution as a model for the percentage yield disc ...

Estimates based on the sample values given by Table 8.1 are, following Equations (9.64) and (9.65), where xj, j 1, 2,... , 200, ...

Answer: in this case, and, following the method of moments, the simplest estimator, ,for is obtained from H ence, the desired es ...

Ex ample 9. 12. Suppose that population X has a uniform distribution over the range (0, ) and we wish to estimate parameter from ...