(being a Borel function a random vector x)any information of what we mean by a"most representative values "must be in terms of the distribution of 0, say f(e). The obvious property to require a ' good'estimator 8 of 8 to satisfy is that f(e) is centered around 8 Definition 7. An estimator 0 of 0 is said to be an unbiased estimator of e if E(O)=6f6=0 That is, the distribution of 0 has mean equal to the unknown parameter to estimated Note that an alternative, but equivalent, way to define e(8)is E(6 h(x)f(x;6) where f(a: 0)=f(a1, 2,,In; 0) is the distribution of the sample, x It must be remembered that unbiasedness is a property based on the distri- bution of 0. This distribution is often called sampling distribution of 0 in order distinguish it from any other distribution of function of r v's 1.2 Effie Although unbiasedness seems at first sight to be a highly desirable property it turns out in most situations there are too many unbiased estimators for this prop- erty to be used as the sole criterion for judging estimators. The question which naturally arises is "How can we choose among unbiased estimators ". Given that the variance is a measure of dispersion, intuition suggests that the estimator with the smallest variance is in a sense better because its distribution is more c Definition 8: An unbiased estimator 0 of 0 is said to be relatively more efficient than some(being a Borel function a random vector x) any information of what we mean by a ”most representative values” must be in terms of the distribution of θˆ, say f(θˆ). The obvious property to require a ’good’ estimator θˆ of θ to satisfy is that f(θˆ) is centered around θ. Definition 7: An estimator θˆ of θ is said to be an unbiased estimator of θ if E(θˆ) = Z ∞ −∞ θˆf(θˆ)dθˆ = θ. That is, the distribution of θˆ has mean equal to the unknown parameter to be estimated. Note that an alternative, but equivalent, way to define E(θˆ) is E(θˆ) = Z ∞ −∞ · · · Z ∞ −∞ h(x)f(x; θ)dx where f(x; θ) = f(x1, x2, ..., xn; θ) is the distribution of the sample, x. It must be remembered that unbiasedness is a property based on the distri￾bution of θˆ. This distribution is often called sampling distribution of θˆ in order to distinguish it from any other distribution of function of r.v.’s. 2.1.2 Efficiency Although unbiasedness seems at first sight to be a highly desirable property it turns out in most situations there are too many unbiased estimators for this prop￾erty to be used as the sole criterion for judging estimators. The question which naturally arises is ” How can we choose among unbiased estimators ?”. Given that the variance is a measure of dispersion, intuition suggests that the estimator with the smallest variance is in a sense better because its distribution is more ’concentrated’ around θ. Definition 8: An unbiased estimator θˆ of θ is said to be relatively more efficient than some 7