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## C19: Unbiased Estimators

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**MATH 3033 based onDekking et al. A Modern Introduction to**Probability and Statistics. 2007Slides by Nathan WeiserFormat by Tim Birbeck Instructor Longin Jan Latecki C19: Unbiased Estimators**19.1 – Estimators**• The parameters that determine the model distribution are called the model parameters • We focus on a situation where a parameter correspond to a feature of the model distribution that can be described by the model parameters themselves or by some function of the model parameters. This is known as the parameter of interest.**19.1 – Estimators**• ESTIMATE: an estimate is a value t that only depends on the dataset x1, x2,…,xn, i.e., t is some function that of the dataset only: • t = h(x1, x2,…,xn) • ESTIMATOR: Let t = h(x1, x2,…,xn) be an estimate based on the dataset x1, x2,…,xn.Then t is a realization of the random variable • T= h(X1, X2,…,Xn). • The random variable T is called an estimator. • Estimator refers to the method or device for estimation • Estimate refers to the actual value computed from a dataset**19.2 Investigating the behavior of an estimator**Estimating the probability p0 of zero arrivals, which is an unknown number between 0 and 1. Possible estimators:**19.3 The Sampling Distribution and Unbiasedness**• Desireable: E[S]=p0 • The Sampling Distribution: Let T= h(X1, X2,…,Xn) be an estimator based on a random sample X1, X2,…,Xn. The probability distribution of T is called the sampling distribution of T. • Sampling Distribution of S: Where Y is the number Xi equal to zero • Thus is follows that:**19.3 The Sampling Distribution and Unbiasedness**Definition: An estimator T is called an unbiased estimator for the parameter Ө,if E[T] = Ө Irrespective of the value of Ө. The difference E[T] – Ө is called the bias of T; if this difference is nonzero, then T is called biased**19.3 The Sampling Distribution and Unbiasedness**Definition: An estimator T is called an unbiased estimator for the parameter Ө,if E[T] = Ө Irrespective of the value of Ө. The difference E[T] – Ө is called the bias of T; if this difference is nonzero, then T is called biased**19.4 Unbiased Estimators for Expectation and Variance**Suppose X1, X2,…,Xnis a random sample from a distribution with finite expectation µ and finite variance σ2. Then: Is an unbiased estimator for µ and Is an unbiased estimator for σ2