slide2 - Parameterestimation information Example:...

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1 Parameter estimation The word “estimation” is normally understood to  mean “approximation” or “educated guessing” of  some value of interest given incomplete or indirect  information. Example: one could try to estimate a person’s age by  his or her visual appearance. In statistics, one usually is interested in estimating  population parameters  based on a  random sample  from the population.
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2 Parameter estimation Population – collection of all possible observations of  the value of interest. Can be either finite (but usually  large) or infinite. We can think of every observation as coming from  some (possibly unknown) probability distribution – the  distribution of the population. That distribution typically has some parameters (e.g.  mean     and variance     ).  These parameters are often unknown. So one might  want to  estimate  them, i.e. find an approximate value. μ 2 σ
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3 Parameter estimation To estimate a parameter, we can take a  random  sample  from the population. For a sample to be a random sample two conditions  have to be satisfied. The members of the sample have to be drawn  independently. Each one of them has to be drawn at random from the whole  overall population (and not from some subset of it). Question:  give an example of a sample which is  not  random.
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4 Parameter estimation Mathematically we can describe a generic sample of  size  n  as a collection of  n  random variables Then any specific sample would be a  realization  of  that: The sample is a random sample if The random variables                       are mutually  independent They all have the same distribution which is the distribution  of the whole population   n X X X ,..., , 2 1 n x x x ,..., , 2 1 n X X X ,..., , 2 1
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5 Parameter estimation statistic  is any function of a random sample, i.e. of  random variables This means that it is also a random variable. Question:  give examples of different statistics. n X X X ,..., , 2 1
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6 Parameter estimation We would like to use a random sample to estimate a  value of an unknown population distribution  parameter. An  estimator  is a prescription of how to do that. So it has to be a function of the random sample, i.e. a  statistic, i.e. a random variable. A specific realization of the estimator gives us an  estimate  – a numerical proxy for the unknown  parameter that can be used in its stead.
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7 Parameter estimation Strictly speaking any statistic can serve as an  estimator for any parameter. Obviously, not all statistics are going to work equally 
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This note was uploaded on 02/07/2011 for the course IE 121 taught by Professor Perevalov during the Spring '08 term at Lehigh University .

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slide2 - Parameterestimation information Example:...

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