经管学会网盘概统lecture

1、Lecture 14Descriptive Statistics Statistical InferenceThe Likelihood Function and Maximum Likelihood EstimatorsWhat is Statistics?Statistics:Collecting dataOrganizing and analyzing dataDrawing conclusion from dataBasic ConceptsPopulation & SamplePopulation: the collection of all units of interestSam

2、ple: the observed unitsParameter & StatisticsParameter: measurement of populationStatistics: measurement s of sample为什么需要抽样(sampling)？ 1）总体(population)无法得到。例：光临麦当劳的所有顾客（无限总体）。 2）时间和成本不允许。例：美国总统选举的民意测验。 3）实验具有破坏性。例：测量产品的寿命。Classification of StatisticsStatistics Descriptive StatisticsStatistical Infer

3、enceDescriptive StatisticsTo show and describe the information of the data by using forms, charts or values location No. of calls weeklyservice typegendercity20Brand Amalerural area20Brand Amalerural area40Brand Amalecity30Brand Bmalecity10Brand Bmalecity20Brand Bfemalecity20Brand Cmalecity40Brand A

4、malecity60Brand Bmalecity20Brand Bmalecity20Brand Bmalecity20Brand Bmalecity20Brand Bmalecity20Brand Cmalecity20Brand Bmalecity25Brand Bmalecity30Brand Bmalecity7Brand Bfemalecity20Brand Bmalecity10Brand BmaleExample. Three Services for Mobile Phones8Describing “Service Type”service typetotalrelativ

5、e frequencypercentagecumulative percentageBrand A40.22020Brand B140.77090Brand C20.110100total201100Describing “Service Type”Sample Mean:Sample Variance:Sample standard deviation: Describing “No. of Calls Weekly”Describing “No. of Calls Weekly”Describing “No. of Calls Weekly”Describing the Relations

6、hip between Two VariablesDiscrete Case: Contingency Table service typemalefemaletotalBrand A404Brand B12214Brand C202Total18220Continuous Case: Scatter PlotThe sample correlation coefficient is valued between：-1，1.Sample Correlation Coefficient1617181920Statistical InferenceTo understand the populat

7、ion parameters through the sample statistics. 统计分析的任务通过样本的统计量来了解总体的参数。总体参数p样本统计量Parametric InferenceThe probability distribution which generated the observed data is completely known except for the values of one or more parameters.E.g. The length of life of a certain type of nuclear pacemaker has an

8、 exponential distribution with parameter b, but the exact value of b is unknown.If we observe the lifetimes of several pacemakers of this type, we can make inference about the value of b.- What is the best estimate of the value of b?- Specify an interval in which we think the value of b is likely to

9、 lie.- Decide whether or not b is smaller than some specified value.E.g. The distribution of the heights of the individuals in a certain population is assumed to be a normal distribution with mean and variance , but the exact values of and are unknown.If we observe the heights of a random sample of

10、individuals, we can make inferences about the values of and . A characteristic or combination of characteristics that determine the distribution generating the observed data is called a parameter of the distribution.The set of all possible values of a parameter or a vector of parameters is called th

11、e parameter space.Examples of the Parameter SpaceE.g. The length of life of a certain type of nuclear pacemaker has an exponential distribution with parameter b. The parameter space will be the set of all positive numbers, soE.g. The distribution of the heights in inches of the individuals in a cert

12、ain population is known to be a normal distribution with mean and variance . We might be certain that and So When the joint p.d.f. of the observations in a random sample is regarded as the function of given values of x1,xn, it is called the likelihood function.The Likelihood FunctionMaximum Likeliho

13、od EstimatorsFor each possible observed vector x=(x1,xn), let denote a value of for which the likelihood function is a maximum.Let be the estimator of q. This estimator is called the maximum likelihood estimator of q.Abbreviation: M.L.E. - maximum likelihood estimator or maximum likelihood estimateE

14、xample. Sampling from a Bernoulli DistributionSuppose that random variables X1,Xn form a random sample from a Bernoulli distribution for which the parameter q is unknown.For any observed values x1,xn, the likelihood function is The logarithm of the likelihood function is To find the value of q which

15、 maximizes and thus maximizes L(q), So the M.L.E. of q is Example. Sampling from a Normal DistributionSuppose that random variables X1,Xn form a random sample from a normal distribution for which the mean m is unknown and the variance is known.For any observed values x1,xn, the likelihood function i

16、s The logarithm of the likelihood function is To find the value of m which maximizes and thus maximizes L(m), So the M.L.E. of m is Example. Sampling from a Normal Distribution with Unknown VarianceSuppose that random variables X1,Xn form a random sample from a normal distribution for which both the

17、 mean m and the variance are unknown.For any observed values x1,xn, the likelihood function is The logarithm of the likelihood function is To find the value of m and which maximizes and thus maximizesSo the M.L.E.s of m and are Example. Sampling from a Uniform DistributionSuppose that random variables X1,Xn form a random sample from a