概率与统计(英文)chapter-6-Point-Estimation

单击此处编辑母版标题样式,单击此处编辑母版文本样式,第二级,第三级,第四级,第五级,*,6 Point Estimation,6.1 Some general concepts of point estimation,6.2 Methods of point estimation,Introduction,Given a parameter of interest,such as a population mean or population proportion,p,the objective point estimating is to use a sample to compute a number that represents in some sense a good guess for the true value of the parameter.The resulting number is called,a point estimate,.In Section 6.2,we describe and illustrate two important methods for obtaining point estimates:the method of moments and the method of maximum likelihood.,6.1 Some General Concepts of Point Estimation,Statistical inference,is almost always directed toward drawing some type of conclusion about one or more parameters(population characteristics).To do so requires that an investigator obtain sample data from each of the populations under study.Conclusions can then be based on the computed values of various sample quantities.,When discussing general concepts and methods of inference,it is convenient to have a generic symbol for the parameter of interest.We will use the Greek letter for this purpose.The objective of point estimation is to select a single number,based on sample data,that represents a sensible value for.,Definition,A,point estimate,of a parameter is a single number that can be regarded as a sensible value for .A point estimate is obtained by selecting a suitable statistic and computing its value from the given sample data.The selected statistic is called the,point estimator,of.,Example 6.1,An automobile manufacturer has developed a new type of bumper,which is supposed to absorb impacts with less damage than previous bumpers.The manufacturer has used this bumper in a sequence of 25 controlled crashes against a wall,each at 10 mph,using one of its compact car models.Let X=the number of crashes that result in no visible damage to the automobile.The parameter to be estimate is p=the proportion of all such crashes that result in no damage alternatively,p=P(no damage in a single crash).,Solution:,If X is observed to be,x,=15,the most reasonable estimator and estimate are,estimator,estimate=,Example 6.2,Reconsider the accompanying 20 observations on dielectric breakdown voltage for pieces of epoxy resin first introduced in Example 4.29,The pattern in the normal probability plot given there is quite straight,so we now assume that the distribution of breakdown voltage is normal with mean value.Because normal distribution are symmetric,is also the median lifetime of the distribution.The given observation are then assumed to be the result of a random sample X,1,X,2,X,20,.from this normal distribution.,Consider the following estimators and resulting estimates for,24.46,25.61,26.25,26.42,26.66,27.15,27.31,27.54,27.74,27.94,27.98,28.04,28.28,28.49,28.50,28.87,29.11,29.13,29.50,30.88,a.,Estimator=,estimate=,b.,Estimator=,estimate=,c.,Estimator min(X,i,)+max(X,j,)/2=the average of the two extreme lifetimes,estimate=min(x,i,)+max(x,i,)/2=(24.46+30.88)/2=27.670,d.,Estimator=,the 10%trimmed mean(discard the smallest and largest 10%of the sample and then average),Example 6.3,In the near future there will be increasing interest in developing low-cost Mg-based alloys for various casting processes.It is therefore important to have practical ways of determining various mechanical properties of such alloys.Assume that the observations of a random sample X,1,X,2,X,8,from the population distribution of elastic modulus under such circumstances.We want to estimate the population variance,2,Solution:,A natural estimator is the sample variance:,The corresponding estimate is,Another estimator is,And estimate is,Unbiased Estimators,Definition,In the best of all possible words,we could find an estimator for which always.However,is a function of the sample X,i,s,so it is a random variable.For some samples,will yield a value large than ,whereas for other samples will underestimate .If we write,+error of estimation,Then an accurate estimator would be one resulting in small estimation errors,so that estimated values will be near the true value.An estimator that has the properties of,unbiasedness,and minimum variance will often be accurate in this sense.,A point estimator is said to be an,unbiased estimator,of if for every possible value of.If is not unbiased,the difference is called the,bias,of,Thus,is unbiased if its probability distribution is always“centered”at the true value of the parameter.Figure 6.1 pictures the distribution of several biased and unbiased estimators.Note that“centered”here means that the expected value,not the median,of the distribution of is equal to,pdf,of,pdf,of,Bias of,1,pdf,of,pdf,of,Bias of,1,Figure 6.1,the,pdfs,of a biased estimator and an unbiased estimated for a,PROPOSITION,When X is a binomial,rv,with parameters,n,and,p,the sample proportion =X/,n,is an unbiased estimator of,p,.,Example 6.4,Suppose that X,the reaction time to a certain stimulus,has a uniform distribution on th