The Bayesian Method For Statistical Indices And Its Application

The index is an old and modern concept in statistics, the theories of indices already have a history of more than 350 years so far from the putting forward of index concept to now. The establishment of the index , especially price indices, there is long history economically. As one of the important tools of economic analysis , it concern one's own accuracy of economic analysis directly that count research deepenning , system of the indices. And only in this way we just can correctly understand change law of things overallly, in order to make the macro economic policy , especially suppress the inflation and manage the deflation to offer the reliable basis.I think the individual index is a random variable, current statistics indices are all sample mean that collect in population. The essence fore making indices is to make a point estimator of some phenomenon. Modern Bayesian statistical method has offered statistical method which is convenient and ocular and pay attention to the historical materials and experiences in the past as well as realistic data from the survey of sample.This point estimator can use bayesian method to finish. We can study the quality of the indices method and study the indices of counting about character and other relevant problems based on this.There are altogether six parts in this thesis.First, the review of the index theory and its essenceThe so—called index is a relative numbers to reflect the quantity changes of social and economical phenomena. According to different weights, there are many kinds of indices such as Laspeyres Index , Passche Index ,and so on.Because difference of study angle and thinking of indices, foreign theory worker have formed two shcools in practice: Stastistic Index School (such as the randomized method) and Economic Index School (function method ). In recent years, there are scholars to begin to use mathematics statistical methods in our country to study indices , they regard individual index as random variable and regard every index formula as stastistical estimator study the index problem.I have put forward my own views about the essence of index :(1) The individual index is a random variable; (2)The calculation of the index is a weighted average to the individual index in essence; (3)The total index calculated according to the sample is to estimate one of the overall parameter in fact; (4)In practice in every kind of indices , a large number of historical datums have been accumulated ,which has accumulated a large amount of beneficial experience for the establishment of the indices and function of the full play index,etc..Scond ,bayesian theory and the connection with statistics indicesBayes school's most basic view:The unknown parameter 3 of population distribution is a random variable ,we describe the unknown state with probability distribution ,which is prior information about 3 before sampling, known as prior distribution. Statistical inference is on the basis of population information , sample information and prior information for BayesIf the sample distribute density of random variable X is p(x |θ), andÏ€(θ) is priori distribute of parameterθ, then:m(x)This is a density function of Bayesian formula, or posterior distribution .It is the key to carrying on Bayesian inference, the method is mainly as follows.no information prior,conjugate prior, histogram method , relative likelihood method , multi-layer priori distribute Jeffy principle,MIDL and so on .Bayesian estimator include point estimator and interval estimator. Bayesian statistics based on population information , sample information and prior information ,propose new studying methods and handling problems for indices.Third , Choice of the population distribution and prior distribution of indicesIn this chapter the author introduce how to find prior distribution and population distribution, combining Shanghai Stock Exchange 180 index mainly. In order to make it convenient for the empirical study which bayesian estimation needs, after introducing 180 indexes and current algorithms of Shanghai Stock Exchange, the author proposed the algorithm of this text. Using SPSS,Eviews,Statistics,etc.software,through NPP picture and Kolmogorov—Smimov test and qualitative analysis, the author thinks that it is more rational to regard normal distribution as the population distribution whether the fixed base 180 index or chain 180 of Shanghai Stock Exchange. By histograms and conjugate method the author confirm normal distribution as prior distribution of the index.Forth,the Bayesian estimator of stastistical indexIn this chapter, the author mainly introduce the mathematica model of point and interval estimator, and empirical study of Bayesian estimator.Under the posterior squared error principle ,the index K' point estimator is :Especially,if the population distribution is N(K,σ²), prior distribution of Kis N(μ,Ï„²) ,the K' point estimator is:Consideration on the empirical study, the author use 180 indices of Shanghai Stock Exchange to estimate .The sample stocks must be listed before Dec.31th 2005,and don not include loss share. Choosing sample stocks must follow the following principle: typical industries principle; mobility and development principle; and scale principle. Through analysing , the author thinks that 50 is a proper number. When confirm the individual sample concretly , pick out those abnormal stocks at first,.Then, assign the sample to only count according to the proportions of market value of circulation of every industry. Concrete until every trade, the author adopt comprehensive appraisal method and qualitative analysis method to select sample stocks.On making the fixed base index , investigates were carried at first to get 180 stock closing prices from January 4 to February 28 of Shanghai Stock Exchange,and investigate the situations of 32 bargain days of March regarding the closing prices of January 2 as the base period' closing prices.(1) Get the prior distribution N(μ,Ï„²);(2) Confirm the population distribution N(K,σ²),K is unknown_parameter,σ² isunknown ,but we can regard S² = 1/nΣ(x₁- x)² as its estimator;(3) Confirm samples and sample mean , n =50;(4) Estimate index K according toFifth, the character of statistical indices and the system of indices assessmentIn this chapter, the author recommends test theory of Irving.Fisher and mathematics evaluation criterion of index at first. On the basis of appraising to above-mentioned theories, the author thinks that a good estimator should satisfy identically test, commensurability test,proportionality test, determinan -cy test and association test on one hand, and should have average , comprehesivity , effectiveness and uniformity on the other hand.After analysing and proving the indices making by Bayesian method,the author think that it is a good statistical indice.Sixth, application of Bayesian method of statistical indicesResident's consumer price index is an index which reflects the important degree of national economy, it is a reduction index in the national business accounting too.It influence the government to make such policies as the currency , finance , consumption , price , salary , social security ,etc. and the appraisal of resident's living standard directly at the same time.Our country current price indices have certain limitations as follows: (1) The representativeness of the sample;( 2) Science of the choose of weight number; ( 3) Historical data or the experience are under-utilized.Bayesian statistical method has provided a kind of new thinking and method for us, especially indrawing up macroscopical price index . While working out the price index, according to bayesian method ,so long as we have abundant prior information, proper sample (scientific select sample in other words ), we can get more accurate indices needn't to choose the weight numbers everywhere. I have investigated the data for 2000-2004 years for explaining the application of Bayesian method in price indices. Government department or enterprise need find out about price tendency, using Bayesian method can not only estimate the price indices accurately but also save the manpower, financial resources and material resources . In addition, it is simple and clear to calculate the price indices using the conjugate normal distribution .Stock index futures, is a new financial future instruments transaction based on stock price. It is the most important and successful financial derivatives in the wave of financial innovation in 1980s, also it is the youngest and most developed financial derivatives. Stock index futures are playing more and more important role in the international capital market as a risk management instrument. In China, with the development of the size of stock market and its rising role insocial and economic activities, the demand for applying stock index futures todo arbitrage is larger and larger, however, after China's accession to WTO,china's capital market will be more and more open, introduction of stock index futures become very urgent.But several main stock indices of our country still have a serious defect as the subject matter of futures stock index futures. Considering the experience of foreign countries and ripe market of Hong Kong, and our country actual conditions of stock market at the same time, the authorthinks that futures of subject matter should meet following principle (1)Representativeness principle; 2)Mobility and stability principle; 3)Higher hedging efficiency and lower hedging cost principle; 4)Growth principle; 5)Sample stock principle.Making stock price index by Bayesian method , scientific sampling not only can solve representativeness and mobility but also guarantee the stock index having higher hedging efficiency and lower hedging cost Bayesian estimator adopt quantitative analysis combining with qualitative analysis to choose samples,which may winkle loss share and guarantee the growth of the share. In addition,Bayesian estimator sovle the choice problem of weight number because it based on prior and posterior information and investigating probability distribution . The share price indices made by bayeesian method , can meet the need that people's arbitrage and speculation , further more its fluctuation is relatively little,so it is relatively stable and it is difficult to be handled by people.