Limit Behaviors Research Of Sample Quantiles

The time series data are widely used in the economic, finance, and biologicalstatistics and so on. However, due to the time series data often are heavy-tailed, sosubject to the moment conditions, the classical statistical method could not use.While quantile can be used for describing some properties of random variables, andthere are not the restrictions of moment conditions. As a result, it is being widelyemployed in diverse problems in finance, such as, quantile-hedging, optimalportfolio allocation, risk management, and so on. Probability limit theory is one ofthe main branches of probability theory; it is also one of the important theoreticalbases of mathematical statistics. So the research of limit properties for samplesquantile is an important direction in probability limit theory research in recent years.In practice, it is also an important approach for the analysis of statistical problems.This paper mainly studies the large deviations and moderate deviations of samplequantile.In recent years, due to the improvement of computer performance, independentidentically distributed samples of the related properties of the quantile estimatorhave been extensive research. However, in practice, we get the sample often cannotmeet the needs of independent identically distributed condition; the overalldistribution is not necessarily continuous. So we need the samples in the moregeneral condition of hypothesis to study the related properties of sample quantile.In this paper, we mainly studied the following two problems:(1) The large sample properties of m-dependent random variables sequencessamples quantile. Using the classical Cramér theorem, under the condition of them-dependence, we get the large deviations principle of sample quantile. Accordingto proof of exponentially tight, we get the large deviations principle.(2) This paper discussed large deviations principle for samples with discontinuous distributions. Using the Chebyshev inequality and Stirling formula,we proved the large deviations principle of sample quantile for samples withdiscontinuous distribution.Based on the continuous case of population distribution and under the conditionof the independent identically distributed, we generalized the large deviationprinciple and moderate deviation principle of samples quantile estimates to them-dependent sequence. Under the situation of population distribution isdiscontinuous, we considered the large deviation properties of general quantileestimates. Those large sample properties lay a foundation for the application ofquantile.