Consistency Of Conditional Density Estimation On The Single Index Model For Functional Time Series Data

Nowadays statistical functional data model about theories and application isone of the hot fields in the research of numerical statistics. Well-developedinformation technology and high-powered computers provide great convenience forus on collecting, storing and transferring the functional data，such as curve data,curved surface data, which will vary continuously along with the variation of timeand space. In addition, the functional data is also facilitated by scientificcalculation. As we know, much more information is supplied by functional data.Meanwhile, huger challenges and opportunities about carrying on the analysis ofdata, constructing model and acquiring effective information also comes functionaldata is different from the traditional finite dimensional data analysis, its dimensionis infinite. Therefore, we need to focus our attention on the structure and variationof the data of function type.The research of this thesis is based on the conditional density functionalestimation of the single-functional index regression model. In fact, the infinitedimensional function variables can be transformed into one-dimensional index byapplying the single-functional index regression model, we can also acquireimportant characteristics of infinite dimensional functional data.Therefore, thiskind of model get more and more attention. The main achievements of this thesisare as follows:1. Based on α-mixing dependence of functional time series data, we constructalmost complete convergence and rate of the double kernel functional estimation ofconditional density on the single index model for functional time series data.2. Based on α-mixing dependence of functional time series data, we acquireuniform almost complete convergence and rate of the double kernel functionalestimation of conditional density on the single index model for functional timeseries data.3. As an application of the uniform convergence rate of the conditional density, wecan also obtain the uniform convergence rate for the estimation of the mode, whichis based on α-mixing dependence of functional time series data.