Statistic Inference For High-Dimensional Matrices And SDEs

Statistical inference is a method of extracting samples from the population and making reasonable analysis of the samples,so as to infer the characteristics of the population from the characteristics of the sample.Statistical inference is widely used in practice.Hypothesis testing and parameter estimation are two main research contents of statistical inference.In this thesis,we study the statistical inference of high dimensional matrices and stochastic differential equations.The classical likelihood ratio test of high-dimensional matrices,parameter estimation of stochastic differential equations,and in order to estimate the parameters of stochastic partial differential equations,we study the well-posedness of the solutions of stochastic partial differential equations.The emergence of high-dimensional data has brought a huge impact on the traditional hypothesis testing.In the past,many scholars analyzed the test statistics when the sample size n tends to infinity and the dimension m remains fixed.However,with the increasing demand for high dimensions in data analysis,fixed or small dimensions are no longer suitable for the amount of data and sample size collected in reality.Therefore,in view of the fact that both the sample size and dimension tend to infinity,Bai et al.used the central limit theorem of the linear spectrum statistics of the sample covariance matrix and random F matrix to make corresponding amendments to the likelihood ratio test.To deal with high-dimensional effects,Jiang et al.studied the likelihood ratio test for normal distributions of two m dimensional variables with equal covariance matrices.On the premise that the limit of the ratio of dimension to sample size belongs to（0,1）,Jiang and Yang show that the test statistic W_n converges to a normal distribution in the case of high dimensions.We utilize the properties of polygamma functions and the conclusions of Jiang and Yang obtained succinctly by Lyapunov’s central limit theorem.In the case of higher dimensions,when the sample size n and the dimension m approach infinity,and m/n→y∈(0,1],the likelihood ratio test statistic will converge to a normal distribution.Stochastic differential equations can be used to simulate various behaviors of stochastic models,such as stock prices,stochastic growth models,or physical systems affected by thermal fluctuations.Due to the wide application of stochastic differential equations,the research on parameter estimation of stochastic differential equations has attracted been much attention,and scholars have obtained a series of effective methods to study parameter estimation of stochastic differential equations.Florens studied the diffusion coefficient using kernel estimation method.Pedersen gave a maximum likelihood estimation method based on discrete observations for a class of stochastic differential equations,and derived an approximate sequence of likelihood functions.The convergence of sequence is proved.Ait-Sahalia uses polynomial approximation method to study stochastic differential equations and proves that the maximum likelihood estimator has convergence.For parameter estimation of stochastic differential equations,we consider that under the condition that the observed time is sparse and irregular,the asymptotic normal estimation is obtained by means of maximum likelihood estimation,and then a kernel weighted score function is proposed for parameters in the drift term by using kernel estimation,and the consistency and asymptotic normality of the estimators are proved.Simulation results show that our method performs well when the sample size is large.We consider the following linear stochastic differential equation,（?）,where μ is an unknown parameter,σ is a constant,and W（t）are independent standard Brownian motions.Our aim is to estimate μ using observations where the observations are expressed as y={y_i（t_ik）;i=1,…,n;k=1,…,d_i} and di<∞.The data yi（t）,i=1,…,n are usually not continuously observed,and it is virtually impossible to observe each individual at t*.Therefore,log-likelihood function ln（y*,θ）is not computable from the observations.For estimation,we construct a smoothed log-likelihood function by using kernel estimation as follows.（?）where ∑_i²（t*）<∞ is the variance of y_i*,Khn（t）=K（（t-t*）/hn）/hn,hn is the bandwidth,and the kernel function K（t）is a symmetric probability density with support[-1,1],mean 0,and bounded the first derivative.Assume that the following conditions hold:（1）Θ_μ0 is an open sets of R,and Θ_μ0={μ:|μ-μ₀|<ρ} for p>0 and μ₀ is the true parameter.（2）λ*（t）is twice continuously differentiable.)3_K（z）is a symmetric density function satisfying ∫_-∞^∞ K（z）2dz<∞.In addition,hn→0,nhn→∞,nhn5→0,when n→∞.By an appropriate choice of the bandwidth,we prove the consistency and the asymptotic normality of the estimator μn.Numerical simulation results show that our proposed method performs well in practice with large samples.Stochastic partial differential equations are widely used in the fields of fluid mechanics,quantum field theory,biology,financial mathematics and so on,among which Kardar-Parisi-Zhang equation and parabolic Anderson model are typical representatives.In order to better study the parameter estimation problem of stochastic partial differential equations,we first study the well-posedness of the solutions of a class of stochastic partial differential equations.We consider the asymptotic property of the total mass of the solution of the parabolic Anderson model.The parabolic Anderson model is the heat equation with random potential,which describes the influence of random or irregular environment on the diffusion process.Assumeξ=ξ（z）:z ∈Z^d)is an i.i.d random potential with values in（-∞,∞）.Let u:[0,∞)×Z^d→[0,∞)be the solution of the heat equation with random potential and the Cauchy problem with localised initial datum.（?）where Δ be the discrete Laplace operator,（?） for all（t,z）∈（0,∞）×Z^d,here y～x means that y and x are the nearest neighbours.By the Feynman-Kac representation of the total mass and the related properties of local time and self-intersection local time,we obtain the asymptotic property of the total mass for the solution of the parabolic Anderson model.The innovations of this thesis are:（1）For the likelihood ratio test statistics in the case of higher dimensions,by using the properties of the polygamma function as well as Lyapunov’s central limit theorem,we proved that the statistic converges to a normal distribution when the dimension and the sample size tend to infinity and the ratio of them belongs to(0,1].（2）We propose a new method for the parameter estimation for linear stochastic differential equations with independent experiments,observed at infrequent and irregularly spaced follow-up times.Maximum likelihood method is used to obtain an asymptotically consistent estimator.A kernel weighted score function is proposed for the parameter in drift term.At the heart of the proposed approach is to“smooth”an individual’s contributions to the likelihood based on the distance of their observed time to the time of interest.The smoothing methods employed where smoothing happens at the individual basis as compared to the population level where all individuals are given the same weights.With a suitable choice of bandwidth,the consistency and asymptotic normality for the proposed estimator can be obtained.Numerical results show that the proposed estimator performs well with large sample sizes.（3）For the total mass of the unique solution of the parabolic Anderson model,we transfer the Feynman-Kac representation of the total mass to the expression containing local time and self-intersection local time,and then by using the exponential moment theory,we obtain the asymptotic property of the total mass.