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如何编辑数学公式

范例： Kernel Density estimator :$\hat{f}(x)=\frac{1}{nh}\sum _{i=1}^{n}K\left(\frac{x-X_i}{h}\right)$ Variable Kernel Density estimator :$\hat{f}(x)=\frac{1}{n}\sum _{i=1}^n\frac{1}{h_i}K\left(\frac{x-X_i}{h_i}\right)$

NormalKerdnfunktion :$k(x)=(2\pi {)}^{-\frac{1}{2}}{e}^{-\frac{x^2}{2}}$ Uniform :$k(x)=\frac{1}{2}{\mathrm{𝟏}}_{\{|x|<1\}}$ Triangular : $k(x)=(1-|x|){\mathrm{𝟏}}_{\{|x|<1\}}$

Epannecknikov $k(x)=2^3𝐁(2,2{)}^{-1}(1-x^2){\mathrm{𝟏}}_{\{|x|<1\}}=\frac{3}{4}(1-x^2){\mathrm{𝟏}}_{\{|x|<1\}}$

Biweight: $k(x)=2^5𝐁(3,3{)}^{-1}(1-x^2{)}^2{\mathrm{𝟏}}_{\{|x|<1\}}=\frac{15}{16}(1-x^2{)}^2{\mathrm{𝟏}}_{\{|x|<1\}}$

Triwegiht:$k(x)=2^7𝐁(4,4{)}^{-1}(1-x^2{)}^3{\mathrm{𝟏}}_{\{|x|<1\}}=\frac{35}{32}(1-x^2{)}^3{\mathrm{𝟏}}_{\{|x|<1\}}$

MISE: $𝐌𝐈𝐒𝐄(\hat{f})=𝔼\int \{\hat{f}(x)-f(x){\}}^2\mathrm{d}x=\int \{𝔼\hat{f}(x)-f(x){\}}^2\mathrm{d}x+\int 𝐯𝐚𝐫\hat{f}(x)\mathrm{d}x$

da : $𝔼\hat{f}(x)-f(x)=\frac{1}{2}{h}^2{\mu }_2(K)f^{″}(x)+o(h^2)$ und $𝐯𝐚𝐫\hat{f}(x)=\frac{1}{nh}R(K)f(x)+o(\frac{1}{nh})$

wobei: ${\mu }_2(g)=\int t^{2}g(t)dt$ und $R(g)=\int g(t{)}^{2}dt$ sind. $𝐌𝐈𝐒𝐄(\hat{f})\approx \frac{1}{4}{h}^4{\mu }_2^2(K)R(f^{″})+\frac{1}{nh}R(K)$

$\text{minimiere }𝐌𝐈𝐒𝐄(\hat{f})⟹h_{opt}={\left[\frac{R(K)}{{\mu }_2^2(K)R(f^{″})n}\right]}^{\frac{1}{5}}$

conditional density estimator:

$\hat{f}(y|x)=\frac{{\sum }_{i=1}^{n}K\left(\frac{\Vert x-X_i{\Vert }_x}{a}\right)K\left(\frac{\Vert y-Y_i{\Vert }_y}{b}\right)}{b{\sum }_{j=1}^{n}K\left(\frac{\Vert x-X_j{\Vert }_x}{a}\right)}$

wobei: $\int K(x)\mathrm{d}x=1,\int xK(x)\mathrm{d}x=0,{\mu }_2(K)=\int x^{2}K(x)\mathrm{d}x<\mathrm{\infty }$

Refenrence Rule:

$𝐌𝐈𝐒𝐄(\hat{f})=𝔼\iint \{\hat{f}(y|x)-f(y|x){\}}^{2}h(x)\mathrm{d}x\mathrm{d}y\approx \frac{c_1}{nab}-\frac{c_2}{na}+c_3{a}^4+c_4{b}^4+c_5{a}^2{b}^2$ , wobei

$c_1=\int R^2(K)\mathrm{d}x$ , $c_2=\iint R(K)f(y|x)\mathrm{d}y\mathrm{d}x$ , $c_3=\iint \frac{{\mu }_4(K)h(x)}{4}{\{2\frac{h^{\prime }(x)}{h(x)}\frac{\partial f(y|x)}{\partial x}+\frac{{\partial }^{2}f(y|x)}{\partial x^2}\}}^2\mathrm{d}y\mathrm{d}x$ ,

$c_4=\iint \frac{{\mu }_4(K)h(x)}{4}{\left\{\frac{{\partial }^{2}f(y|x)}{\partial y^2}\right\}}^2\mathrm{d}y\mathrm{d}x$ ,

$c_5=\iint \frac{{\mu }_4(K)h(x)}{2}\{2\frac{h^{\prime }(x)}{h(x)}\frac{\partial f(y|x)}{\partial x}+\frac{{\partial }^{2}f(y|x)}{\partial x^2}\}\left\{\frac{{\partial }^{2}f(y|x)}{\partial y^2}\right\}\mathrm{d}y\mathrm{d}x$ ,

Normal Refenrence Rule:

$a_N={\left\{\frac{16k{R}^2(K)p^5(288{\pi }^9{\sigma }_h^{58}{\lambda }^2(k){)}^{1/8}}{n{\sigma }_K^4{d}^{5/2}{\nu }^{3/4}(k)[{\nu }^{1/2}(k)+d(18\pi {\sigma }_h^{10}{\lambda }^2(k){)}^{1/4}]}\right\}}^{1/6}$, und

$b_N={\left\{\frac{d^2\nu (k)}{3\sqrt{2\pi }{\sigma }_h^5\lambda (k)}\right\}}^{1/4}{a}_N$,

wobei $\lambda (k)={\int }_{-k}^k\varphi (t)dt,\nu (k)=\sqrt{2\pi }{\sigma }_h^3(3d^2{\sigma }_h^2+8p^2)\lambda (k)-16k{\sigma }_h^2{p}^2{e}^{-k^2/2}$

Bivariate density estimator: $\hat{f}(x)=\frac{1}{n{h}^2}\sum _{i=1}^{n}K\left(\frac{x-X_i}{h}\right)$ , wobei $\iint K(x)\mathrm{d}x=1$

polynomial Kernel: $\hat{f}(x)=\frac{1}{nh}\left[\text{Anzahl der }{X}_i\text{, die in das Intervall }(x-\frac{h}{2},x+\frac{h}{2})\text{ fallen}\right]$

Bashtannyk-Hydermann-Regression method: $\text{minimiere }{Q}_b(a)=\frac{\mathrm{\Delta }}{n}\sum _{k=1}^N\sum _{i=1}^n{({\nu }_i(y{\text{'}}_k)-\sum _{j=1}^n{w}_j(x_i){\nu }_j(y{\text{'}}_k))}^{2}p(w_i(X_i))$

wobei: $Y^{\prime }=\{y{\text{'}}_1,\cdots ,y{\text{'}}_N\}$ eine Folge, die den Raum von Y mit gleicher Distanz $\mathrm{\Delta }$, einteilen kann, und${\nu }_i(y)=\frac{1}{b}K\left(\frac{|Y_i-y|}{b}\right),w_j(x)=\frac{K\left(\frac{\Vert x-X_j\Vert }{a}\right)}{{\sum }_{i=1}^{n}K\left(\frac{\Vert x-X_i\Vert }{a}\right)}$.

Bashtannyk-Hydermann_Bootstrap methode: $\text{minimiere }M(a,b;m,y^{\prime },\tilde{f})=\frac{1}{m}\sum _{l=1}^n\left\{\frac{\mathrm{\Delta }}{n}\sum _{j=1}^N\sum _{i=1}^n{[\hat{f}(y{\text{'}}_j|X_i)-\tilde{f}(y{\text{'}}_j|X_i)]}^2\right\}$

Hier: $\tilde{f}(y|x)$ ist ein parametrischer Schätzer aus der linearen Modell $Y_i=\sum _{j=1}^n{\beta }_j{X}_i^j+\sigma {ϵ}_i$, und der Vektor $Y^{(l)}=\{Y_i^{(l)},\cdots ,Y_n^{(l)}\}$ ist durch Bootstrapping basierend Stichproben X simuliert.

ASH: $\hat{f}(x)=\frac{{\sum }_{i=1-m}^{m-1}K(i/m){\nu }_{k+i}}{nh{\sum }_{j=1-m}^{m-1}K(j/m)},x\in B_k$

统计和金融部分常见公式

Description	Formula
Future value of $P invested today for n periods at a periodic interest rate of r%	$F_n=P(1+r{)}^n$
Present value of a $F cashflow received in n periods time if periodic interest rate is r%	$P=\frac{F_n}{(1+r{)}^n}$
Future value of an $R annuity lasting n periods at periodic interest rate r%	$F_n=R\left[\begin{array}{c}\frac{(1+r{)}^n-1}{r}\end{array}\right]$
Present value of an $R annuity lasting n periods at periodic interest rate r%	$P=R\left[\begin{array}{c}\frac{1-(1+r{)}^{-n}}{r}\end{array}\right]$
Present value of a perpetual cashflow of $R per period at periodic interest rate of r%	$P=\frac{R}{r}$
Present value of a perpetual cashflow growing at g% when interest rate is r% per period. The first payment is $di and occurs in one period's time.	$P=\frac{d_1}{r-g}$

${\beta }_j=\frac{\text{cov}(r_j,r_M)}{\text{var}(r_M)}=\frac{{\sigma }_{j,M}}{{\sigma }_M^2}$

$CAPM:E(r_j)=r_f+{\beta }_j\lfloor E(r_M)-r_f\rfloor$

$\text{var}(r_p)={\sigma }_p^2={\omega }_1^2{\sigma }_1^2+{\omega }_2^2{\sigma }_2^2+2{\omega }_1{\omega }_2{\rho }_{1,2}{\sigma }_1{\sigma }_2$

${\rho }_{1,2}=\frac{\text{cov}(r_1,r_2)}{{\sigma }_1{\sigma }_2}=\frac{{\sigma }_{1,2}}{{\sigma }_1{\sigma }_2}$

$r_{\ast }^L=r_e^L\left(\frac{E^L}{V^L}\right)+r_d(1-\tau )\left(\frac{D^L}{V^L}\right)$

$r_e^L=r_{\ast }^U+(r_{\ast }^U-r_d)(1-\tau )\left(\frac{D^L}{E^L}\right)$