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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 :$k(x)={\frac {1}{2}}\mathbf {1} _{\left\{|x|<1\right\}}$ Triangular : $k(x)=(1-|x|)\mathbf {1} _{\left\{|x|<1\right\}}$

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

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

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

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

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

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

${\mbox{minimiere }}\mathbf {MISE} ({\hat {f}})\Longrightarrow h_{opt}=\left[{\frac {R(K)}{\mu _{2}^{2}(K)R(f^{\prime \prime })n}}\right]^{\frac {1}{5}}$

conditional density estimator:

${\hat {f}}(y\,|x)={\frac {\sum _{i=1}^{n}K\left({\frac {\|x-X_{i}\|_{x}}{a}}\right)K\left({\frac {\|y-Y_{i}\|_{y}}{b}}\right)}{b\sum _{j=1}^{n}K\left({\frac {\|x-X_{j}\|_{x}}{a}}\right)}}$

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

Refenrence Rule:

$\mathbf {MISE} ({\hat {f}})=\mathbb {E} \iint \{{\hat {f}}(y|x)-f(y|x)\}^{2}h(x)\mathrm {d} x\mathrm {d} y\approx {\frac {c_{1}}{nab}}-{\frac {c_{2}}{na}}+c_{3}a^{4}+c_{4}b^{4}+c_{5}a^{2}b^{2}$ , wobei

$c_{1}=\int R^{2}(K)\mathrm {d} x$ , $c_{2}=\iint R(K)f(y|x)\mathrm {d} y\mathrm {d} x$ , $c_{3}=\iint {\frac {\mu _{4}(K)h(x)}{4}}\left\{2{\frac {h^{\prime }(x)}{h(x)}}{\frac {\partial f(y|x)}{\partial x}}+{\frac {\partial ^{2}f(y|x)}{\partial x^{2}}}\right\}^{2}\mathrm {d} y\mathrm {d} x$ ,

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

$c_{5}=\iint {\frac {\mu _{4}(K)h(x)}{2}}\left\{2{\frac {h^{\prime }(x)}{h(x)}}{\frac {\partial f(y|x)}{\partial x}}+{\frac {\partial ^{2}f(y|x)}{\partial x^{2}}}\right\}\left\{{\frac {\partial ^{2}f(y|x)}{\partial y^{2}}}\right\}\mathrm {d} y\mathrm {d} x$ ,

Normal Refenrence Rule:

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

$b_{N}=\left\{{\frac {d^{2}\nu (k)}{3{\sqrt {2\pi }}\sigma _{h}^{5}\lambda (k)}}\right\}^{1/4}a_{N}$,

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

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

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

Bashtannyk-Hydermann-Regression method: ${\mbox{minimiere }}Q_{b}(a)={\frac {\Delta }{n}}\sum _{k=1}^{N}\sum _{i=1}^{n}\left(\nu _{i}(y'_{k})-\sum _{j=1}^{n}w_{j}(x_{i})\nu _{j}(y'_{k})\right)^{2}p(w_{i}(X_{i}))$

wobei: $Y'=\{y'_{1},\cdots ,y'_{N}\}$ eine Folge, die den Raum von Y mit gleicher Distanz $\Delta$, einteilen kann, und$\nu _{i}(y)={\frac {1}{b}}K\left({\frac {|Y_{i}-y|}{b}}\right)\ ,\ w_{j}(x)={\frac {K\left({\frac {\|x-X_{j}\|}{a}}\right)}{\sum _{i=1}^{n}K\left({\frac {\|x-X_{i}\|}{a}}\right)}}$.

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

Hier: ${\tilde {f}}(y|x)$ ist ein parametrischer Schätzer aus der linearen Modell $Y_{i}=\sum _{j=1}^{n}\beta _{j}X_{i}^{j}+\sigma \epsilon _{i}$, 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%	$F_{n}=R{\begin{bmatrix}{\frac {(1+r)^{n}-1}{r}}\end{bmatrix}}$
Present value of an $R annuity lasting n periods at periodic interest rate r%	$P=R{\begin{bmatrix}{\frac {1-(1+r)^{-n}}{r}}\end{bmatrix}}$
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.	$P={\frac {d_{1}}{r-g}}$

$\beta _{j}={\frac {{\mbox{cov}}(r_{j},r_{M})}{{\mbox{var}}(r_{M})}}={\frac {\sigma _{j,M}}{\sigma _{M}^{2}}}$

$CAPM:E(r_{j})=r_{f}+\beta _{j}\lfloor E(r_{M})-r_{f}\rfloor$

${\mbox{var}}(r_{p})=\sigma _{p}^{2}=\omega _{1}^{2}\sigma _{1}^{2}+\omega _{2}^{2}\sigma _{2}^{2}+2\omega _{1}\omega _{2}\rho _{1,2}\sigma _{1}\sigma _{2}$

$\rho _{1,2}={\frac {{\mbox{cov}}(r_{1},r_{2})}{\sigma _{1}\sigma _{2}}}={\frac {\sigma _{1,2}}{\sigma _{1}\sigma _{2}}}$

$r_{*}^{L}=r_{e}^{L}\left({\frac {E^{L}}{V^{L}}}\right)+r_{d}(1-\tau )\left({\frac {D^{L}}{V^{L}}}\right)$

$r_{e}^{L}=r_{*}^{U}+(r_{*}^{U}-r_{d})(1-\tau )\left({\frac {D^{L}}{E^{L}}}\right)$