应用非参数统计-第4讲 概率分布和多元概率密度的非参数估计课件

Let X1 , ..., Xn be a random sample from a population X with density f (x). The kernel density estimator of f (x) is: 1 ˆ f h (x) = n where
n
Kh (x − Xi ),
The mean integrated squared error (MISE): ˆ MISE(f h) = = ˆ MSE{f h (x)}dx h4 1 K 2 + [µ2 (K )]2 f (x) 2 nh 4 1 +o + o(h4 ), nh as h → 0, nh → ∞.
2 2
where f (x)
Chapter 4: Nonparametric density estimation
Baisen Liu School of Statistics, Dongbei University of Finance & Economics September 23, 2014
Contents
Chapter 4: Nonparametric density estimation Baisen Liu Contents Histogram Smoother univariate density estimation The choice of smoothing parameter The constructing the confidence intervals Univariate cumulative distribution function estimation
where µ2 (K ) =
s2 K (s)ds and K
2 2
=
K 2 (s)ds.
The optimal local bandwidth hopt (x): ˆ hopt (x) = arg min MSE{f h (x)}.
The statistical properties
Chapter 4: Nonparametric density estimation Baisen Liu Contents Histogram Smoother univariate density estimation The choice of smoothing parameter The constructing the confidence intervals Univariate cumulative distribution function estimation
− u2 )I (|u| ≤ 1) − u2 )2 I (|u| ≤ 1) − u2 )3 I (|u| ≤ 1)
1 2 exp(− 2 u )
√1 2π
cos( π 2 u)I (|u| ≤ 1)
The kernel density estimator of f (x)
Chapter 4: Nonparametric density estimation Baisen Liu Contents Histogram Smoother univariate density estimation The choice of smoothing parameter The constructing the confidence intervals Univariate cumulative distribution function estimation
The definition of the density f (x):
f (x) ≡ d F (x + h) − F (x − h) F (x) ≡ limh→0 . dx 2h
The histogram estimate of f (x): ˆ(x) = #{xi ∈ (x − h, x + h]} . f 2nh The kernel density estimator of f (x): ˆ(x) = 1 f nh where K (u) =
2 2
= [f (x)]2 dx.
The approximate MISE (AMISE): ˆ AMISE(f h) = 1 K nh
2 2
+
h4 [µ2 (K )]2 f (x) 2 2. 4
The statistical properties
Chapter 4: Nonparametric density estimation Baisen Liu Contents Histogram Smoother univariate density estimation The choice of smoothing parameter The constructing the confidence intervals Univariate cumulative distribution function estimation
1
Histogram Smoother univariate density estimation The choice of smoothing parameter The constructing the confidence intervals Univariate cumulative distribution function estimation Multivariate kernel density estimation
Chapter 4: Nonparametric density estimation Baisen Liu Contents Histogram Smoother univariate density estimation The choice of smoothing parameter The constructing the confidence intervals Univariate cumulative distribution function estimation
Kernel density estimation
Chapter 4: Nonparametric density estimation Baisen Liu Contents Histogram Smoother univariate density estimation The choice of smoothing parameter The constructing the confidence intervals Univariate cumulative distribution function estimation
L: Kernel functions
Kernel Uniform Triangle Epanechnikov Quartic(Biweight) Triweight Gaussian Cosine
π 4
K (u)
1 2 I (|u|
≤ 1)
(1 − |u|)I (|u| ≤ 1)
3 4 (1 15 16 (1 35 32 (1
2
3
4
5
6
Histogram
Chapter 4: Nonparametric density estimation Baisen Liu Contents Histogram Smoother univariate density estimation The choice of smoothing parameter The constructing the confidence intervals Univariate cumulative distribution function estimation
i=1
1 K (·/h). h K (·) is called kernel function, and h is call bandwith. Kh (·) =
The construction of kernel density estimate
Chapter 4: Nonparametric density estimation Baisen Liu Contents Histogram Smoother univariate density estimation The choice of smoothing parameter The constructing the confidence intervals Univariate cumulative distribution function estimation
Bias: ˆ Bias{f h (x)} = =
h→0
ˆ E{f h (x) − f (x)}
1 n n
E{Kh (x − Xi )} − f (x) f (x) s2 K (s)ds + o(h2 ).
=
i=1 h2
2
Variance: ˆ Var{f h (x)} = =
nh→∞
Var
=
1 n Var{Kh (x − Xi )} n2 i=1 1 1 f (x) K 2 (s)ds + o nh nh
The statistical properties
Chapter 4: Nonparametric density estimation Baisen Liu Contents Histogram Smoother univariate density estimation The choice of smoothing parameter The constructing the confidence intervals Univariate cumulative distribution function estimation
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The definition of the density f (x): F (x + h) − F (x) d F (x) ≡ limh→0 . dx h The empirical distribution: f (x) ≡ ˆ (x) = #{xi ≤ x} . F n The histogram estimate of f (x): ˆ(x) = (#{xi ≤ bj +1 } − #{xi ≤ bj })/n , x ∈ (bj , bj +1 ], f h where h = bj +1 − bj is called binwidth.
The mean squared error (MSE):
2 ˆ ˆ ˆ MSE{f h (x)} = Var{fh (x)} + [Bias{fh (x)}] 4 1 h = K 2 [f (x)]2 [µ2 (K )]2 2 f (x) + nh 4 1 + o(h4 ) + o . nh as h → 0, nh → ∞.
1 n Kh (x − Xi ) n i=1
.
The statistical properties
Chapter 4: Nonparametric density estimation Baisen Liu Contents Histogram Smoother univariate density estimation The choice of smoothing parameter The constructing the confidence intervals Univariate cumulative distribution function estimation
n
K
i=1
x − xi h
,
1 2,
0
if − 1 < u ≤ 1, otherwise.
is called the uniform kernel function.
Chapter 4: Nonparametric density estimation Baisen Liu Contents Histogram Smoother univariate density estimation The choice of smoothing parameter The constructing the confidence intervals Univariate cumulative distribution function estimation