高斯到均匀分布转换在均匀分布的边缘有误差

Question

我有一些从高斯分布生成的样本，我想转换它们，使它们具有均匀分布。我按照以下步骤操作：

1-标准化高斯分布，使其平均值为0，标准值为1。

2-使用此标准化高斯分布的CDF来获得均匀分布。

我使用here解释的方法来估算CDF。但最终结果并不统一。它看起来像中间的均匀，但在边缘（接近0和1）有错误。

这是我的代码：

import numpy as np
import matplotlib.pyplot as plt

# ================= Parameers ===================

mu = 5.0
sig = 2.0
data_cnt = 10**6
bin_cnt = 10**3


# ================ Functions ===================

# Estimates gaussian distribution parameters from data using ML
def gaussian_estimation(vector):
    mu = np.mean(vector)
    sig = np.std(vector)

    return (mu,sig)

# Adjusts the data so it forms a gaussian with mean of 0 and std of 1
def gaussian_normalization(vector, char = None):

    if char is None:
        mu , sig = gaussian_estimation(vector)
    else:
        mu = char[2]
        sig = char[3]

    normalized = (vector-mu)/np.sqrt(sig)

    return normalized

# Taken from https://en.wikipedia.org/wiki/Normal_distribution#Cumulative_distribution_function
def CDF(x, max_i = 100):
    sum = x
    value = x
    for i in np.arange(max_i)+1:
        value = value*x*x/(2.0*i+1)
        sum = sum + value

    return 0.5 + (sum/np.sqrt(2*np.pi))*np.exp(-1*(x*x)/2)

def gaussian_to_uniform(vector, if_normal = False):

    if (if_normal == False):
        vector = gaussian_normalization(vector)

    uni = np.apply_along_axis(CDF, 0, vector)

    return uni


# ========== Step1: Data Generation ============

data = np.random.normal(mu, sig, data_cnt)
plt.hist(data,bin_cnt)


# ========= Step2: Data Normalization ==========

data2 = gaussian_normalization(data)
plt.hist(data2,bin_cnt)


# ======== Step3: Data Uniformization ==========

data3 = gaussian_to_uniform(data2, if_normal = True)
plt.hist(data3,bin_cnt)

所以有人可以帮我解决这个问题吗？也许近似方法不那么准确？