Talk slides msast2016

Nonparametric Estimation of Distribution and Density
Functions Using q-Bernstein Polynomials
Yogendra P. Chaubey
Department of Mathematics and Statistics
Concordia University, Montreal, Canada H3G 1M8
E-mail:yogen.chaubey@concordia.ca
Talk to be presented at the 6th International Conference
of IMBIC, Kolkata, India, December 21-23, 2016
Acknowledgment: The computations were performed by Qi Zhang as
part of his BSc (Honours) project.
Yogendra Chaubey (Concordia University) Department of Mathematics & Statistics December 21-23, 2016 1 / 47

Abstract
Here we consider the smooth density estimation for the random variables
that have a compact support. Without loss of generality, we restrict the
support to the closed interval [0, 1], for which Bernstein polynomial
estimator, originally proposed by Vitale (1973), is known to have some
good properties, such as being a genuine density estimator and being free
from boundary bias (see Babu, Canty and Chaubey (2002) and Kakizawa
(2004)). Generalized version of these estimators were proposed by Prakasa
Rao (2005) and Kakizawa (2011) by considering the generalized Bernstein
polynomials introduced in Cao (1997). Some work on improving
convergence rate of the Bernstein polynomial density estimator were
carried out in Leblanc (2009, 2012) and Igarshi and Kakizawa (2014). In
this paper, we consider yet another generalization of the Bernstein
polynomials introduced by Phillips (1997), that may have some advantage
over the usual Bernstein polynomial estimator.

1 Introduction
2 Preliminaries
3 Smooth Estimators of the Distribution and Density Functions
4 Asymptotic Properties of Estimators
Asymptotic Properties of ˜Fn,m,q
Asymptotic Properties of ˜fn,m,q
5 Numerical Studies
Cross Validation Method to Determine the Smoothing Parameter q
Likelihood Based Cross Validation
Integrated Square Error Cross Validation
Least Square Cross Validation
Illustration for Distribution Function Estimator
Illustration for Density Estimator
Comparing Estimators by Average Squared Error

1. Introduction
Let h be a bounded continuous function defined on the interval [0, 1],
then the corresponding Bernstein polynomial of degree m is defined as
Bm(x; h) =
m
k=0
h
k
m
bk(x, m), (1.1)
where
bk(x, m) =
m
k
xk
(1 − x)m−k
, k = 0, 1, .... (1.2)
This polynomial was proposed as an uniform approximation to
bounded continuous functions defined on [0, 1] by S.N. Bernstein
(1912), that provided an existential proof of Weierstrass
approximation theorem.
The monograph by Lorentz (1986) makes an excellent reference
regarding the properties of the Bernstein polynomials.

1. Introduction
Assuming that the function h has a continuous derivative, the
derivative Bm(x; h) given by
Bm(x; h) = m
m−1
k=0
h
k + 1
m
− h
k
m
bk(x, m − 1) (1.3)
uniformly approximates the derivative h (x).
Given a random sample {X1, ..., Xn} from an absolutely continuous
distribution function F, we can estimate it by the empirical
distribution function
Fn(x) =
1
n
n
i=1
I{Ti ≤ x}. (1.4)
Since the above function is a step function, it is not appropriate for
ﬁnding the density function f = F , thus smoothing of this function
is desired.

1. Introduction
Replacing h by the empirical distribution function Fn, the Bernstein
polynomial approximation mentioned earlier motivates the following
smooth estimator of the distribution function
˜Fn,m(x) =
m
k=0
Fn
k
m
bk(x, m). (1.5)
This estimator is a genuine distribution function and therefore its
derivative serves as a smooth density estimator given by
˜fn,m(x) = Bm(x; Fn)
= m
m−1
k=0
Fn
k + 1
m
− Fn
k
m
bk(x, m − 1)
(1.6)

1. Introduction
Bernstein polynomial density estimator: Originally proposed by Vitale
(1973) and later investigated by many authors
Gowronski and Stadm¨uler (1981): Smoothing of histogram using
Feller’s Theorem
Stadtm¨uller, (1983): Convergence of the density estimator
Petrone (1999): Nonparametric Bayesian prior
Ghosal, 2001: Convergence of nonparametric posteriors
Babu, Canty and Chaubey (2002): Further studies
Leblanc (2009, 2012): Smooth distribution function estimator.
Prakasa Rao (2005) and Kakizawa (2011): Generalized Bernstein
polynomials introduced by Cao (1997).
Another generalization is introduced by Phillips (1997) that is the
subject of this talk.

2. Preliminaries
Let q > 0, then for any k ∈ N ∪ {0}, the q−integer [k]q and
q−factorial [k]q! are respectively defined by
[k]q =
(1 − qk)/(1 − q), for q = 1
k, for q = 1,
(2.1)
and
[k]q! =
[k]q[k − 1]q...[1]q, for k ≥ 1
1, for k = 0.
(2.2)
For integers 0 ≤ k ≤ m, the q−binomial coefficients are defined by
m
k q
=
[m]q!
[k]q![m − k]q!
(2.3)
For q = 1, we clearly have [m]q = m, [m]q! = m! and
m
k q
=
m
k
,
q−factorials and q−binomial coefficients reduce to the usual factorials
and binomial coefficients, respectively.

2. Preliminaries
Deﬁnition
(Phillips 1997). Let h ∈ C[0, 1]. The generalized Bernstein polynomial
based on the q−integers is
Bm,q(x; h) =
m
k=0
h
[m]q
[k]q
bk,q(x, m) (2.4)
where
bk,q(x, m) =
m
k q
xk
m−k−1
s=0
(1 − qs
x), m = 1, 2, ...
and an empty product is taken to be equal to 1.
The polynomials deﬁned in (2.4) reduce to the usual Bernstein
polynomials for q = 1, and hence they are referred to as the
generalized Bernstein polynomials, however, in the sequel we will refer
them as the q−Bernstein polynomials.

2. Preliminaries
The following analog of Bernstein’s Theorem for q−Bernstein
polynomials was proved by Phillips (1997).
Theorem
Let a sequence {qm}∞
m=1 satisfy 0 < qm < 1 and qm → 1 as m → ∞.
Then for any function h ∈ C[0, 1],
Bm,qm (x; h) → h(x) [x ∈ [0, 1]; m → ∞].
Here the expression gm(x) → g(x)[x ∈ [0, 1]; m → ∞] denotes
uniform convergence of gm to g with respect to x ∈ [0, 1].

2. Preliminaries
The following theorem from Phillips (1996) gives convergence results
of the derivative of the q−Bernstein polynomial
Theorem
Let h ∈ C1[0, 1] and let the sequence (qm) be chosen so that the sequence
( m) converges to zero from above faster than 1/3m, where
m =
m
1 + qm + q2
m + ... + qm−1
m
− 1.
Then the sequence of derivatives of the q−Bernstein polynomials
converges uniformly on [0, 1] to h (x).
Phillips (1996) showed that the condition on ( m) can be satisﬁed by
choosing (qm) such that
1 −
a−m
m
≤ qm < 1 with a > 3.

3. Estimators of the Distribution and Density Functions
Similar to the estimator given in (1.5) using the Bernstein polynomial,
the q− Bernstein polynomial may be used to deﬁne the following
non-parametric smooth estimator of F. Thus, the q−Bernstein
polynomial estimator of the distribution function F is given by
˜Fn;m,q(x) =
m
k=0
Fn
[k]q
[m]q
bk,q(x, m), x ∈ [0, 1], (3.1)
where
bk,q(x, m) =
m
k q
xk
m−k−1
s=0
(1 − qs
x). (3.2)
For q = 1, the above estimator reduces to the usual Bernstein
polynomial smooth estimator of the distribution function studied in
Babu, Canty and Chaubey (2004).

By Euler identity (see Philips 1997) the product in (3.2) can be
written as a sum and thus taking the derivative of ˜Fn;m,q, the
following smooth estimator of f is obtained that will be called
q−Bernstein density estimator:
˜fn;m,q(x) =
m
k=0
m−k
s=0
Fn
[k]q
[m]q
ωs,k(x; m, q) (3.3)
where
ωs,k(x; m, q) =
m
k q
(−1)s
q
s(s−1)
2
m − k
s q
(k + s)xk+s−1
.
This paper presents a comparative study of the resulting smooth
estimators with Bernstein polynomial estimators while choosing the
value of q by cross validation and the value of m = n/ log(n) as
suggested in Babu, Canty and Chaubey (2004).

Since, the approximation improves as m → ∞, one may propose the
use of limiting q−Bernstein polynomials, that was studies in Il’inskii
and Ostrovska (2002). They noted that as m → ∞ for 0 < q < 1
lim
m→∞
[k]q
[m]q
= 1 − qk
lim
m→∞
bk,q(x, m) =
xk
(1 − qk)[k]q!
∞
s=0
(1 − qs
x) =: bk,∞(x, q)
the limiting q−Bernstein polynomial becomes
B∞,q(x, h) =
∞
k=0 h(1 − qk)bk,∞(x, q), if x ∈ [0, 1),
h(1) if x = 1.
Theorem
For any f ∈ C[0, 1],
B∞,q(x, h) → f(x)[x ∈ [0, 1]; q ↑ 1].

Theorem
For any f ∈ C[0, 1],
B∞,q(x, h) → f(x)[x ∈ [0, 1]; q ↑ 1].
Theorem
Let 0 < α < 1. Then for any f ∈ C[0, 1],
Bm,q(x, h) → h(x)[x ∈ [0, 1]; q ∈ [α, 1]; m → ∞].
The above theorems basically recast Phillip’s results by explicitly
exhibiting the limiting form of Bm,q(x; h) as m → ∞. So, for general
approximation, the problem of determining the constants m and q
reduces to determining q only. In our experience, the corresponding
smooth estimator of F is meaningful in practice, however, the
corresponding density estimator may not be very smooth.

In our experience, the corresponding smooth estimator of F is
meaningful in practice, however, the corresponding density estimator
may not be very smooth.
Thus, for numerical studies, we propose to consider m = n/ log(n)
[the choice suggested in Babu, Canty and Chaubey (2004)] and
determine q by cross-validation.
In the next section we provide the almost sure convergence results for
the smooth estimators similar to those obtained in Babu, Canty and
Chaubey (2004) using Bernstein polynomials. But, ﬁrst some pictures
of the estimators.

Distribution Function Estimator for Beta(3, 3) density
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
q−Bernstein Smooth Distribution
Sample from Beta(3,3) Density, n=50
x
Fn(x)
Bernstein
q−Bernstein,q=.99
Figure: 1. Distribution Function Estimators for Beta(3, 3) density; n = 50.

Distribution Function Estimator for Beta(3, 3) density
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
q−Bernstein Smooth Distribution
x
Fn(x)
Bernstein
q−Bernstein,q=.99,m=200
Figure: 2. Distribution Function Estimators for Beta(3, 3) density; n = 50.

Density Function Estimator for Beta(3, 3) density
q−Bernstein Smooth Desnity estimator
x
fn(x)
0.0 0.2 0.4 0.6 0.8 1.0
0.00.51.01.52.0
Bernstein
q−Bernstein,q=.99
Figure: 3. Density Function Estimators for Beta(3, 3) density; n = 50.

x
fn(x)
0.0 0.2 0.4 0.6 0.8 1.0
0.00.51.01.52.0
Bernstein

4. Asymptotic Properties of Estimators
Throughout this paper we use the notation
G = sup
x∈[0,1]
|G(x)|,
for a bounded function G on [0, 1],
an = (n−1
log n)1/2
, and bn,m = (n−1
log n)1/2
(m−1
log m)1/4
.
(4.1)
The following theorem shows that ˜Fn,m is strongly consistent.
Theorem
If m, n → ∞, then ˜Fn,m,q − F → 0 a.s. as q → 1.

4.1 Asymptotic Properties of Distribution Function
Estimator
Next theorem gives a result depicting the closeness of the smooth
estimator with the empirical distribution function.
Theorem
Let F be continuous and diﬀerentiable on the interval [0, 1] with density f.
If f is Lipschitz of order 1, then for n2/3 ≤ m ≤ (n/ log n)2, we have a.s.
as n → ∞ ,
lim
q↑1
˜Fn,m − Fn = O (n−1
log n)1/2
(m−1
log m)1/4
. (4.2)
For m = n, we have
lim
q↑1
˜Fn,m − Fn = O(n−3/4
(log n)3/4
) a.s.. (4.3)

4.2. Asymptotic Properties of Density Function Estimator
We now establish the strong convergence of ˜fn(t) similar to that of
˜Fn(t).
Theorem
Let F be continuous on the interval [0, 1] with a continuous density f.
Then for 2 ≤ m ≤ (n/ log n), we have a.s. as n → ∞ and q → 1,
˜fn,m,q − f = O(m1/2
an) + O( F∗
m − f ), (4.4)
a.s. as n → ∞ , where F∗
m denotes the derivative of F∗
m. Consequently, if
m = o(n/ log n), then ˜fn,m,q − f → 0 a.s. as m, n → ∞.

4.2. Asymptotic Properties of Density Function Estimator
The following theorem establishes the asymptotic normality of the
˜fn,m,q(x).
Theorem
Under conditions in theorem 3.1, if f(x) > 0 then as q → 1
n1/2
m1/4
( ˜fn,m,q(x) − f(x))
D
→ N 0,
f(x)
2 πx(1 − x)
(4.5)
as m, n → ∞ such that m ≤ (n/ log n).
Remark: If mn−2/5 → δ−2 > 0, then as q → 1,
n(2/5)
( ˜fn,m,q(x) − f(x))
D
→ N 0,
δf(x)
2 πx(1 − x)
(4.6)

4.3. Estimation for Other Support
The case of supports other than [0, 1] may be handled by
transformation.
1. For a ﬁnite support [a, b], transform the sample
values to Y1, ...., Yn, where Yi = (Xi − a)/(b − a).
Denoting the ˜Gn(y) as the smooth distribution function
of the transformed sample, the smooth distribution
˜Fn(x) is given by
˜Fn(x) = ˜Gn(y) where y = (x − a)/(b − a).
2.For the interval (−∞, ∞), the transformation
Yi = (1/2) + (1/π) tan−1 Xi is useful. In this case
˜Fn(x) = ˜Gn(y) where y = (1/2)(1/π) tan−1
x.
3. For the non-negative support [0, ∞) :, we may use
the transformation: y = x/(1 + x)
˜Fn(x) = ˜Gn(y) where y = x/(1 + x).

5. Numerical Studies
Here we illustrate the results for the estimation of distribution and
density for the following three cases: (i) Beta (3,3) distribution, (ii)
Beta(2,4) distribution and (iii) 0.65×Beta(2,10)+0.35×Beta(10,2)
distribution.
The three cases are chosen to depict three scenarios, namely that of
symmetric unimodal distribution, asymmetric unimodal distribution
and a bimodal distribution. We investigate the possibility of choosing
q diﬀerent than 1, for the choice of m = n/ log(n), a choice
recommended in Babu, Canty and Chaubey (2002) for Bernstein
polynomials. We would also like to investigate the optimal choice of q
using the limiting form of the q−Bernstein polynomials, however, this
is not pursued here. For a given m, the choice of q is obtained by
cross-validation that is explained below.

5.1 Cross Validation
For the q−Bernstein polynomial CDF and PDF estimators, in order to
choose an appropriate parameter q, we introduce a numerical method
called Cross Validation, detail of relative equations are referenced
from Chaubey and Sen (2009).
Kullback-Liebler divergence between the estimated density ˜fn and the
true density f is given by
KL( ˜fn; f) = E log
f(x)
˜fn(x)
dF(x)
In practice, the optimum cross-validation method estimates such
divergence from the data for a given smoothing parameter and
chooses one which gives the smallest estimated divergence. Bowman
(1984) shows that this procedure is equivalent to the minimization of
the negative likelihood,
CVKL(q) = − log
n
i=1
˜fn(Xi; Di) = −
n
i=1
log( ˜fn(Xi; Di)) (5.1)
where D denotes data with X removed from D, which is the wholeYogendra Chaubey (Concordia University) Department of Mathematics & Statistics December 21-23, 2016 28 / 47

5.1 ISE Cross Validation
According to this criterion we determine λn that minimizes the
criterion related to the mean integrated squared error,
MISE(q) = E ( ˜fn(x) − f(x))2
dx
Estimating this from the data and minimizing it is equivalent to the
minimization of (see Silverman (1986),
CVISE(q) = ˜f2
n(x; q, D)dx −
2
n
n
i=1
˜fn−1(Xi; q, Di) (5.2)
The ﬁrst part can be obtain by numerical integration, say composite
Simpson Method. And again, Di denotes data with Xi removed from
D.

5.1 LS Cross Validation
The above cross validation works well if f is a density estimator. For
CDF, it is better to use least square cross validation (see H¨ardle,
1991).
CVLS(q) =
n
i=1
( ˜Fn(Xi; q, D) − ˜Fn(Xi; q, Di))2
(5.3)

Distribution Function-Illustration
For a given sample of size n we start with m = n and determine the
optimum value of the smoothing parameter as illustrated in Figures
6-8 for a random sample generated from each of the distributions.
We also investigate the eﬀect of increasing the value of m. This is
done by selecting a grid of size n on the interval [0, 1], and we let M
denote the value of m that gives the average error of estimation
between the successive values of M and M + 1 less than ε, in order
to minimize the amount of computation. [In order to control the
individual errors, M will depend on the value of x also.].

0.0 0.2 0.4 0.6 0.8 1.0
0.050.100.150.200.25
q
SumofSquareErros(SSE)
(a) n = 20, q = 0.94, m = 20
0.2 0.4 0.6 0.8
0.00.20.40.60.81.0
x
cdf
(b) M = 147, ε = 9.96 × 10−5
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.8
q
(c) n = 50, q = 1, m = 50
0.2 0.4 0.6 0.8
0.00.20.40.60.81.0
x
cdf
(d) M = 1024, ε = 5.31 × 10−5
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.01.2
q
(e) n = 100; q = 1, m = 100
0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
x
cdf
(f) M = 801, ε = 7.56 × 10−5
Figure 6: LSCV plot for q−Bernstein estimator (Left Panel); CDF Estima-
tor for Beta(3, 3) Samples (Right Panel).
Remark: red - true density; dash - q−Bernstein with m = n; blue -
q−Bernstein with parameter q, M
12

0.0 0.2 0.4 0.6 0.8 1.0
0.20.40.60.8
q
(a) n = 20, q = 0.96, m = 20
0.1 0.2 0.3 0.4 0.5 0.6 0.7
0.00.20.40.60.81.0
x
cdf
(b) M = 206, ε = 9.60 × 10−5
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.01.21.4
q
(c) n = 50, q = 0.99, m = 50
0.2 0.4 0.6 0.8
0.00.20.40.60.81.0
x
cdf
(d) M = 105, ε = 8.28 × 10−5
0.0 0.2 0.4 0.6 0.8 1.0
0.00.51.01.52.02.53.03.5
q
(e) n = 100; q = 1, m = 100
0.0 0.2 0.4 0.6 0.8
0.00.20.40.60.81.0
x
cdf
(f) M = 611, ε = 6.64 × 10−5
Figure 7: Smooth Bernstein(Left) and q−Bernstein(Right) CDF Estimator
for Beta(2, 4) Samples.
13

0.0 0.2 0.4 0.6 0.8 1.0
0.20.40.60.81.01.2
q
(a) n = 20, q = 0.99, m = 20
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
x
cdf
(b) M = 73, ε = 6.88 × 10−5
0.0 0.2 0.4 0.6 0.8 1.0
0.00.51.01.52.02.5
q
(c) n = 50, q = 1, m = 50
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
x
cdf
(d) M = 355, ε = 4.41 × 10−5
0.0 0.2 0.4 0.6 0.8 1.0
012345
q
(e) n = 100; q = 1, m = 100
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
x
cdf
(f) M = 371, ε = 8.98 × 10−5
for 0.65 · Beta(2, 10) + 0.35 · Beta(10, 2) Samples.
14

0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
x
cdf
(a) n = 20, q = 0.94, m = 7, M = 147
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
x
cdf
(b) n = 20, q = 0.96, m = 7, M = 206
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
x
cdf
(c) n = 50, q = 1, m = 13, M = 1024
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
x
cdf
(d) n = 50, q = 0.99, m = 13, M = 105
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
x
cdf
(e) n = 100, q = 1, m = 22, M = 801
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
x
cdf
(f) n = 100, q = 1, m = 22, M = 611
Figure 9: Smooth Bernstein and q−Bernstein CDF Estimator for
Beta(3, 3)(Left) and Beta(2, 4)(Right) Samples.
Remark: red is theoretical, black is Bernstein ,dash is q−Bernstein with
M = m, blue is q−Bernstein with parameter q, M
15

0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
x
cdf
(a) n = 20, q = 0.99, m = 7, M = 73
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
x
cdf
(b) n = 50, q = 1, m = 13, M = 355
0.0 0.2 0.4 0.6 0.8 1.0
0.00.20.40.60.81.0
x
cdf
(c) n = 100, q = 1, m = 22, M = 371
for 0.65 · Beta(2, 10) + 0.35 · Beta(10, 2) Samples.
Remark: red is theoretical, dash is q−Bernstein with M = m, blue is
16

Density Function-Illustration
In Figures 11-12, we present the plots of density estimators corresponding
to the three distributions based on a random sample for sample sizes
n = 20, 50, 100, where m = n/ log(n) for diﬀerent values of q ∈ [.85, 1].
We will use the following colors to highlight speciﬁc q’s.
We will use red for the theoretical density, while use green3 for q = 0.85,
lightseagreen for q = 0.9, olivedrab for q = 0.95 and darkorange3 for
optimized q and blue for the Bernstein density estimator. For optimized
q ∈ {0.85, 0.9, 0.95}, we will display it in darkorange3. Note that we have
not considered q−Bernstein polynomial density estimator for the M, q
case, as in this situation the density shape is very rough. These are
illustrated in Figures 9-10.

0.0 0.2 0.4 0.6 0.8 1.0
0.00.51.01.5
x
density
(a) n = 20, m = 7, q = 0.85
0.0 0.2 0.4 0.6 0.8 1.0
0.00.51.01.52.0
x
density
(b) n = 20, m = 7, q = 0.85
0.0 0.2 0.4 0.6 0.8 1.0
0.00.51.01.5
x
density
(c) n = 50, m = 13, q = 0.96
0.0 0.2 0.4 0.6 0.8 1.0
0.00.51.01.52.0
x
density
(d) n = 50, m = 13, q = 0.96
0.0 0.2 0.4 0.6 0.8 1.0
0.00.51.01.5
x
density
(e) n = 100, m = 22, q = 0.99
0.0 0.2 0.4 0.6 0.8 1.0
0.00.51.01.52.0
x
density
(f) n = 100, m = 22, q = 0.99
Figure 11: Comparison of Bernstein and q−Bernstein PDF estimator with
q ∈ [0.85, 1] for Beta(3, 3)(left) and Beta(2, 4)(right)
18

0.0 0.2 0.4 0.6 0.8 1.0
0.00.51.01.52.02.5
xcdf
(a) n = 20, m = 7, q = 0.95
0.0 0.2 0.4 0.6 0.8 1.0
0.00.51.01.52.02.5
x
cdf
(b) n = 50, m = 13, q = 0.99
0.0 0.2 0.4 0.6 0.8 1.0
0.00.51.01.52.02.5
x
cdf
(c) n = 100, m = 22, q = 1
Figure 12: Comparison of Bernstein and q−Bernstein PDF estimator with
q ∈ [0.85, 1] for 0.65 · Beta(2, 10) + 0.35 · Beta(10, 2)

Average Squared Error Comparison
The following tables display the average squared errors (ASE) based on
B = 40 comparisons, where ASE(g) = 1
B (˜g(xi) − g(xi))2, where g is
the true distribution or density, and ˜g is its nonparametric estimator.
Distribution Sample size Bernstein q−Bernstein
20 0.0020684 0.0004371
Beta(3,3) 50 0.0037048 0.0043542
100 0.0003380 0.0001946
20 0.0036687 0.0021716
Beta(2,4) 50 0.0049030 0.0041611
100 0.0013453 0.0006329
20 0.0014412 0.0036309
Mixture 50 0.0012856 0.0032519
100 0.0041389 0.0046857
Table: Average Square Error for Bernstein and q−Bernstein CDF estimator with
optimized q

Average Squared Error Comparison
Distribution Sample size Bernstein q−Bernstein
20 0.1388213 0.0760121
Beta(3,3) 50 0.0278798 0.0669346
100 0.0182203 0.0185668
20 0.1669993 0.1340892
Beta(2,4) 50 0.0488355 0.0780358
100 0.0221844 0.0219737
20 0.4132526 0.5370016
Mixture 50 0.1991615 0.2336240
100 0.2113997 0.1938492
Table: Average Square Error for Bernstein and q−Bernstein PDF estimator with
optimized q

By looking at the ﬁgures, we see that as the sample size increase, the ASE
has a decreasing trend. The distribution function estimator may have
smaller errors specially for small sample sizes. For density estimators, q−
Bernstein polynomial may have slight advantage over the Bernstein
polynomial that too for small samples and unimodal distributions. Overall,
we can say that the classical Bernstein polynomial estimator for estimating
density may still be considered superior over the general q− Bernstein
polynomial estimator, however, the q−Bernstein polynomial estimator for
the distribution function might be potentially better. This requires further
investigation, especially in the manner in which we have selected M. The
typical question is that of ﬁnding optimal q when m = ∞?

References
1 Babu, G.J., Canty, A.J. and Chaubey, Y.P. (2002). Application of
Bernstein polynomials for smooth estimation of a distribution and
density function. Journal of Statistical Planning and Inference 105,
377-392.
2 Bernstein, S.N. (1912). Démonstration du théorème de Weierstrass,
fondeé sur le calcul des probabilités. Commun. Soc. Math. Kharkov,
No.1. Series XIII, 1–2.
3 Bowman, A.W. (1984). An alternative method of cross-validation for
the smoothing of density estimates.Biometrika 71, 353-360.
4 Cao, J.D. (1997). A generalization of the Bernstein polynomials.
Journal Of Mathematical Analysis And Applications 209, 140–146.
5 Chaubey, Y. P., & Sen, P. K. (2009). On the selection of the
smoothing parameter in poisson smoothing of histogram estimator:
Computational aspects. Pakistan Journal of Statistics 25, 385-401.

References
6 Gawronski,W. and StadtM¨uller, U. (1981). Smoothing histograms by
means of lattice- and continuous distributions. Metrika 28, 155164.
7 Ghosal, S. (2001). Convergence rates for density estimation with
bernstein polynomials. The Annals of Statistics 29, 1264–1280.
8 H¨ardle, W. (1991). Smoothing Techniques with Implementation in S.
Springer-Verlag: New York.
9 Igarashi, G. and Yoshihide Kakizawa, Y. (2014). On improving
convergence rate of Bernstein polynomial density estimator. Journal
of Nonparametric Statistics 26, 6184.
10 Il’inskii, A. and Ostrovska, S. (2002). Convergence of generalized
Bernstein polynomials. Journal of Approximation Theory 116,
100–112.

References
11 Kakizawa, Y. (2004). Bernstein polynomial probability density
estimation. Journal of Nonparametric Statistics 16, 709–729.
12 Kakizawa, Y. (2011). A note on generalized bernstein polynomial
density estimators. Statistical Methodology 8, 136153.
13 Leblanc, A. (2009). ChungSmirnov property for Bernstein estimators
of distribution functions. Journal of Nonparametric Statistics 21,
133142.
14 Leblanc, A. (2012). On estimating distribution functions using
Bernstein polynomials. Ann Inst Stat Math 64, 919943.
15 Lorentz, G. G. (1986). Bernstein Polynomials (2nd ed.) New York:
Chelsea Publishing.

References
16 Phillips, G. M. (1997). Bernstein polynomials based on the q-integers.
The heritage of P. L. Chebyshev: a Festschrift in honor of the 70th
birthday of T. J. Rivlin. Ann. Numer. Math. 14, 511–518.
17 Prakasa Rao, B. L. S. (2005). Estimation of distribution and density
functions by generalized Bernstein polynomials. Indian Journal of
Pure and Applied Mathematics 36, 63–88.
18 Silverman, B. W. (1986). Density estimation for statistics and data
analysis, Chapman and Hall, London.
19 StadtM¨uller, U. (1983). Asymptotic distributions of smoothed
histograms. Metrika 30, 145–158.
20 Vitale, R.A. (1973). A Bernstein polynomial approach to density
estimation. Comm. Statist. 2, 493–506.

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