Slides csm

On Nonparametric Density Estimation for Size Biased
Data
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
5e Recontre scientiﬁques Sherbrooke-Montpelier:
Colloque de statistique et de biostatistique
June 10-11, 2015
Yogendra Chaubey (Concordia University) Department of Mathematics & Statistics June 11, 2015 1 / 75

Abstract
This talk will highlight some recent developments in the area of
nonparametric functional estimation with emphasis on nonparametric
density estimation for size biased data. Such data entail constraints that
many traditional nonparametric density estimators may not satisfy. A
lemma attributed to Hille, and its generalization [see Lemma 1, Feller
(1965) An Introduction to Probability Theory and Applications, §VII.1)] is
used to propose estimators in this context from two diﬀerent perspectives.
After describing the asymptotic properties of the estimators, we present
the results of a simulation study to compare various nonparametric density
estimators. The optimal data driven approach of selecting the smoothing
parameter is also outlined.

Outline
1 1. Introduction/Motivation
1.1 Kernel Density Estimator
1.2. Smooth Estimation of Densities on R+
2 2. An Approximation Lemma and Some Alternative Smooth Density
Estimators
2.1 Some Alternative Smooth Density Estimators on R+
2.2 Asymptotic Properties of the New Estimator
2.3 Extensions Non-iid cases
3 3. Estimation of Density in Length-biased Data
3.1 Smooth Estimators Based on the Estimators of G
3.2 Smooth Estimators Based on the Estimators of F
4 4. Choice of Smoothing Parameters
4.1 Unbiased Cross Validation
4.2 Biased Cross Validation
5 5. A Comparison Between Diﬀerent Estimators: Simulation Studies
5.1 Simulation for χ2
2
5.2 Simulation for χ2
6
5.3 Simulation for Some Other Standard Distributions
6 6. References

1. Introduction/Motivation
1.1 Kernel Density Estimator
Consider X as a non-negative random variable with density f(x) and
distribution function
F(x) =
x
0
f(t)dt for x > 0. (1.1)
Such random variables are more frequent in practice in life testing and
reliability.

Based on a random sample (X1, X2, ..., Xn), from a univariate
density f(.), the empirical distribution function (edf) is deﬁned as
Fn(x) =
1
n
n
i=1
I(Xi ≤ x). (1.2)
edf is not smooth enough to provide an estimator of f(x).
Various methods (viz., kernel smoothing, histogram methods, spline,
orthogonal functionals)
The most popular is the Kernel method (Rosenblatt, 1956).
[See the text Nonparametric Functional Estimation by Prakasa Rao
(1983) for a theoretical treatment of the subject or Silverman (1986)].

ˆfn(x) =
1
n
n
i=1
kh(x − Xi) =
1
nh
n
i=1
k
x − Xi
h
) (1.3)
where the function k(.) called the Kernel function has the following
properties;
(i)k(−x) = k(x)
(ii)
∞
−∞
k(x)dx = 1
and
kh(x) =
1
h
k
x
h
h is known as bandwidth and is made to depend on n, i.e. h ≡ hn,
such that hn → 0 and nhn → ∞ as n → ∞.
Basically k is a symmetric probability density function on the entire
real line. This may present problems in estimating the densities of
non-negative random variables.

Kernel Density Estimators for Suicide Data
0 200 400 600
0.0000.0020.0040.006
x
Default
SJ
UCV
BCV
Figure 1. Kernel Density Estimators for Suicide Study Data

1.2. Smooth Estimation of densities on R+
ˆfn(x) might take positive values even for x ∈ (−∞, 0], which is not
desirable if the random variable X is positive. Silverman (1986)
mentions some adaptations of the existing methods when the support
of the density to be estimated is not the whole real line, through
transformation and other methods.
1.2.1 Bagai-Prakasa Rao Estimator
Bagai and Prakasa Rao (1996) proposed the following adaptation of
the Kernel Density estimator for non-negative support [which does
not require any transformation or corrective strategy].
fn(x) =
1
nhn
n
i=1
k
x − Xi
hn
, x ≥ 0. (1.4)

Here k(.) is a bounded density function with support (0, ∞), satisfying
∞
0
x2
k(x)dx < ∞
and
hn is a sequence such that hn → 0 and nhn → ∞ as n → ∞.
The only diﬀerence between ˆfn(x) and fn(x) is that the former is
based on a kernel possibly with support extending beyond (0, ∞).
One undesirable property of this estimator is that that for x such that
for X(r) < x ≤ X(r+1), only the ﬁrst r order statistics contribute
towards the estimator fn(x).

Bagai-Prakasa Rao Density Estimators for Suicide Data
0 200 400 600
0.0000.0020.0040.006
x
Default
SJ
UCV
BCV
Figure 2. Bagai-Prakasa Rao Density Estimators for Suicide Study Data
Silverman (1986)

2.1 An approximation Lemma
The following discussion gives a general approach to density
estimation which may be specialized to the case of non-negative data.
The key result for the proposal is the following Lemma given in Feller
(1965, §VII.1).
Lemma 1: Let u be any bounded and continuous function and
Gx,n, n = 1, 2, ... be a family of distributions with mean µn(x) and
variance h2
n(x) such that µn(x) → x and hn(x) → 0. Then
˜u(x) =
∞
−∞
u(t)dGx,n(t) → u(x). (2.1)
The convergence is uniform in every subinterval in which hn(x) → 0
uniformly and u is uniformly continuous.

This generalization may be adapted for smooth estimation of the
distribution function by replacing u(x) by the empirical distribution
function Fn(x) as given below ;
˜Fn(x) =
∞
−∞
Fn(t)dGx,n(t). (2.2)
Note that Fn(x) is not a continuous function as desired by the above
lemma, hence the above lemma is not directly used in proposing the
estimator but it works as a motivation for the proposal. It can be
considered as the stochastic adaptation in light of the fact that the
mathematical convergence is transformed into stochastic convergence
that parallels to that of the strong convergence of the empirical
distribution function as stated in the following theorem.

Theorem 1: Let hn(x) be the variance of Gx,n as in Lemma 1 such
that hn(x) → 0 as n → ∞ for every ﬁxed x as n → ∞, then we have
sup
x
| ˜Fn(x) − F(x)|
a.s.
→ 0 (2.3)
as n → ∞.
Technically, Gx,n can have any support but it may be prudent to
choose it so that it has the same support as the random variable
under consideration; because this will get rid of the problem of the
estimator assigning positive mass to undesired region.
For ˜Fn(x) to be a proper distribution function, Gx,n(t) must be a
decreasing function of x, which can be shown using an alternative
form of ˜Fn(x) :
˜Fn(x) = 1 −
1
n
n
i=1
Gx,n(Xi). (2.4)

Equation (2.4) suggests a smooth density estimator given by
˜fn(x) =
d ˜Fn(x)
dx
= −
1
n
n
i=1
d
dx
Gx,n(Xi). (2.5)
The potential of this lemma for smooth density estimation was
recognized by Gawronski (1980) in his doctoral thesis written at Ulm.
Gawronski and Stadmuller (1980, Skand. J. Stat.) investigated mean
square error properties of the density estimator when Gx,n is obtained
by putting Poisson weight
pk(xλn) = e−λnx (λnx)k
k!
(2.6)
to the lattice points k/λn, k = 0, 1, 2, ...

Other developments:
This lemma has been further used to motivate the Bernstein
Polynomial estimator (Vitale, 1975) for densities on [0, 1] by Babu,
Canty and Chaubey (1999). Gawronski (1985, Period. Hung.)
investigates other lattice distributions such as negative binomial
distribution.
Some other developments:
Chaubey and Sen (1996, Statist. Dec.): survival functions, though in a
truncated form.
Chaubey and Sen (1999, JSPI): Mean Residual Life; Chaubey and Sen
(1998a, Persp. Stat., Narosa Pub.): Hazard and Cumulative Hazard
Functions; Chaubey and Sen (1998b): Censored Data;
(Chaubey and Sen, 2002a, 2002b): Multivariate density estimation
Smooth density estimation under some constraints: Chaubey and
Kochar (2000, 2006); Chaubey and Xu (2007, JSPI).
Babu and Chaubey (2006): Density estimation on hypercubes. [see
also Prakasa Rao(2005)and Kakizawa (2011), Bouezmarni et al. (2010,
JMVA) for Generalised Bernstein Polynomials and Bernstein copulas]

A Generalised Kernel Estimator for Densities with
Non-Negative Support
Lemma 1 motivates the generalised kernel estimator of Földes and
Révész (1974):
fnGK(x) =
1
n
n
i=1
hn(x, Xi)
Chaubey et al. (2012, J. Ind. Stat. Assoc.) show the following
adaptation using asymmetric kernels for estimation of densities with
non-negative support.
Let Qv(x) represent a distribution on [0, ∞) with mean 1 and
variance v2, then an estimator of F(x) is given by
F+
n (x) = 1 −
1
n
n
i=1
Qvn
Xi
x
, (2.7)
where vn → 0 as n → ∞.

Obviously, this choice uses Gx,n(t) = Qvn (t/x) which is a decreasing
function of x.
This leads to the following density estimator
d
dx
(F+
n (x)) =
1
nx2
n
i=1
Xi qvn
Xi
x
, (2.8)
where qv(.) denotes the density corresponding to the distribution
function Qv(.).

However, the above estimator may not be deﬁned at x = 0, except in
cases where limx→0
d
dx (F+
n (x)) exists. Moreover, this limit is
typically zero, which is acceptable only when we are estimating a
density f with f(0) = 0.
Thus with a view of the more general case where 0 ≤ f(0) < ∞, we
considered the following perturbed version of the above density
estimator:
f+
n (x) =
1
n(x + n)2
n
i=1
Xi qvn
Xi
x + n
, x ≥ 0 (2.9)
where n ↓ 0 at an appropriate (suﬃciently slow) rate as n → ∞. In
the sequel, we illustrate our method by taking Qv(.) to be the
Gamma (α = 1/v2, β = v2) distribution function.

Remark:
Note that if we believe that the density is zero at zero, we set n ≡ 0,
however in general, it may be determined using the cross-validation
methods. For n > 0, this modiﬁcation results in a defective distribution
F+
n (x + n). A corrected density estimator f∗
n(x) is therefore proposed:
f∗
n(x) =
f+
n (x)
cn
, (2.10)
where cn is a constant given by
cn =
1
n
n
i=1
Qvn
Xi
n
.
Note that, since for large n, n → 0, f∗
n(x) and f+
n (x) are asymptotically
equivalent, we study the asymptotic properties of f+
n (x) only.

Next we present a comparison of our approach with some existing
estimators.
Kernel Estimator.
The usual kernel estimator is a special case of the representation
given by Eq. (2.5), by taking Gx,n(.) as
Gx,n(t) = K
t − x
h
, (2.11)
where K(.) is a distribution function with mean zero and variance 1.

Transformation Estimator of Wand et al.
The well known logarithmic transformation approach of Wand,
Marron and Ruppert (1991) leads to the following density estimator:
˜f(L)
n (x) =
1
nhnx
n
i=1
k(
1
hn
log(Xi/x)), (2.12)
where k(.) is a density function (kernel) with mean zero and variance
1.

This is easily seen to be a special case of Eq. (2.5), taking Gx,n again
as in Eq. (2.11) but applied to log x. This approach, however, creates
problem at the boundary which led Marron and Ruppert (1994) to
propose modiﬁcations that are computationally intensive.
Estimators of Chen and Scaillet.
Chen’s (2000) estimator is of the form
ˆfC(x) =
1
n
n
i=1
gx,n(Xi), (2.13)
where gx,n(.) is the Gamma(α = a(x, b), β = b) density with b → 0
and ba(x, b) → x.

This also can be motivated from Eq. (2.1) as follows: take
u(t) = f(t) and note that the integral f(t)gx,n(t)dt can be
estimated by n−1 n
i=1 gx,n(Xi). This approach controls the
boundary bias at x = 0; however, the variance blows up at x = 0, and
computation of mean integrated squared error (MISE) is not
tractable. Moreover, estimators of derivatives of the density are not
easily obtainable because of the appearance of x as argument of the
Gamma function.

Scaillet’s (2004) estimators replace the Gamma kernel by inverse
Gaussian (IG) and reciprocal inverse Gaussian (RIG) kernels. These
estimators are more tractable than Chen’s; however, the IG-kernel
estimator assumes value zero at x = 0, which is not desirable when
f(0) > 0, and the variances of the IG as well as the RIG estimators
blow up at x = 0.
Bouezmarni and Scaillet (2005), however, demonstrate good
ﬁnite-sample performance of these estimators.

It is interesting to note that one can immediately deﬁne a
Chen-Scaillet version of our estimator, namely,
f+
n,C(x) =
1
n
n
i=1
1
x
qvn
Xi
x
.
On the other hand, our version (i.e., perturbed version) this estimator
would be
ˆf+
C (x) =
1
n
n
i=1
gx+ n,n(Xi);
that should not have the problem of variance blowing up at x = 0.

It may also be remarked that the idea used here may be extended to
the case of densities supported on an arbitrary interval
[a, b], − ∞ < a < b < ∞, by choosing for instance a Beta kernel
(extended to the interval [a, b]) as in Chen (1999). Without loss of
generality, suppose a = 0 and b = 1. Then we can choose, for
instance, qv(.) as the density of Y/µ, where
Y ∼ Beta(α, β), µ = α/(α + β), such that α → ∞ and β/α → 0, so
that Var(Y/µ) → 0.

2.2 Asymptotic Properties of the New Estimator
2.2.1 Asymptotic Properties of ˜F+
n (x)
The strong consistency holds in general for the estimator ˜F+
n (x). We can
easily prove the following theorem parallel to the strong convergence of the
empirical distribution function.
Theorem:
If λn → ∞ as n → ∞ we have
sup
x
| ˜F+
n (x) − F(x)|
a.s.
→ 0.
as n → ∞.

We can also show that for large n, the smooth estimator can be arbitrarily
close to the edf by proper choice of λn, as given in the following theorem.
Theorem:Assuming that f has a bounded derivative, and λn = o(n),
then for some δ > 0, we have, with probability one,
sup
x≥0
| ˜F+
n (x) − Fn(x)| = O n−3/4
(log n)1+δ
.

2.2.2 Asymptotic Properties of ˜f+
n (x)
Under some regularity conditions, they obtained
Theorem:
sup
x≥0
| ˜f+
n (x) − f(x)|
a.s.
−→ 0
as n → ∞.
Theorem:
(a) If nvn → ∞, nv3
n → 0, nvnε2
n → 0 as n → ∞, we have
√
nvn(f+
n (x) − f(x)) → N 0, I2(q)
µf(x)
x2
, for x > 0.
(b) If nvn
2
n → ∞ and nvnε4
n → 0 as n → ∞, we have
nvn
2
n(f+
n (0) − f(0)) → N 0, I2(q)f(0) .

2.3. Extensions to Non-iid cases
We can extend the technique to non-iid cases where a version of Fn(x) is
available.
Chaubey, Y.P., Dewan, I. and Li, J. (2012) – Density estimation for
stationary associated sequences. Comm. Stat.- Simula. Computa.
41(4), 554- 572 –Using generalised kernel approach Chaubey
Chaubey, Yogendra P., Dewan, Isha and Li, Jun (2011) – Density
estimation for stationary associated sequences using Poisson weights.
Statist. Probab. Lett. 81, 267-276.
Chaubey, Y.P. and Dewan, I. (2010). A review for smooth estimation
of survival and density functions for stationary associated sequences:
Some recent developments – J. Ind. Soc. Agr. Stat. 64(2), 261-272.
Chaubey, Y.P., La¨ıb, N. and Sen, A. (2010). Generalised kernel
smoothing for non-negative stationary ergodic processes – Journal of
Nonparametric Statistics, 22, 973-997

3. Estimation of Density in Length-biased Data
In general, when the probability that an item is sampled is
proportional to its size, size biased data emerges.
The density g of the size biased observation for the underlying density
f, is given by
g(x) =
w(x)f(x)
µw
, x > 0, (3.1)
where w(x) denotes the size measure and µw = w(x)f(x).
In the area of forestry, the size measure is usually proportional to
either length or area (see Muttlak and McDonald, 1990).
Another important application occurs in renewal theory where
inter-event times data are of this type if they are obtained by
sampling lifetimes in progress at a randomly chosen point in time (see
Cox, 1969).

Here we will talk about the length biased case where we can write
f(x) =
1
x
g(x)/µ. (3.2)
In principle any smooth estimator of the density function g may be
transformed into that of the density function f as follows:
ˆf(x) =
1
x
ˆg(x)/ˆµ, (3.3)
where ˆµ is an estimator of µ.
Note that 1/µ = Eg(1/X), hence a strongly consistent estimator of µ
is given by
ˆµ = n{
n
i=1
X−1
i }−1

Bhattacharyya et al. (1988) used this strategy in proposing the
following smooth estimator of f,
ˆfB(x) = ˆµ(nx)−1
n
i=1
kh(x − Xi). (3.4)
Also since, F(x) = µ Eg(X−11(X≤x)), Cox (1969) proposed the
following as an estimator of the distribution function F(x) :
ˆFn(x) = ˆµ
1
n
n
i=1
X−1
i 1(Xi≤x). (3.5)
So there are two competing strategies for density estimation for LB
data. One is to estimate g(x) and then use the relation (3.3) (i.e.
smooth Gn as in Bhattacharyya et al. (1988)). The other is to
smooth the Cox estimator ˆFn(x) directly and use the derivative as the
smooth estimator of f(x).

Jones (1991) studied the behaviour of the estimator fB(x) in contrast
to smooth estimator obtained directly by smoothing the estimator
Fn(x), by Kernel method:
ˆfJ (x) = n−1
ˆµ
n
i=1
X−1
i kh(x − Xi). (3.6)
He noted that this estimator is a proper density function when
considered with the support on the whole real line, where as ˆfB(y)
may be not. He compared the two estimators based on simulations,
and using the asymptotic arguments, concluded that the latter
estimator may be preferable in practical applications.
Also using Jensen’s inequality we ﬁnd that
Eg(ˆµ) ≥ 1/Eg{
1
n
n
i=1
X−1
i } = µ,
hence the estimator ˆµ may be positively biased which would transfer
into increased bias in the above density estimators.

If g(x)/x is integrable, the deﬁciency of fB(x) of not being a proper
density may be corrected by considering the alternative estimator
ˆfa(x) =
ˆg(x)/x
(ˆg(x)/x)dx
, (3.7)
and this may also eliminate the increase in bias to some extent.
However, in these situations, since X is typically a non-negative
random variable, the estimator must satisfy the following two
conditions:
(i) ˆg(x) = 0 for x ≤ 0,
(ii) ˆg(x)/x is integrable.
Here both of the estimators ˆfB(x) and ˆfJ (x) do not satisfy these
properties.
We have a host of alternatives, those based on smoothing Gn and
those based on smoothing Fn, that we are going to talk about next.

3.1.1 Poisson Smoothing of Gn
Here, we would like to see the application of the weights generated by
the Poisson probability mass function as motivated in Chaubey and
Sen (1996, 2000). However, a modification is necessary in the present
situation which is also outlined here.
Using Poisson smoothing, an estimator of g(x) may be given by,
˜gnP (x) = λn
∞
k=0
pk(λnx) Gn
k + 1
λn
− Gn
k
λn
, (3.8)
however, note that limx→0˜gnP (x) = λnGn(1/λn) which may
converge to 0 as λn → ∞, however for finite samples it may not be
zero, hence the density f at x = 0 may not be defined. Furthermore,
˜gnP (x)/x is not integrable.

A simple modiﬁcation by attaching the weight pk(λnx) to
Gn((k − 1)/λn), rather than to Gn(k/λn), the above problem is
avoided. This results in the following smooth estimator of G(x) :
˜Gn(x) =
k≥0
pk(xλn)Gn
k − 1
λn
, (3.9)
The basic nature of the smoothing estimator is not changed, however
this provides an alternative estimator of the density function as its
derivative is given by
˜gn(x) = λn
k≥1
pk(xλn) Gn
k
λn
− Gn
k − 1
λn
, (3.10)
such that ˜gn(0) = 0 and that ˜gn(x)/x is integrable.

Since,
∞
0
˜gn(x)
x
dx = λn
k≥1
[Gn
k
λn
− Gn
k − 1
λn
]
∞
0
pk(xλn)
x
dx
= λn
k≥1
[Gn
k
λn
− Gn
k − 1
λn
]
1
k
= λn
k≥1
1
k(k + 1)
Gn
k
λn
,
The new smooth estimator of the length biased density f(x) is given
by
˜fn(x) =
k≥1
pk−1(xλn)
k Gn
k
λn
− Gn
k−1
λn
k≥1
1
k(k+1) Gn
k
λn
. (3.11)

The corresponding smooth estimator of the distribution function
F(x) is given by
˜Fn(x) =
k≥1(1/k)Wk(xλn)[Gn
k
λn
− Gn
k−1
λn
]
k≥1
1
k(k+1)Gn
k
λn
(3.12)
where
Wk(λnx) =
1
Γ(k)
λnx
0
e−y
yk−1
dy =
j≥k
pj(λnx).

An equivalent expression for the above estimator is given by
˜Fn(x) =
k≥1 Gn
k
λn
Wk(λnx)
k −
Wk+1(λnx)
k+1
k≥1
1
k(k+1) Gn
k
λn
= 1 +
k≥1 Gn
k
λn
Pk(λnx)
k+1 −
Pk−1(λnx)
k
k≥1
1
k(k+1) Gn
k
λn
,
where Pk(µ) =
k
j=0
pj(µ)
denotes the cumulative probability corresponding to the Poisson(µ)
distribution.
The properties of above estimators can be established in an analogous
way to those in the regular case.

3.1.2 Gamma Smoothing of Gn
The smooth estimator using the log-normal density may typically have
a spike at zero, however the gamma density may be appropriate, since
it typically has the density estimator ˆg(x) such that ˆg(0) = 0, so that
no perturbation is required. The smooth density estimator in this case
is simply given by
g+
n (x) =
1
nx2
n
i=1
Xi qvn
Xi
x
, (3.13)
where qv(.) denotes the density corresponding to a
Gamma(α = 1/vn, β = vn). Since
∞
0
gn(x)
x dx = 1
n
n
i=1
1
Xi
, this
gives the density estimator
fn(x) =
1
x3
n
i=1 Xiqvn
Xi
x
n
i=1
1
Xi
. (3.14)

Note that the above estimator may be reasonable for a density f(x) with
f(0) = 0, but may need a modiﬁcation for general density functions. We
use the idea of perturbation as before to arrive at an acceptable estimator
through this approach as given by
ˆf+
n (x) =
1
(x+εn)3
n
i=1 Xiqvn
Xi
x+εn
n
i=1
1
Xi
(3.15)

3.2.1 Poisson Smoothing of Fn
smoothing Fn directly using Poisson weights, an estimator of f(x)
may be given by,
˜fnP (x) = λn
∞
k=0
pk(λnx) Fn
k + 1
λn
− Fn
k
λn
. (3.16)
No modiﬁcations are necessary.

3.2.2 Gamma Smoothing of Fn
The gamma based smooth estimate of F(x) is given by
F+
n (x) = 1 −
n
i=1
1
Xi
Qvn (Xi
x )
n
i=1
1
Xi
, (3.17)
and that for the density f in this case is simply given by
˜f+
n (x) =
1
(x+εn)2
n
i=1 qvn ( Xi
x+εn
)
n
i=1
1
Xi
. (3.18)
where qv(.) denotes the density corresponding to a
Gamma(α = 1/vn, β = vn).
Note that the above estimator is computationally intensive as two
smoothing parameters have to be computed using bivariate cross
validation.

4. Choice of Smoothing Parameters
4.1 Unbiased Cross Validation
This criterion evolves from minimizing an unbiased estimate of the
integrated squared error (ISE) of fn with respect to f given by
ISE(fn; f) =
∞
0
(fn(x) − f(x))2
dx
=
∞
0
f2
n(x) − 2
∞
0
fn(x)f(x)dx +
∞
0
f2
(x)dx
The last term being free of the smoothing parameter (b), we minimize
UCV (b) =
∞
0
f2
n(x; b, D)dx − 2
n
i=1
fn−1(Xi; b, Di)/Zi,
that estimates the ﬁrst two terms of ISE(fn; f), where Zi =
j=i
Xi
Xj
and fn(x; b, D) denotes the estimator of f(x) that involves data D.

This criterion involves minimization of an estimate of the asymptotic
mean integrated squared error (AMISE).
For AMISE( ˆfn), we use the formula as obtained in Chaubey et al.
(2012), that is given by
AMISE[ ˆf+
n (x)] =
I2(q)µ
nvn
∞
0
f(x)
(x + εn)2
dx
+
∞
0
[v2
nf(x) + (2v2
nx + εn)f (x) + v2
n
x2
2
f (x)]2
dx
(4.1)
where I2(q) lim
v→0
v
∞
0 (qvn (t))2dt.

The AMISE( ˜fn), is given by in Chaubey et al. (2012), that is given
by
AMISE[ ˜f+
n ] =
∞
0
[(xv2
n + εn)f (x) +
x2
2
f (x)v2
n]2
dx
+
I2(q)µ
nvn
∞
0
f(x)
(x + εn)2
dx.
(4.2)
The Biased Cross Validation (BCV) criterion is deﬁned as
BCV (vn, εn) = AMISE(fn). (4.3)

5. A Simulation Study
Here we consider parent distributions to estimate as exponential (χ2
2),
χ2
6, lognormal, Weibull and mixture of exponential densities.
Since the computation is very extensive for obtaining the smoothing
parameters, we compute approximations to MISE and MSE by
computing
ISE(fn, f) =
∞
0
[fn(x) − f(x)]2
dx
and
SE (fn(x), f(x)) = [fn(x) − f(x)]2
for 1000 samples.

Here, MISE give us the global performance of density estimator.
MSE let us to see how the density estimator performs locally at the
points in which we might be interested. It is no doubt that we
particularly want to know the behavior of density estimators near the
lower boundary. We illustrate only MISE values.
Optimal values of smoothing parameters are obtained using either
BCV or UCV criterion, that roughly approximates Mean Integrated
Squared Error.
For Poisson smoothing as well as for Gamma smoothing BCV
criterion is found to be better, where as for Chen and Scaillet
method, use of BCV method is not tractable as the derivative of their
density is not explicitly available..

Next table gives the values of MISE for exponential density using new
estimators as compared with Chen’s and Scaillet estimators. Note
that we include the simulation results for Scaillet’s estimator using
RIG kernel only.
Inverse Gaussian kernel is known not to perform well for direct data
[see Kulasekera and Padgett (2006)]. Similar observations were noted
for LB data.

Table : Simulated MISE for χ2
2
Distribution Estimator
Sample Size
30 50 100 200 300 500
χ2
2
Chen-1 0.13358 0.08336 0.07671 0.03900 0.03056 0.02554
Chen-2 0.11195 0.08592 0.05642 0.03990 0.03301 0.02298
RIG 0.14392 0.11268 0.07762 0.06588 0.05466 0.04734
Poisson(F) 0.04562 0.03623 0.02673 0.01888 0.01350 0.01220
Poisson(G) 0.08898 0.06653 0.04594 0.03127 0.02487 0.01885
Gamma(F) 0.06791 0.05863 0.03989 0.03135 0.02323 0.01589
Gamma*(F) 0.02821 0.01964 0.01224 0.00796 0.00609 0.00440
Gamma(G) 0.09861 0.07663 0.05168 0.03000 0.02007 0.01317
Gamma*(G) 0.02370 0.01244 0.00782 0.00537 0.00465 0.00356

Table : Simulated MSE for χ2
2
Sample Size Estimator
x
0 0.1 1 2 5
n=30
I 0.1307 0.2040 0.0181 0.0044 0.0003
II 0.1187 0.2499 0.0173 0.0045 0.0012
III 0.2222 0.1823 0.0250 0.0074 0.0022
IV 0.1487 0.1001 0.0049 0.0015 0.0005
V 0.3003 0.2438 0.0286 0.0148 0.0013
VI 0.1936 0.1447 0.0117 0.0042 0.0002
VI* 0.0329 0.0286 0.0090 0.0030 9.8 × 10−5
VII 0.1893 0.1720 0.0209 0.0066 0.0003
VII* 0.0528 0.0410 0.0032 0.0020 8.4 × 10−5
n=50
I 0.1370 0.1493 0.0121 0.0030 0.0002
II 0.1279 0.1894 0.0112 0.0032 0.0008
III 0.2193 0.1774 0.0161 0.0046 0.0046
IV 0.1393 0.0885 0.0034 0.0012 0.0003
V 0.2939 0.1924 0.0218 0.0094 0.0007
VI 0.1808 0.1365 0.0101 0.0036 0.0001
VI* 0.0196 0.0172 0.0070 0.0024 6.8 × 10−5
VII 0.1584 0.1440 0.0168 0.0060 0.0002
VII* 0.0322 0.0236 0.0014 0.0012 4.8 × 10−5
I-Chen-1, II-Chen-2, III-RIG, IV-Poisson(F), V-Poisson(G), VI-Gamma(F), VI*-Corrected Gamma(F), VII-Gamma(G),
VII*-Corrected Gamma(G)

2
x
0 0.1 1 2 5
n=100
I 0.1442 0.8201 0.0070 0.0017 0.0001
II 0.1142 0.1391 0.0054 0.0019 0.0005
III 0.2151 0.1631 0.0091 0.0030 0.0020
IV 0.1335 0.0724 0.0023 0.0008 0.0002
V 0.2498 0.1267 0.0116 0.0050 0.0003
VI 0.1332 0.0823 0.0090 0.0032 0.0001
VI* 0.0105 0.0094 0.0047 0.0015 5.9 × 10−5
VII 0.1078 0.0980 0.0121 0.0051 0.0002
VII* 0.0280 0.0184 0.0006 0.0007 3.8 × 10−5
n=200
I 0.3327 0.0901 0.0046 0.0012 6.6 × 10−5
II 0.2111 0.0943 0.0027 0.0012 0.0003
III 0.2127 0.1896 0.0067 0.0019 0.0080
IV 0.1139 0.0545 0.0015 0.0005 0.0001
V 0.1908 0.0703 0.0056 0.0026 0.0001
VI 0.0995 0.0782 0.0065 0.0024 7.4 × 10−5
VI* 0.0137 0.0125 0.0031 0.0010 5.8 × 10−5
VII 0.0636 0.0560 0.0072 0.0038 0.0002
VII* 0.0217 0.0134 0.0002 0.0005 2.9 × 10−5
I-Chen-1, II-Chen-2, III-RIG, IV-Poisson(F), V-Poisson(G),VI-Gamma(F), VI*-Corrected Gamma(F),VII-Gamma(G),

Table : Simulated MISE for χ2
6
Sample Size
30 50 100 200 300 500
χ2
6
Chen-1 0.01592 0.01038 0.00578 0.00338 0.00246 0.00165
Chen-2 0.01419 0.00973 0.00528 0.00303 0.00224 0.00153
RIG 0.01438 0.00871 0.00482 0.00281 0.00208 0.00148
Poisson(F) 0.00827 0.00582 0.00382 0.00241 0.00178 0.00119
Poisson(G) 0.00834 0.00562 0.00356 0.00216 0.00166 0.00117
Gamma(F) 0.01109 0.00805 0.00542 0.00327 0.00249 0.00181
Gamma*(F) 0.01141 0.00844 0.00578 0.00345 0.00264 0.00193
Gamma(G) 0.01536 0.01063 0.00688 0.00398 0.00303 0.00213
Gamma*(G) 0.01536 0.01063 0.00688 0.00398 0.00303 0.00213

6
x
0 0.1 1 4 10
n=30
I 0.0017 0.0018 0.0018 0.0019 0.0001
II 0.0018 0.0017 0.0011 0.0017 0.0002
III 5.6 × 10−5
6.7 × 10−5
0.0006 0.0017 0.0002
IV 0.0016 0.0016 0.0012 0.0012 0.0001
V 0.0000 2.6 × 10−5
0.0017 0.0008 0.0001
VI 0.0011 0.0010 0.0019 0.0012 8.5 × 10−5
VI* 0.0015 0.0021 0.0020 0.0012 7.9 × 10−5
VII 0.0000 3.6 × 10−7
0.0058 0.0008 0.0001
VII* 0.0000 3.6 × 10−7
0.0058 0.0008 0.0001
n=50
I 0.0012 0.0013 0.0015 0.0012 0.0001
II 0.0013 0.0012 0.0008 0.0011 0.0001
III 4.6 × 10−5
5.7 × 10−5
0.0006 0.0005 .0001
IV 0.0011 0.0011 0.0010 0.0008 8.4 × 10−5
V 0.0000 6.7 × 10−5
0.0012 0.0005 0.0001
VI 0.0005 0.0005 0.0016 0.0008 5.5 × 10−5
VI* 0.0006 0.0015 0.0016 0.0008 5.3 × 10−5
VII 0.0000 4.3 × 10−6
0.0037 0.0004 7.9 × 10−5
VII* 0.0000 4.3 × 10−6
0.0037 0.0004 7.9 × 10−5
I-Chen-1, II-Chen-2, III-RIG, IV-Poisson(F), V-Poisson(G), VI-Gamma(F), VI*-Corrected Gamma(F), VII-Gamma(G),

For the exponential density, ˆfC2 has smaller MSEs at the boundary
and MISEs than ˆfC1. This means ˆfC2 performs better locally and
globally than ˆfC1. Similar result holds in direct data.
Poisson weight estimator based on Fn is found to be better than that
based on Gn.
Although Poisson weight estimator based on Gn has relatively smaller
MISEs, it has large MSEs at the boundary as well, just like Scaillet
estimator.
Scaillet estimator has huge MSEs at the boundary and the largest
MISEs.
Corrected Gamma estimators have similar and smaller MISE values as
compared to the corresponding Poisson weight estimators.
For χ2
6, all estimators have comparable global results. Poison weight
estimators based on Fn or Gn have similar performances and may be
slightly better than the others.

We have considered following additional distributions for simulation as well:
(i). Lognormal Distribution
f(x) =
1
√
2πx
exp{−(log x − µ)2
/2}I{x > 0};
(ii). Weibull Distribution
f(x) = αxα−1
exp(−xα
)I{x > 0};
(iii). Mixtures of Two Exponential Distribution
f(x) = [π
1
θ1
exp(−x/θ1) + (1 − π)
1
θ2
exp(−x/θ2]I{x > 0}.

Table : Simulated MISE for Lognormal with µ = 0
Sample Size
30 50 100 200 300 500
Lognormal
Chen-1 0.12513 0.08416 0.05109 0.03450 0.02514 0.01727
Chen-2 0.12327 0.08886 0.05200 0.03545 0.02488 0.01717
RIG 0.14371 0.09733 0.05551 0.03308 0.02330 0.01497
Poisson(F) 0.05559 0.04379 0.02767 0.01831 0.01346 0.01001
Poisson(G) 0.06952 0.04820 0.03158 0.01470 0.01474 0.01061
Gamma*(F) 0.06846 0.05614 0.03963 0.02640 0.01998 0.01470
Gamma*(G) 0.16365 0.12277 0.07568 0.04083 0.029913 0.02035

Table : Simulated MSE for Lognormal with µ = 0
x
0 0.1 1 5 8
n=30
I 0.1108 0.0618 0.0211 2.0 × 10−4
2.8 × 10−5
II 0.1045 0.0441 0.0196 6.0 × 10−4
6.4 × 10−5
III 0.0026 0.0494 0.0207 3.0 × 10−4
3.2 × 10−5
IV 0.1307 0.0485 0.0071 3.5 × 10−4
3.8 × 10−5
V 0.0009 0.0810 0.0126 2.2 × 10−4
2.6 × 10−5
VI* 0.0090 0.1546 0.0133 2.0 × 10−4
9.1 × 10−5
VII* 0.0007 0.7321 0.0121 8.2 × 10−5
9.4 × 10−6
n=50
I 0.1123 0.0535 0.0158 1.4 × 10−4
1.1 × 10−5
II 0.1056 0.0442 0.0110 3.7 × 10−4
2.7 × 10−5
III 0.0027 0.0436 0.0133 2.0 × 10−4
1.5 × 10−5
IV 0.1398 0.0482 0.0050 1.8 × 10−4
1.6 × 10−5
V 0.0020 0.0641 0.0090 1.3 × 10−4
1.3 × 10−5
VI* 0.0035 0.1349 0.0111 1.7 × 10−4
6.6 × 10−5
VII* 0.0000 0.5482 0.0080 4.7 × 10−5
4.9 × 10−6
I-Chen-1, II-Chen-2, III-RIG, IV-Poisson(F), V-Poisson(G), VI*-Corrected Gamma(F), VII*-Corrected Gamma(G)

Table : Simulated MSE for Lognormal with µ = 0
x
0 0.1 1 5 8
n=100
I 0.1044 0.0486 0.0086 5.1 × 10−5
8.2 × 10−6
II 0.1038 0.0412 0.0059 1.5 × 10−4
1.3 × 10−5
III 0.0028 0.0383 0.0064 7.2 × 10−5
7.2 × 10−6
IV 0.1053 0.0424 0.0029 4.9 × 10−5
4.7 × 10−6
V 0.0018 0.0541 0.0060 5.7 × 10−5
6.8 × 10−6
VI* 0.0033 0.1011 0.0086 1.1 × 10−4
4.7 × 10−5
VII* 0.0000 0.3237 0.0044 1.9 × 10−5
2.1 × 10−6
n=200
I 0.0854 0.0422 0.0054 2.7 × 10−5
2.9 × 10−6
II 0.0871 0.0387 0.0036 6.5 × 10−5
4.6 × 10−6
III 0.0024 0.0318 0.0042 3.5 × 10−5
3.3 × 10−6
IV 0.0663 0.0320 0.0019 2.3 × 10−5
2.4 × 10−6
V 0.0019 0.0309 0.0027 2.0 × 10−5
2.5 × 10−6
VI* 0.0058 0.0717 0.0058 9.2 × 10−5
3.3 × 10−5
VII* 0.0015 0.1780 0.0026 1.0 × 10−5
1.1 × 10−6

Table : Simulated MISE for Weibull with α = 2
Sample Size
30 50 100 200 300 500
Weibull
Chen-1 0.10495 0.06636 0.03884 0.02312 0.01700 0.01167
Chen-2 0.08651 0.05719 0.03595 0.02225 0.01611 0.01111
RIG 0.08530 0.05532 0.03227 0.01984 0.01470 0.01045
Poisson(F) 0.04993 0.03658 0.02432 0.01459 0.01179 0.00856
Poisson(G) 0.05288 0.03548 0.02268 0.01392 0.01106 0.00810
Gamma*(F) 0.08358 0.06671 0.04935 0.03169 0.02652 0.01694
Gamma*(G) 0.12482 0.08526 0.05545 0.03402 0.02731 0.02188

Table : Simulated MSE for Weibull with α = 2
x
0 0.1 1 2 3
n=30
I 0.0856 0.1343 0.0588 .0030 1.9 × 10−4
II 0.0949 0.0555 0.0398 .0116 1.4 × 10−4
III 0.0025 0.0802 0.0394 .0095 4.6 × 10−4
IV 0.0844 0.0548 0.0280 0.0086 6.9 × 10−4
V 0.0068 0.0636 0.0186 0.0031 3.1 × 10−5
VI* 0.0019 0.1049 0.0682 0.0053 0.0022
VII* 0.0000 0.2852 0.0336 0.0011 1.8 × 10−4
n=50
I 0.0644 0.0576 0.0349 0.0020 1.0 × 10−4
II 0.0679 0.0431 0.0223 0.0077 7.1 × 10−4
III 0.0021 0.0208 0.0218 0.0063 2.2 × 10−4
IV 0.0682 0.0427 0.0217 0.0059 3.7 × 10−4
V 0.0025 0.0453 0.0138 0.0018 1.6 × 10−5
VI* 1.1 × 10−6
0.0763 0.0560 0.0048 0.0018
VII* 0.0000 0.1865 0.0251 0.0008 1.4 × 10−4

Table : Simulated MISE for Mixture of Two Exponential Distributions with
π = 0.4, θ1 = 2 and θ2 = 1
Sample Size
30 50 100 200 300 500
Mixture
Chen-1 0.22876 0.17045 0.08578 0.06718 0.05523 0.03811
Chen-2 0.17564 0.15083 0.07331 0.08029 0.04931 0.03808
RIG 0.25284 0.20900 0.13843 0.10879 0.09344 0.07776
Poisson(F) 0.06838 0.05746 0.04116 0.02612 0.01896 0.01179
Poisson(G) 0.11831 0.09274 0.06863 0.05019 0.03881 0.03044
Gamma*(F) 0.04147 0.02645 0.01375 0.00758 0.00532 0.00361
Gamma*(G) 0.02534 0.01437 0.01091 0.01223 0.01132 0.00994

Table : Simulated MSE for Mixtures of Two Exponential Distributions with
π = 0.4, θ1 = 2 and θ2 = 1
x
0 0.1 1 2 10
n=30
I 0.3499 0.3075 0.0249 0.0037 2.6 × 10−6
II 0.3190 0.3181 0.0245 0.0071 1.3 × 10−5
III 0.5610 0.4423 0.0564 0.0056 2.9 × 10−6
IV 0.3778 0.1907 0.0057 0.0027 1.7 × 10−6
V 0.6409 0.3237 0.0156 0.0043 2.1 × 10−6
VI* 0.0652 0.0549 0.0098 0.0006 1.1 × 10−4
VII* 0.0696 0.0539 0.0065 0.0009 1.4 × 10−5
n=50
I 0.3158 0.7921 0.0128 0.0023 1.1 × 10−6
II 0.2848 0.7600 0.0143 0.0051 2.3 × 10−6
III 0.5582 0.8473 0.0364 0.0041 1.3 × 10−6
IV 0.3840 0.1633 0.0051 0.0020 1.0 × 10−6
V 0.6228 0.2673 0.0121 0.0028 1.3 × 10−6
VI* 0.0489 0.0414 0.0066 0.0004 7.7 × 10−5
VII* 0.0500 0.0336 0.0030 0.0007 9.2 × 10−6

The basic conclusion is that the smoothing based on Fn using
Poisson weights or corrected Gamma perform in a similar way and
produce better boundary correction as compared to Chen or Scaillet
asymmetric kernel estimators.
The smoothing based on Gn may have large local MSE near the
boundary and hence is not preferable over smoothing of Fn. A similar
message is given in Jones and Karunamuni (1997, Austr. J. Stat.).

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Acknowledgement of Collaborators
Most of the results in this talk are taken from the following papers
Chaubey, Y.P. and Li, J. (2013). Asymmetric kernel density estimator
for length biased data. In Contemporary Topics in Mathematics and
Statistics with Applications, Vol. 1, (Ed.: A. Adhikari, M.R. Adhikari
and Y.P. Chaubey), Pub: Asian Books Pvt. Ltd., New Delhi, India.
Chaubey, Y. P. , Li, J., Sen, A. and Sen, P. K. (2012). A New
Smooth Density Estimator for Non-Negative Random Variables.
Journal of the Indian Statistical Association 50, 83-104.
Chaubey, Y. P. and Sen, P. K. (1996). On Smooth Estimation of
Survival and Density Function. Statistics and Decision, 14, 1-22.
Chaubey, Y. P. , Sen, P. K. and Li, J. (2010a). Smooth Density
Estimation for Length Biased Data. Journal of the Indian Society of
Agricultural Statistics, 64(2), 145-155.

Talk slides are available on SlideShare:
http://www.slideshare.net/YogendraChaubey/Slides-CSM
THANKS!!

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