Abstract
This paper considers non-parametric density estimation in the context of multiplicative censoring. A new estimator for the density function is proposed and consistency of the proposed estimator is investigated. Simulations are drawn to illustrate the results and to show how the estimator behaves for finite samples.
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Appendices
Appendix A: Proofs of the main results
Proof of Theorem 1
Under \({\mathbf {K(1)}},\) \(\mathbf {H(2)}\) and according to Lemma 7.1 we have
Let
be an ordinary kernel density estimator function of \(q\), that is defiend by (2.1), and based on the sample of \((T_1,\ldots ,T_n)\) where \(T_i=-\ln Z_i.\) An interchange of expectation and integration, justified by Fubini’s Theorem and \(\mathbf { K(3)},\) show that
Then it follows from (6.2) that
Thus \(q_{n}\) has the same bias as the ordinary kernel density estimator \(q^{*}_{n}\). \({\mathbf {K(1)}}, {{\mathbf {H(1)}}}\) and Lemma 7.1 follow
hence
Therefore from (6.1), (6.3) and Assumption \({\mathbf {P}}\), we have
Assumption \({{\mathbf {H(2)}}}\) and Lemma 7.2 follow that
By proof of Theorem 2.1 of Stefanski and Carroll (1990), \({\mathbf {K(2)-K(4)}}\) and \({\mathbf {H(1)}}\), we have
It can be easily seen that
so by combining (6.5), (6.6) we have
Finally, (6.4) and (6.8) complete the proof. \(\square \)
Proof of Theorem 2
It is easy to see that
From G(1), G(2), H(3), K(1) and Lemma 7.3 we have
For \(I_{2}\) first we have
Let \(W_i\) has density function \(f^*\) with characteristic function \(\Psi _{f^{*}}.\) Since \(\Psi _{K_{2}}\) is vanished out of \([-1,1]\), we have
where
hence
where
Lemma 7.4 implies that
hence using Assumption \({\mathbf {K(3)}}\) we have
Using \(\mathbf G(1) \) and \( \mathbf K(1) \),
Now (6.11), (6.13) and (6.14) yield
(6.9), (6.10), and (6.15) complete the proof. \(\square \)
Appendix B
In this section, we consider the kernel density estimator \(f_n\) of a univariate density \(f\) introduced by Roenblatt (1956),
where \(\xi _{1},\xi _{2},\ldots ,\xi _{n}\) are independent observations from distribution \(F\) with density function \(f\). \(K,\) \(a_n\) and \(F_n\) are kernel function, a bandwidth and empirical distribution function, respectively. Also, the characteristic function of \(f\) and its empirical are denoted by
and
Lemma 7.1
(Corollary 1A, Parzen (1962)). The estimator defined by (7.1) is asymptotically unbiased at all points \(x\) at which the probability density function is continuous if the constants \(a_{n}\) satisfy \(\lim _{n \longrightarrow \infty }a_{n}=0\) and if the function \(K(\cdot )\) satisfies
and
\(\square \)
Lemma 7.2
(Theorem 2A, Parzen (1962)) The estimates defined by (7.1) have variances satisfying
at all points \(x\) of continuity of \(f(x),\) if \(\lim _{n \longrightarrow \infty } a_{n}=0\) and the function \(K\) satisfies (7.2) and (7.3). \(\square \)
Lemma 7.3
(Theorem 1.2., Stute (1982)) Let \(K\) be of bounded variation and suppose that \(K(x)=0\) outside some finite interval \([r,s)\). If
with \(J=(a, b),\) then for each \(\epsilon >0\) and every bandsequence \((a_{n})\), where \(\lim _{n\rightarrow \infty }\) \(a_{n}=0\) and \(\lim _{n \rightarrow \infty }\frac{\ln ({a^{-1}_{n}})}{na_{n}}=0,\) we have
where \(J_{\epsilon }=(a+\epsilon , b-\epsilon )\) and \(C\) is a constant. \(\square \)
Lemma 7.4
(Example (1), Csörgő (1985)) Suppose that \(P\left( |\xi |>x\right) \le Lx^{-\alpha }\) for all large enough \(x,\) where \(L\) and \(\alpha \) are arbitrary positive constants. Then for any \(A>0\)
where \(C\) is a positive constant and
for any extended number \(0<T\le \infty \). \(\square \)
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Zamini, R., Fakoor, V. & Sarmad, M. On estimation of a density function in multiplicative censoring. Stat Papers 56, 661–676 (2015). https://doi.org/10.1007/s00362-014-0602-x
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DOI: https://doi.org/10.1007/s00362-014-0602-x