Asymptotic properties of conditional distribution estimator with truncated, censored and dependent data

Liang, Han-Ying; Uña‐Álvarez, Jacobo de; Iglesias-Pérez, María del Carmen

doi:10.1007/s11749-012-0281-7

Cited by 29 publications

(28 citation statements)

References 22 publications

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“…The consistency and asymptotic normality of Theorem 1 follows directly from Wu & Yin () and thus the proofs are omitted. Under conditions C1–C4 and C6, and following Theorem 2.3 of Liang, de Una‐Alvarez, & Iglesias‐Perez (), we obtain the Bahadur representation of

{\overset{trueˆ}{F}}_{T^{*}} false(t false| z false),

{\overset{trueˆ}{F}}_{T^{*}} false(t false| z false) - F_{0 T^{*}} false(t false| z false) = \frac{1}{{nh}_{n}^{p} f_{boldZ} false(z false)} \sum_{i = 1}^{n} K_{p} (\frac{boldz - Z_{i}}{h_{n}}) ψ false(O_{i}, t, z; {boldγ}_{0}, F_{0 T^{*}} false) + O_{P} false(α_{n} false),

where

ψ false(scriptO, t, z; {boldγ}_{0}, F_{0 T^{*}} false)

is a measurable function with a zero mean and finite variance for any t and

boldz

and

α_{n} = h_{n}^{ℓ} + {log n / ({italicnh}_{n}^{p})}^{3 / 4}

.…”

Section: Asymptotic Propertiesmentioning

confidence: 70%

Cure rate quantile regression accommodating both finite and infinite survival times

Yin

2016

Can J Statistics

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In survival analysis a proportion of patients may be cured by the treatment, and thus they become risk‐free of the event of interest and their survival times change to infinity. The existence of such a survival fraction often makes the underlying population more heterogeneous and heavily right‐skewed. Compared with the traditional mean‐ or hazard‐based regression methods, quantile regression is more suitable for such survival data as it is more robust against outliers or infinite survival times. Moreover, it offers a comprehensive assessment of the covariate effects on the survival times at different quantile levels. We propose a new cure rate quantile regression model for the entire population including both finite and infinite survival times. By invoking non‐parametric functional estimation an iterative algorithm is developed to estimate the cure rate parameters. The scheme of redistribution‐of‐mass to the right for censored data is adopted to estimate the quantile regression parameters. The consistency and asymptotic normality of the proposed estimators are established. Extensive simulation studies are conducted to evaluate the finite‐sample performance of the proposed method, which is further illustrated with a phase III melanoma clinical trial study. The Canadian Journal of Statistics 45: 29–43; 2017 © 2016 Statistical Society of Canada

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{\overset{trueˆ}{F}}_{T^{*}} false(t false| z false),

{\overset{trueˆ}{F}}_{T^{*}} false(t false| z false) - F_{0 T^{*}} false(t false| z false) = \frac{1}{{nh}_{n}^{p} f_{boldZ} false(z false)} \sum_{i = 1}^{n} K_{p} (\frac{boldz - Z_{i}}{h_{n}}) ψ false(O_{i}, t, z; {boldγ}_{0}, F_{0 T^{*}} false) + O_{P} false(α_{n} false),

where

ψ false(scriptO, t, z; {boldγ}_{0}, F_{0 T^{*}} false)

is a measurable function with a zero mean and finite variance for any t and

boldz

and

α_{n} = h_{n}^{ℓ} + {log n / ({italicnh}_{n}^{p})}^{3 / 4}

.…”

Section: Asymptotic Propertiesmentioning

confidence: 70%

Cure rate quantile regression accommodating both finite and infinite survival times

Yin

2016

Can J Statistics

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“…Let {χ i , i ≥ 1} be a stationary α-mixing sequence of real random variables with mixing coefficients {α(k)}. Lemma 5.1 (see [20]) Let α(n) = O(n −λ ) for some λ > 2, and let τ be a finite positive…”

Section: Appendixmentioning

confidence: 99%

“…Lemma 5.2 (see [20]) Set ξ(Z, T, δ, y, x) = I(Z≤y,δ=1) [20]) Let x ∈ D and α(n) = O(n −λ ) for some λ > 6. Suppose that conditions (A1 )-(A4 ), (B1)-(B2), (B3)(ii) and (E2)-(E3) hold.…”

Section: Appendixmentioning

confidence: 99%

Conditional quantile estimation with truncated, censored and dependent data

Liang

Miao

2015

Chin. Ann. Math. Ser. B

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This paper deals with the conditional quantile estimation based on left-truncated and right-censored data. Assuming that the observations with multivariate covariates form a stationary α-mixing sequence, the authors derive the strong convergence with rate, strong representation as well as asymptotic normality of the conditional quantile estimator. Also, a Berry-Esseen-type bound for the estimator is established. In addition, the finite sample behavior of the estimator is investigated via simulations.

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“…Under independent and identically distributed assumptions, Iglesias‐Pérez & González‐Manteiga () obtained an almost sure representation and asymptotic normality of

{\overset{F}{true}}_{n} (MathClass-bin\cdot MathClass-rel| x)

. Liang et al () investigated for the first time the asymptotic properties of

{\overset{F}{true}}_{n} (MathClass-bin\cdot MathClass-rel| x)

with α ‐mixing data.…”

Section: Introductionmentioning

confidence: 99%

A Central Limit Theorem in Non‐parametric Regression with Truncated, Censored and Dependent Data

Liang

Uña‐Álvarez

Iglesias-Pérez

2014

Scandinavian J Statistics

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On the basis of the idea of the Nadaraya–Watson (NW) kernel smoother and the technique of the local linear (LL) smoother, we construct the NW and LL estimators of conditional mean functions and their derivatives for a left‐truncated and right‐censored model. The target function includes the regression function, the conditional moment and the conditional distribution function as special cases. It is assumed that the lifetime observations with covariates form a stationary α‐mixing sequence. Asymptotic normality of the estimators is established. Finite sample behaviour of the estimators is investigated via simulations. A real data illustration is included too.

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Asymptotic properties of conditional distribution estimator with truncated, censored and dependent data

Cited by 29 publications

References 22 publications

Cure rate quantile regression accommodating both finite and infinite survival times

Cure rate quantile regression accommodating both finite and infinite survival times

Conditional quantile estimation with truncated, censored and dependent data

A Central Limit Theorem in Non‐parametric Regression with Truncated, Censored and Dependent Data

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