[Paper Review] Some estimators of the PDF and CDF of the Lindley Distribution

Sudhansu S. Maiti, Indrani Mukherjee|arXiv (Cornell University)|Apr 21, 2016

Statistical Distribution Estimation and Applications34 references3 citations

TL;DR

This paper proposes and compares multiple estimators—maximum likelihood, uniformly minimum variance unbiased (UMVUE), percentile, least squares, weighted least squares, Cramér-von-Mises, and Anderson-Darling—for the probability density function (PDF) and cumulative distribution function (CDF) of the Lindley distribution. Using Monte Carlo simulations, it finds that while UMVUE is optimal in the unbiased class due to minimum variance, MLE performs better in terms of mean squared error (MSE), making it preferable overall despite bias.

ABSTRACT

This article addresses the different methods of estimation of the probability density function (PDF) and the cumulative distribution function (CDF) for the Lindley distribution. Following estimation methods are considered: uniformly minimum variance unbiased estimator (UMVUE), maximum likelihood estimator (MLE), percentile estimator (PCE), least square estimator (LSE), weighted least square estimator (WLSE), Cramér-von-Mises estimator (CVME), Anderson-Darling estimator (ADE). Monte Carlo simulations are performed to compare the performances of the proposed methods of estimation.

Motivation & Objective

• To develop and compare various estimators for the PDF and CDF of the Lindley distribution, shifting focus from parameter estimation to function estimation.
• To evaluate the performance of different estimation methods—MLE, UMVUE, PCE, LSE, WLSE, CVME, ADE—on the PDF and CDF.
• To assess the finite-sample performance of these estimators via Monte Carlo simulation, particularly in terms of mean squared error (MSE).
• To provide analytical expressions for UMVUE of the PDF and CDF based on sufficient statistics and conditional distributions.
• To demonstrate that MLE, though biased, offers superior MSE performance compared to UMVUE, justifying its practical preference.

Proposed method

• Derives the maximum likelihood estimator (MLE) for the parameter θ using the invariance principle, then applies it to estimate the PDF and CDF via substitution into the Lindley distribution formulas.
• Constructs the UMVUE for the PDF and CDF using the conditional distribution of X₁ given the complete sufficient statistic T = ΣXᵢ, leveraging Basu’s theorem and the Lehmann-Scheffé theorem.
• Derives the PDF and CDF estimators under the least squares (LSE) and weighted least squares (WLSE) methods by minimizing the sum of squared differences between empirical and theoretical values.
• Proposes percentile estimators (PCE) by inverting the CDF and minimizing a weighted sum of squared deviations between observed and theoretical percentiles.
• Develops Cramér-von-Mises (CVME) and Anderson-Darling (ADE) estimators by minimizing distance functions between empirical and theoretical distribution functions.
• Employs Monte Carlo simulation with 1,000 replications to compute and compare the mean squared errors (MSEs) of all estimators across varying sample sizes.

Experimental results

Research questions

• RQ1 Which estimator yields the lowest mean squared error (MSE) for the PDF and CDF of the Lindley distribution in finite samples?
• RQ2 How does the performance of the uniformly minimum variance unbiased estimator (UMVUE) compare to the maximum likelihood estimator (MLE) in terms of MSE and variance?
• RQ3 To what extent do alternative estimators—percentile, least squares, weighted least squares, Cramér-von-Mises, and Anderson-Darling—perform relative to MLE and UMVUE?
• RQ4 Can analytical expressions for UMVUE of the PDF and CDF be derived using the conditional distribution of order statistics given the sufficient statistic?
• RQ5 Does the MLE, despite being biased, offer a better trade-off between bias and variance than the unbiased UMVUE for the PDF and CDF?

Key findings

• The MLE of the PDF and CDF outperforms the UMVUE in terms of mean squared error (MSE), even though the MLE is biased.
• The UMVUE for the PDF and CDF is derived analytically using the conditional distribution of X₁ given the sum T, and is optimal within the unbiased class.
• As sample size increases, the MSEs of all estimators decrease, confirming their consistency.
• The simulation study shows that MSEs for all estimators decrease with increasing sample size, supporting the asymptotic properties of the estimators.
• Among the considered methods, MLE provides the smallest MSE for both the PDF and CDF, indicating superior finite-sample performance despite bias.
• The Cramér-von-Mises and Anderson-Darling estimators are derived via minimum distance principles, with the latter showing faster convergence to asymptotic behavior.