Abstract: In general, classical methods such as maximum
likelihood (ML) and least squares (LS) estimation methods are used
to estimate the shape parameters of the Burr XII distribution.
However, these estimators are very sensitive to the outliers. To
overcome this problem we propose alternative robust estimators
based on the M-estimation method for the shape parameters of the
Burr XII distribution. We provide a small simulation study and a real
data example to illustrate the performance of the proposed estimators
over the ML and the LS estimators. The simulation results show that
the proposed robust estimators generally outperform the classical
estimators in terms of bias and root mean square errors when there
are outliers in data.
Abstract: The unit root tests based on the robust estimator for the first-order autoregressive process are proposed and compared with the unit root tests based on the ordinary least squares (OLS) estimator. The percentiles of the null distributions of the unit root test are also reported. The empirical probabilities of Type I error and powers of the unit root tests are estimated via Monte Carlo simulation. Simulation results show that all unit root tests can control the probability of Type I error for all situations. The empirical power of the unit root tests based on the robust estimator are higher than the unit root tests based on the OLS estimator.
Abstract: One of the purposes of the robust method of
estimation is to reduce the influence of outliers in the data, on the
estimates. The outliers arise from gross errors or contamination from
distributions with long tails. The trimmed mean is a robust estimate.
This means that it is not sensitive to violation of distributional
assumptions of the data. It is called an adaptive estimate when the
trimming proportion is determined from the data rather than being
fixed a “priori-.
The main objective of this study is to find out the robustness
properties of the adaptive trimmed means in terms of efficiency, high
breakdown point and influence function. Specifically, it seeks to find
out the magnitude of the trimming proportion of the adaptive
trimmed mean which will yield efficient and robust estimates of the
parameter for data which follow a modified Weibull distribution with
parameter λ = 1/2 , where the trimming proportion is determined by a
ratio of two trimmed means defined as the tail length. Secondly, the
asymptotic properties of the tail length and the trimmed means are
also investigated. Finally, a comparison is made on the efficiency of
the adaptive trimmed means in terms of the standard deviation for the
trimming proportions and when these were fixed a “priori".
The asymptotic tail lengths defined as the ratio of two trimmed
means and the asymptotic variances were computed by using the
formulas derived. While the values of the standard deviations for the
derived tail lengths for data of size 40 simulated from a Weibull
distribution were computed for 100 iterations using a computer
program written in Pascal language.
The findings of the study revealed that the tail lengths of the
Weibull distribution increase in magnitudes as the trimming
proportions increase, the measure of the tail length and the adaptive
trimmed mean are asymptotically independent as the number of
observations n becomes very large or approaching infinity, the tail
length is asymptotically distributed as the ratio of two independent
normal random variables, and the asymptotic variances decrease as
the trimming proportions increase. The simulation study revealed
empirically that the standard error of the adaptive trimmed mean
using the ratio of tail lengths is relatively smaller for different values
of trimming proportions than its counterpart when the trimming
proportions were fixed a 'priori'.