Modified minimum distance estimators: definition, properties and applications

Abstract

Estimating the location and scale parameters of a distribution is one of the most crucial issues in Statistics. Therefore, various estimators are proposed for estimating them, such as maximum likelihood, method of moments and minimum distance (e.g. Cramér-von Mises—CvM and Anderson Darling—AD), etc. However, in most of the cases, estimators of the location parameter $\mu$ and scale parameter $\sigma$ cannot be obtained in closed forms because of the nonlinear function(s) included in the corresponding estimating equations. Therefore, numerical methods are used to obtain the estimates of these parameters. However, they may have some drawbacks such as multiple roots, wrong convergency, and non-convergency of iterations. In this study, we adopt the idea of Tiku (Biometrika 54:155–165, 1967) into the CvM and AD methodologies with the intent of eliminating the aforementioned difficulties and obtaining closed form estimators of the parameters $\mu$ and $\sigma$. Resulting estimators are called as modified CvM (MCvM) and modified AD (MAD), respectively. Proposed estimators are expressed as functions of sample observations and thus their calculations are straightforward. This property also allows us to avoid computational cost of iteration. A Monte-Carlo simulation study is conducted to compare the efficiencies of the CvM and AD estimators with their modified counterparts, i.e. the MCvM and MAD, for the normal, extreme value and Weibull distributions for an illustration. Real data sets are used to show the implementation of the proposed estimation methodologies.

Appendix

Let $x_{1}, x_{2}, \ldots , x_{n}$ be independent random variables from a cdf F(x). Without loss of generality, the location parameter $\mu$ and scale parameter $\sigma$ in dF(x) are assumed to be 0 and 1, respectively. Furthermore, $x_{(1)} \le x_{(2)} \le \cdots \le x_{(n)}$ denote the order statistics and $t_{(i)}$ are the expected values of the order statistics, i.e., $t_{(i)}=E(x_{(i)})$, $i=1,2,\ldots ,n$. It is also assumed that

$$\begin{aligned} \int \limits _{-\infty }^{\infty } |x| dF(x) < \infty . \end{aligned}$$

This implies that $t_{(i)}$ values exist and are finite for all $i=1,2,\ldots ,n$.

Theorem 1

(Hoeffding 1953, p. 93) Let h(x) be a real-valued continuous function such that

$$\begin{aligned} |h(x)| \le g(x) \end{aligned}$$

where the function g(x) is convex and

$$\begin{aligned} \int \limits _{-\infty }^{\infty } g(x) dF(x) < \infty . \end{aligned}$$

Then,

$$\begin{aligned} \lim \limits _{n \rightarrow \infty }\frac{1}{n}\sum _{i=1}^{n}h\left( t_{(i)}\right) =\int \limits _{-\infty }^{\infty } h(x) dF(x). \end{aligned}$$