On moment-type estimators for a class of log-symmetric distributions

Abstract

In this paper, we propose three simple closed form estimators for a class of log-symmetric distributions on ${\mathbb {R}}^{+}$. The proposed methods make use of some key properties of this class of distributions. We derive the asymptotic distributions of these estimators. The performance of the proposed estimators are then compared with those of the maximum likelihood estimators through Monte Carlo simulations. Finally, some illustrative examples are presented to illustrate the methods of estimation developed here.

Appendices

Appendix 1: Asymptotic joint distribution of $\widetilde{\nu }$ and $\widetilde{\theta }$

Let $\mathbf{X}=(X_{1},\ldots ,X_{n})^{\top }$ be independent and identically distributed random variables with log-symmetric distribution. Consider

$$\begin{aligned} {S}=\frac{1}{n}\sum _{j=1}^{n}X_{j} \quad {\text {and}} \quad R=\left[ \sum _{j=1}^{n}\frac{1}{X_{j}} \right] ^{-1}. \end{aligned}$$

From the strong law of large numbers, it is known that S and $R^{-1}$ converge almost surely to E[X] and $E[X^{-1}]$, respectively. Also from the central limit theorem, we readily observe that S and $R^{-1}$ are asymptotically normally distributed. Note that the vector $(S,R^{-1})^{\top }$ is bivariate normally distributed, that is,

$$\begin{aligned} \sqrt{n}\left( \begin{array}{l} S-E[X]\\ R^{-1}-E[X^{-1}] \end{array}\right) \sim {\text {N}} \left[ \left( \begin{array}{l} 0\\ 0 \end{array}\right) , \left( \begin{array}{c@{\quad }c} {\text {Var}}[X] &{} 1 - {\text {E}}[X]{\text {E}}[X^{-1}]\\ 1 - {\text {E}}[X]{\text {E}}[X^{-1}] &{} {\text {Var}}[X^{-1}] \end{array}\right) \right] . \end{aligned}$$

We now need to find the asymptotic joint distribution of $\widetilde{{\theta }} = f_1(S, R)$ and $\widetilde{\nu } = f_2(S, R)$. For this purpose, we make use of the delta method. Suppose ${\varvec{X}} = (x_1, x_2)^{\top }$, ${\varvec{\eta }}$ and ${\varvec{\Sigma }}$ are such that $\sqrt{n}({\varvec{X}} - {\varvec{\eta }})\mathop {\rightarrow }\limits ^{d}\mathrm{N}\left( 0,{\varvec{\Sigma }}\right)$. Let $f = (f_1(x_1, x_2), f_2(x_1, x_2))^{\top }$ be a mapping from ${\mathbb {R}}^{2}$ to ${\mathbb {R}}^2$, where each $f_i$ is differentiable at ${\varvec{\eta }}$. Let ${\mathbf {D}}$ be the Jacobian matrix of f with respect to ${\varvec{X}}$. Then, $\sqrt{n}(f({\varvec{X}}) - f({\varvec{\eta }}))\mathop {\rightarrow }\limits ^{d}\mathrm{N}\left( 0,{\mathbf {D}} {\varvec{\Sigma }} {\mathbf {D}}^{\top }\right)$.

Thus, $\sqrt{n}\left( f\left( R, S\right) - {(\theta , \nu )}^{\top }\right) \mathop {\rightarrow }\limits ^{d}\mathrm{N}\left( 0,{\mathbf {D}} {\varvec{\Sigma }} {\mathbf {D}}^{\top }\right)$, with

$$\begin{aligned} \mathbf{D} = \left( \begin{array}{c@{\quad }c} \frac{\partial \, f_1(x_1,x_2)}{\partial \, x_1} &{} \frac{\partial \, f_1(x_1,x_2)}{\partial \, x_2}\\ \frac{\partial \, f_2(x_1,x_2)}{\partial \, x_1} &{} \frac{\partial \, f_2(x_1,x_2)}{\partial \, x_2} \end{array}\right) \Bigg |_{x_1=E[X], \, x_2=E[X^{-1}]}. \end{aligned}$$

Appendix 2: Asymptotic distribution of $\widetilde{\nu }_{\star }$

First, note that

$$\begin{aligned} E[\,\overline{Z}\,]=E\left[ {X}\right] E\left[ \frac{1}{X}\right] \end{aligned}$$

and

$$\begin{aligned} E[\,\overline{Z}^{2}\,]= & {} \frac{1}{n^{2}(n-1)^{2}}E\left[ \sum _{1\le {i}\ne {j}\ne {k}\ne {l}\le {n}}\frac{{X}_{i}X_{j}}{X_{k}X_{l}} +\sum _{1\le {i}\ne {j}\ne {k}\le {n}}\frac{X_{i}^{2}}{X_{j}X_{k}}\right. \\&\left. +\sum _{1\le {i}\ne {j}\ne {k}\le {n}}\frac{X_{j}X_{k}}{X_{i}^{2}}\right. \\&\left. +2\sum _{1\le {i}\ne {j}\ne {k}\le {n}}\frac{X_{i}X_{j}}{X_{i}X_{k}} + \sum _{1\le {i}\ne {j}\le {n}}\frac{X_{i}^{2}}{X_{j}^{2}}+ \sum _{1\le {i}\ne {j}\le {n}}\frac{X_{i}X_{j}}{X_{j}X_{i}} \right] \\= & {} \frac{1}{n^{2}(n-1)^{2}}\left[ n(n-1)(n-2)(n-3)\left( E[X]E\left[ \frac{1}{X}\right] \right) ^2 \right. \\&+2n(n-1)(n-2)E[X]E\left[ \frac{1}{X}\right] \\&+ 2n(n-1)(n-2)E[X^2]\left( E\left[ \frac{1}{X}\right] \right) ^2 \\&+ \left. 2n(n-1) \right] . \end{aligned}$$

These expressions yield

$$\begin{aligned} {\text {Var}}[\,\overline{Z}\,]\approx \frac{2}{n}\left[ -2\left( E[X]E\left[ \frac{1}{X}\right] \right) ^2 + E[X]E\left[ \frac{1}{X}\right] +E[X^2]\left( E\left[ \frac{1}{X}\right] \right) ^2\right] . \end{aligned}$$

Upon using Taylor expansion, we have

$$\begin{aligned} \widetilde{\nu }_{\star } = g(\overline{z})=g(E[\,\overline{Z}\,]) +(\overline{z}-E[\,\overline{Z}\,])g'(E[\,\overline{Z}\,])+\frac{(\overline{z}-E[\,\overline{Z}\,])^2}{2}g''(E[\,\overline{Z}\,])+\cdots , \end{aligned}$$

where $g'(\cdot )$ and $g''(\cdot )$ denote the first and second derivatives of the function of $g(\cdot )$. We thus obtain the asymptotic distribution of $\widetilde{\nu }_{\star }$ as

$$\begin{aligned} \widetilde{\nu }_{\star }\xrightarrow [n\rightarrow \infty ]{}{\text {N}} \left( \nu ,[g'(E[\,\overline{Z}\,])]^2{\text {Var}}[\,\overline{Z}\,]\right) . \end{aligned}$$

We also obtain the bias of $\widetilde{\nu }_{\star }$ as

$$\begin{aligned} {\text {Bias}}(\widetilde{\nu }_{\star })\approx \frac{1}{2}g''(E[\,\overline{Z}\,]){\text {Var}}[\,\overline{Z}\,]. \end{aligned}$$

Appendix 3: R scripts

In this appendix, we present some R codes used in Sect. 5.