Nonparametric two-sample estimation of location and scale parameters from empirical characteristic functions

ABSTRACT

Two random variables X and Y are said to belong to the same location-scale family when $Y\stackrel{D}{=}\mu +\sigma X$ with unknown constants $\mu \in \mathbb{R}$ and $\sigma >0$. Given iid observations $X_i$, $i=1,\dots ,n$ and $Y_i$, $i=1,\dots ,m$ satisfying this location-scale assumption, we wish to estimate μ and σ with high efficiency in the absence of knowledge of the functional form of the underlying common family of distributions of X and Y. Here, ‘high efficiency’ means that the estimator is asymptotically unbiased and that its asymptotic variance is close to the asymptotic variance of the maximum likelihood estimator that would be used had the form of the underlying location-scale family of distributions been known. We propose in the present paper two methods for estimating these parameters based on the empirical characteristic function (ECF). The first approach considered minimizes a weighted $L^2$ distance between the ECFs of the X and Y data. The second approach constructs a quadratic form comparing the real and imaginary parts of the X- and Y-sample ECFs at a preselected number of points. In both approaches, the constructed distance metric is minimized to estimate μ and σ. The asymptotic distributions of the estimators are found, and small sample performance is investigated via a Monte Carlo simulation study.

KEYWORDS:

• Empirical characteristic functions
• location-scale families
• semiparametric estimation
• bootstrap

AMS SUBJECT CLASSIFICATION:

• 62G05
• 62G09
• 62G20
• 62G30

Disclosure statement

No potential conflict of interest was reported by the authors.