Nonparametric Mean Estimation for Big-but-Biased Data ^†

Abstract

Some authors have recently warned about the risks of the sentence with enough data, the numbers speak for themselves. The problem of nonparametric statistical inference in big data under the presence of sampling bias is considered in this work. The mean estimation problem is studied in this setup, in a nonparametric framework, when the biasing weight function is unknown (realistic). The problem of ignoring the weight function is remedied by having a small SRS of the real population. This problem is related to nonparametric density estimation. The asymptotic expression for the MSE of the estimator proposed is considered. Some simulations illustrate the performance of the nonparametric method proposed in this work.

1. Introduction

At certain times a large sample is not representative of the population, but it is biased (B3D). Some of the problems coming from ignoring sampling bias in big data statistical analysis have been recently reported by Cao [1]. A good example cited by Crawford [2] is the data collected in the city of Boston through the StreetBump smartphone app that underestimates the number of potholes in some neighborhoods of the city, with the consequent deficient management of resources. Another example is the database of more than 20 million tweets generated by Hurricane Sandy. These data come from a biased sample of the population, since most of the tweets came from Manhattan, while few tweets were originated in the most affected areas by the catastrophe. In other examples, such as those cited in Hargittai [3], survey data show that the use of sites is biased yielding samples that limit the generalizability of findings.

In this context, let us consider a population with CDF F (density f) and consider a SRS, $\mathbf{X}=(X_1,\dots ,X_n)$, of size n from this population. Assume that we are not able to observe this sample but we observe, instead, another sample $\mathbf{Y}=(Y_1,\dots ,Y_N)$, of a much larger sample size ($N>>n$) from a biased distribution G (density g), such that $g\left(x\right)=w\left(x\right)f\left(x\right)$, for some weight function $w\left(x\right)\ge 0$, $\forall x$.

2. Mean Estimation in B3D

To deal with the mean estimation problem in this context, we propose the realistic estimator (unknown w case) whose motivation is explained by Cao and Borrajo [4]:

$${\widehat{\mu }}^{{\widehat{w}}_{h,b}}=\frac{{\displaystyle \frac{1}{N}\sum _{i=1}^N\frac{Y_i}{{\widehat{w}}_{h,b}\left(Y_i\right)}}}{{\displaystyle \frac{1}{N}\sum _{i=1}^N\frac{1}{{\widehat{w}}_{h,b}\left(Y_i\right)}}}=\frac{{\displaystyle \frac{1}{N}\sum _{i=1}^N{Y}_i\frac{{\widehat{f}}_h\left(Y_i\right)}{{\widehat{g}}_b\left(Y_i\right)}}}{{\displaystyle \frac{1}{N}\sum _{i=1}^N\frac{{\widehat{f}}_h\left(Y_i\right)}{{\widehat{g}}_b\left(Y_i\right)}}}.$$

(1)

In order to work with this estimator, extra information is required. We propose a scenario in which, in addition to the biased sample, $\mathbf{Y}$, we also observe a SRS, $\mathbf{X}$, of small size of the real population. The Parzen-Rosenblatt KDE (see [5,6]) based on $\mathbf{X}$ and $\mathbf{Y}$ can be used to estimate f and g.

The final expression of the AMSE of (1) ($h\to 0$, $b\to 0$, $nh\to \infty$, $Nb\to \infty$ and $N/n\to \infty$) is:

$$\begin{array}{ccc}\hfill AMSE\left({\widehat{\mu }}^{{\widehat{w}}_{h,b}}\right)& =& {\left(C_1{b}^2+\frac{C_2}{Nb}\right)}^2+\frac{C_3}{n}+\frac{C_4}{Nn}+\frac{C_5}{N^2}+\frac{C_6}{Nnh}+\frac{C_7}{N^{2}b}\hfill \\ & +& \frac{C_8{h}^2}{N^{2}b}+\frac{C_9{h}^4}{N}+\frac{C_{10}{b}^4}{N}+\frac{C_{11}{h}^2{b}^2}{N}+\frac{C_{12}h}{Nn}+\frac{C_{13}b}{N^2}.\hfill \end{array}$$

3. Case Study with Simulated Data

Let us consider $f\left(x\right)=\frac{3}{14}(x^2+1){\mathbf{1}}_{\left[0,2\right]}\left(x\right)$ and $w\left(x\right)=1.5{\mathbf{1}}_{\left[0,1.5\right]}\left(x\right)+x{\mathbf{1}}_{\left(1.5,2\right]}\left(x\right)$ (Figure 1a):

Figure 1b shows that the proposed estimator improves the estimation performed using the SRS, $\overline{X}$, and the biased sample, $\overline{Y}$, for a large number of combinations of h and b. Looking at

Table 1. MSE of the different estimators and optimal bandwidths obtained from the simulation study.

n	N	$\mathbf{MSE}\left(\overline{\mathit{X}}\right)$	$\mathbf{MSE}\left(\overline{\mathit{Y}}\right)$	$\mathbf{MSE}\left({\widehat{\mathit{\mu }}}^{{\mathit{w}}_{\mathit{h},\mathit{b}}}\right)$	h	b
10	100	$2.9\times {10}^{-2}$	$4.4\times {10}^{-3}$	$2.4\times {10}^{-3}$	1.99	1.05
50	2500	$5.6\times {10}^{-3}$	$1.7\times {10}^{-3}$	$9.9\times {10}^{-5}$	3.97	1.18
100	$10,000$	$2.9\times {10}^{-3}$	$1.6\times {10}^{-3}$	$2.5\times {10}^{-5}$	5.00	1.20
500	$250,000$	$5.0\times {10}^{-4}$	$1.6\times {10}^{-3}$	$1.1\times {10}^{-6}$	12.22	1.23
1000	$1,000,000$	$2.0\times {10}^{-4}$	$1.6\times {10}^{-3}$	$2.7\times {10}^{-7}$	12.22	1.24

, we observe that the best choice for h and b based on the simulation study contradicts the assumption ($h\to 0$, $b\to 0$) used in obtaining the asymptotic results. The AMSE for (1) under these non-standard asymptotic conditions ($h\to h_0$, $b\to b_0$) is:

$$AMSE\left({\widehat{\mu }}^{{\widehat{w}}_{h_0,b_0}}\right)=\frac{D_1}{N}+\frac{D_2}{Nn}+\frac{D_3}{N^2}+\frac{D_4}{N^3}.$$

4. Conclusions

Big Data brings new statistical challenges since bias is much more present. Ideas from length-biased data and nonparametric smoothing techniques are important in this context, testing for bias is a relevant problem in Big Data and smoothing parameter selection may be paradoxical in B3D.