Although the use of UWB signals can resolve these multipath components, however, the large number of multipath components in harsh environments makes direct path (DP) detection nontrivial. In addition, both the relative clock offsets as well as response delay strongly affect ranging accuracy [14]. UWB ranging with the appropriate error calibration methods is able to provide indoor positioning of high accuracy in the centimeter level, but it requires to perform the calibration for each indoor area of interest in advance [15]. A nonparametric machine learning approach is employed to estimate the ranging error directly from the received waveform without any a priori or a posteriori knowledge, and it has the potential to significantly improve localization performance [16]. Ranging based on the RSS is very sensitive to the path loss model, and a better placement of the base stations could bring better accuracy through better geometrical condition [17]. Moreover, an accurate indoor positioning system requires more than just accurate range measurements. Its performance is affected by the number and the configuration of the anchor, along with the accuracy of the anchor positions. In general, the error in the functional model will have more serious consequences for the parameter estimation than that of the stochastic model. The functional model of UWB positioning is nonlinear. The best method is [PERSON]'s expansion of the nonlinear distance equations, and then the approximate solutions can be obtained iteratively [18,19]. As a standard method for solving general nonlinear equations, the Gauss-Newton iteration is efficient and has a linear convergence rate for points close to the solution [20]. However, the estimation is usually inherently biased by linearizing the nonlinear expression, and the bias is derived from non-zero higher-order terms [21]. The bias in nonlinear least-squares estimator is studied to show that the relatively high precision of geodetic measurements often makes bias negligible, despite strong non-linearity in the functional model [22; 23]. The amount of nonlinearity of the model is related to the curvature of the manifold and the parameter curves with a differential geometry point of view [24]. As a differential geometric interpretation, the bias in parameter vector depends on the parametrization which is known as geodetic coordinates, and the bias in residual vector is determined by the mean curvature of manifold [25]. For short-distance indoor positioning, the bias in the nonlinear least-squares solution should be detected and the estimation can obtain unbiased values by removing the bias from the nonlinear least-squares solution [26; 27]. Therefore, we deem it essential to discuss the factors contributing to the bias and how the Taylor remainder affects the parameter estimators in nonlinear least-squares problems. Statistical testing for modelling errors and biases is an important component in data quality control [28]. The DIA (detection, identification and adaptation) method of model misspecifications combines parameter estimation with a hypothesis test with the aim to remove any biases that may be present [29; 30]. The minimal detectable bias (MDB) of an alternative hypothesis is the smallest outlier that can lead to rejection of a null hypothesis. The MDB describes the internal reliability, and the value propagated into the parameters is said to describe the external reliability [31]. Under the assumption with Gaussian distribution, the Chi-square test is performed to identify and exclude the suspected observations corresponding to these faults [32; 33]. A judging index is defined as the square of the Mahalanobis distance from the observation to its prediction, and the hypothesis test is performed to the actual observation by treating the judging index as the test statistic [34]. In particular, the remainder of this paper is organized as follows: In Section 2, the nonlinear least-squares solution of overdetermined distance equations is introduced.