Robust bootstrap procedures for the chain-ladder method

Insurers are faced with the challenge of estimating the future reserves needed to handle historic and outstanding claims that are not fully settled. A well-known and widely used technique is the chain-ladder method, which is a deterministic algorithm. To include a stochastic component one may apply generalized linear models to the run-off triangles based on past claims data. Analytical expressions for the standard deviation of the resulting reserve estimates are typically difficult to derive. A popular alternative approach to obtain inference is to use the bootstrap technique. However, the standard procedures are very sensitive to the possible presence of outliers. These atypical observations, deviating from the pattern of the majority of the data, may both inflate or deflate traditional reserve estimates and corresponding inference such as their standard errors. Even when paired with a robust chain-ladder method, classical bootstrap inference may break down. Therefore, we discuss and implement several robust bootstrap procedures in the claims reserving framework and we investigate and compare their performance on both simulated and real data. We also illustrate their use for obtaining the distribution of one year risk measures

An Inferential Method for Determining Which of Two Independent Variables Is Most Important When There Is Curvature

Consider three random variables Y, X1 and X2, where the typical value of Y, given X1 and X2, is given by some unknown function m(X1, X2). A goal is to determine which of the two independent variables is most important when both variables are included in the model. Let τ1 denote the strength of the association associated with Y and X1, when X2 is included in the model, and let τ2 be defined in an analogous manner. If it is assumed that m(X1, X2) is given by Y = β0 + β1X1 + β2X2 for some unknown parameters β0, β1 and β2, a robust method for testing H0 : τ1 = τ2 is now available. However, the usual linear model might not provide an adequate approximation of the regression surface. Many smoothers (nonparametric regression estimators) were proposed for estimating the regression surface in a more flexible manner. A robust method is proposed for assessing the strength of the empirical evidence that a decision can be made about which independent variable is most important when using a smoother. The focus is on LOESS, but it is readily extended to any nonparametric regression estimator of interest

Regression: Determining Which of p Independent Variables Has the Largest or Smallest Correlation With the Dependent Variable, Plus Results on Ordering the Correlations Winsorized

In a regression context, consider p independent variables and a single dependent variable. The paper addresses two goals. The first is to determine the extent it is reasonable to make a decision about whether the largest estimate of the Winsorized correlations corresponds to the independent variable that has the largest population Winsorized correlation. The second is to determine the extent it is reasonable to decide that the order of the estimates of the Winsorized correlations correctly reflects the true ordering. Both goals are addressed by testing relevant hypotheses. Results in Wilcox (in press a) suggest using a multiple comparisons procedure designed specifically for the situations just described, but execution time can be quite high. A modified method for dealing with this issue is proposed

On Flexible Tests of Independence and homoscedasticity

Consider the nonparametric regression model Y = m(X) + τ(X)ε , where X and ε are independent random variables, ε has a mean of zero and variance σ2, τ is some unknown function used to model heteroscedasticity, and m(X) is an unknown function reflecting some conditional measure of location associated with Y, given X. Detecting dependence, by testing the hypothesis that m(X) does not vary with X, has the potential of being more sensitive to a wider range of associations compared to using Pearson\u27s correlation. This note has two goals. The first is to point out situations where a certain variation of an extant test of this hypothesis fails to control the probability of a Type I error, but another variation avoids this problem. The successful variation provides a new test of H0:τ(X) ≡ 1, the hypothesis that the error term is homoscedastic, which has the potential of higher power versus a method recently studied by Wilcox (2006). The second goal is to report some simulation results on how this method performs

TREATMENT OF INFLUENTIAL OBSERVATIONS IN THE CURRENT EMPLOYMENT STATISTICS SURVEY

It is common for many establishment surveys that a sample contains a fraction of observations that may seriously affect survey estimates. Influential observations may appear in the sample due to imperfections of the survey design that cannot fully account for the dynamic and heterogeneous nature of the population of businesses. An observation may become influential due to a relatively large survey weight, extreme value, or combination of the weight and value. We propose a Winsorized estimator with a choice of cutoff points that guarantees that the resulting mean squared error is lower than the variance of the original survey weighted estimator. This estimator is based on very un-restrictive modeling assumptions and can be safely used when the sample is sufficiently large. We consider a different approach when the sample is small. Estimation from small samples generally relies on strict model assumptions. Robustness here is understood as insensitivity of an estimator to model misspecification or to appearance of outliers. The proposed approach is a slight modification of the classical linear mixed model application to small area estimation. The underlying distribution of the random error term is a scale mixture of two normal distributions. This setup can describe outliers in individual observations. It is also suitable for a more general situation where units from two distinct populations are put together for estimation. The mixture group indicator is not observed. The probabilities of observations coming from a group with a smaller or larger variance are estimated from the data. These conditional probabilities can serve as the basis for a formal test on outlyingness at the area level. Simulations are carried out to compare several alternative estimators under different scenarios. Performance of the bootstrap method for prediction confidence intervals is investigated using simulations. We also compare the proposed method with alternative existing methods in a study using data from the Current Employment Statistics Survey conducted by the U.S. Bureau of Labor Statistics

Alexander-Govern test using Winsorized means

Classical tests for testing the equality of independent groups which are based on arithmetic mean can produce invalid results especially when dealing with non-normal data and heterogeneous variances (heteroscedasticity). In alleviating the problem, researchers are working on methods that are more adapt to the aforementioned conditions which include a procedure known as Alexander-Govern test. This procedure is insensitive in the presence of heteroscedasticity under normal distribution. However, the test which employs the arithmetic mean as the central tendency measure is sensitive to non-normal data. This is due to the fact that the arithmetic mean is easily influenced by the shape of distribution. In this study, the arithmetic mean is replaced by robust estimators, namely the Winsorized mean or adaptive Winsorized mean. The proposed Alexander-Govern test with Winsorized mean and with adaptive Winsorized mean are denoted as AGW and AGAW, respectively. For the purpose of comparison, different Winsorization percentages of 5%, 10%, 15% and 20% are considered. A simulation study was conducted to investigate on the performance of the tests which is based on rate of Type I error and power. Four variables; shape of distribution, sample size, level of variance heterogeneity and nature of pairings are manipulated to create the conditions which could highlight the strengths and weaknesses of each test. The performance of the proposed tests is compared with their parametric counterparts, the t-test and ANOVA. The proposed tests show improvement in terms of controlling Type I Error and increasing power under the influence of heteroscedasticity and non-normality. The AGAW test performed best with 10% Winsorization while AGW test performed best with 5% Winsorization. Under most conditions (74%), AGAW tests outperform AGW tests. Therefore, the Winsorized mean and the adaptive Winsorized mean can significantly improve the performance of the original Alexander-Govern test. These proposed procedures are beneficial to statistical practitioners in testing the equality of independent groups even under the influence of non-normality and variance heterogeneity

Regression When There Are Two Covariates: Some Practical Reasons for Considering Quantile Grids

When dealing with the association between some random variable and two covariates, extensive experience with smoothers indicates that often a linear model poorly reflects the nature of the association. A simple approach via quantile grids that reflects the nature of the association is given. The two main goals are to illustrate this approach can make a practical difference, and to describe R functions for applying it. Included are comments on dealing with more than two covariates

Data Transformations for Inference with Linear Regression: Clarifications and Recommendations

Data transformations have been promoted as a popular and easy-to-implement remedy to address the assumption of normally distributed errors (in the population) in linear regression. However, the application of data transformations introduces non-ignorable complexities which should be fully appreciated before their implementation. This paper adds to existing Practical Research and Assessment Evaluation (PARE) publications on data transformations by providing a broad overview underlying the use of data transformations for the specific purpose of statistical inference and interpreting meaningful effect sizes. Data transformations not only potentially change the scale of the transformed variable; they also alter the fundamental relationships among variables while simultaneously changing the distribution of the errors. Given these repercussions, we clarify the nature of certain data transformations and strongly recommend the use of data transformations when they can enhance the interpretation of effect sizes. Accessed 5,515 times on https://pareonline.net from October 11, 2017 to December 31, 2019. For downloads from January 1, 2020 forward, please click on the PlumX Metrics link to the right

A Power Comparison of Robust Test Statistics Based On Adaptive Estimators

Seven test statistics known to be robust to the combined effects of nonnormality and variance heterogeneity were compared for their sensitivity to detect treatment effects in a one-way completely randomized design containing four groups. The six Welch-James-type heteroscedastic tests adopted either symmetric or asymmetric trimmed means, were transformed for skewness, and used a bootstrap method to assess statistical significance. The remaining test, due to Wilcox and Keselman (2003), used a modification of the well-known one-step M-estimator of central tendency rather than trimmed means. The Welch-James-type test is recommended because for nonnormal data likely to be encountered in applied research settings it should be more powerful than the test presented by Wilcox and Keselman. However, the reverse is true for data that are extremely nonnormal

Bivariate modified hotelling’s T2 charts using bootstrap data

The conventional Hotelling’s  charts are evidently inefficient as it has resulted in disorganized data with outliers, and therefore, this study proposed the application of a novel alternative robust Hotelling’s  charts approach. For the robust scale estimator , this approach encompasses the use of the Hodges-Lehmann vector and the covariance matrix in place of the arithmetic mean vector and the covariance matrix, respectively.  The proposed chart was examined performance wise. For the purpose, simulated bivariate bootstrap datasets were used in two conditions, namely independent variables and dependent variables. Then, assessment was made to the modified chart in terms of its robustness. For the purpose, the likelihood of outliers’ detection and false alarms were computed. From the outcomes from the computations made, the proposed charts demonstrated superiority over the conventional ones for all the cases tested