in:

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  is the idiosyncratic return, denoted as $E_{i,d}$, which is estimated by the regression model $R_{i,d} = \alpha_i + \beta_i R_{m,d} + \gamma_i R_{ind,d} + E_{i,d}$. Among them, $R_{i,d}$ is the total return of stock i on day d, $R_{m,d}$ is the return of the market portfolio on day d, and $R_{ind,d}$ is the return of the industry portfolio on day d. $\alpha_i$ is the intercept term, $\beta_i$ is the market risk exposure coefficient, and $\gamma_i$ is the industry risk exposure coefficient. $E_{i,d}$ represents the idiosyncratic return of individual stocks after excluding market and industry factors, which is the return part that really matters in factor construction.

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  is the tail threshold, which is used to distinguish significant tail areas. It can usually be set to a multiple of the standard deviation, such as 1.5 or 2, representing the rate of return beyond 1.5 or 2 times the standard deviation. This parameter determines the range of the tail area we are concerned about. When the k value increases, the tail area of ​​concern will also decrease. Generally speaking, it can be reasonably set by the standard deviation of historical returns.

• For the yield data used in factor calculation, it is recommended to use the daily yield data of the past three months (about 60 trading days) to ensure the adequacy and timeliness of the data. The length of the data window can be adjusted according to the specific research purpose and market environment.

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  is the bandwidth parameter of the kernel density estimation, and its size determines the smoothness of the kernel function, which in turn affects the accuracy of the density estimation. Here, Silverman's rule of thumb (Silverman, 1986) is used to automatically select the bandwidth. The specific formula is $h ≈ 1.06\hat{\sigma}n^{-1/5}$, where $\hat{\sigma}$ is the standard deviation of the yield sample and n is the number of samples. This rule of thumb is widely used in practice and can better balance the bias and variance of the estimate.

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  The kernel density estimation function $\bar{f}(x)$ representing the return distribution is estimated using historical return data.

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  Represents the kernel density estimation function assuming a symmetric distribution. In actual calculations, we can use a translated Gaussian kernel function so that the center of the symmetric distribution is aligned with the mean of the actual returns. This symmetric distribution serves as a benchmark for comparison.

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  Represents the mean of the return distribution. The $ ext{Sign}(E_φ)$ function represents the sign of the mean return, ensuring that the positive and negative directions of the factor are consistent with the direction of the mean return. This sign function makes the factor take positive values ​​when the return is positive and negative values ​​when the return is negative, which is convenient for subsequent analysis.