The Distribution of Products of Independent Random Variables

By M. Springer, W. E. Thompson. 1966

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Citations in the corpus

Computing functions of random variables via reproducing kernel Hilbert space representations (2015, neutral): “…suggests that Fourier transforms may help, and indeed, people have used transforms such as the ones due to Fourier and Mellin to derive the distribution function of either the sum, difference, product, or quotient of iRVs (Epstein 1948; Springer and Thompson 1966; Prasad 1970; Springer 1979).”
Are Psychological and Ecological Well-being Compatible? The Role of Values, Mindfulness, and Lifestyle (2005, background): “…the regression coefficient by its standard error, then multiplying the zs together, and comparing the resulting value (labeled a P statistic) to critical values in the theoretical sampling distribution of the product of two normal random variables (obtained from Springer and Thompson, 1966).”
Distribution of the product confidence limits for the indirect effect: Program PRODCLIN (2007, background): “…(Hanushek & Jackson, 1977), the distribution of the product can be applied to the use of the product as a test of the indirect effect based on the product z z , where z / and z / The distribution of the product of two normal variables is not normal (Lomnicki, 1967; Springer & Thompson, 1966).”
A comparison of methods to test mediation and other intervening variable effects. (2002, methodology): “This test involves converting both the α and the β paths to z scores, multiplying the zs, and using a critical value based on the distribution of the product of random variables, P = zαzβ, from Craig (1936; see also Meeker et al., 1981; Springer & Thompson, 1966) to determine significance.”
The role of mindfulness in romantic relationship satisfaction and responses to relationship stress. (2007, unknown): “The P value is then compared to critical values in the sampling distribution of the product of two normal random variables (Springer & Thompson, 1966).”
Review and Recommendations for Integrating Mediation and Moderation (2017, background): “As the description suggests, this estimator assumes that the sampling distribution of the indirect effect is normal, which is not true unless the indirect effect is zero (Springer & Thompson, 1966).”
Multilevel Mediation With Small Samples: A Cautionary Note on the Multilevel Structural Equation Modeling Framework (2017, methodology): “Although easy to calculate, the approximation of the product of the mediation paths (as noted by the dot above the tilde in Equation 8) as a normal distribution is not always appropriate (MacKinnon et al., 2002; Springer & Thompson, 1966).”
Confidence Limits for the Indirect Effect: Distribution of the Product and Resampling Methods (2004, methodology): “Although computation of these values is complex, Springer and Thompson (1966)”
The Importance of Indirect Effects in Media Effects Research: Testing for Mediation in Structural Equation Modeling (2003, background): “Researchers can then look to a product of two random, normal variables table to establish statistical 564 Journal of Broadcasting & Electronic MedidDecember 2003 significance (see Craig, 1936; Meeker, Cornwell, & Aroian, 1981; Springer & Thompson, 1966).”
Resampling and Distribution of the Product Methods for Testing Indirect Effects in Complex Models (2008, background): “Springer and Thompson (1966) presented tables of the product when both coefficients are equal to zero.”