Abstract:This paper proposes the singly and doubly correlated bivariate noncentral F (BNCF) distributions. The probability density function (pdf) and the cumulative distribution function (cdf) of the distributions are derived for arbitrary values of the parameters. The pdf and cdf of the distributions for different arbitrary values of the parameters are computed, and their graphs are plotted by writing and implementing new R codes. An application of the correlated BNCF distribution is illustrated in the computations of the power function of the pretest test for the multivariate simple regression model (MSRM).Note: The following files were submitted by the author for peer review, but cannot be converted to PDF. You must view these files (e.g. movies) online. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 The bivariate central F (BCF) distribution has been studied by many authors, including Krishaniah (1965a), Amos and Bulgren (1972), Schuurmann et al. (1975), Johnson et al. (1995) and El-Bassiouny and Jones (2009). Krishnaiah (1965b) described the use of the BCF distribution in a problem of simultaneous statistical inference. Krishnaiah (1965c) and Krishnaiah and Armitage (1965) later studied the multivariate central F distribution. Hewett and Bulgren (1971) studied the prediction interval for failure times in certain life testing experiments using the multivariate central F distribution.Many authors have also studied the univariate noncentral F distribution, including Mudholkar et al. (1976), Muirhead (1982, Johnson et al. (1995), and Shao (2005). Johnson et al. (1995 provided the definition of the univariate noncentral F distribution known as the singly noncentral F distribution. The authors also described the doubly noncentral F distribution with (ν 1 , ν 2 ) degrees of freedom and noncentrality parameters λ 1 and λ 2 as the ratio of two independent noncentral chi-square variables, χ Tiku (1966) proposed an approximation to the multivariate noncentral F distribution.In the study of improving the power of a statistical test by pre-testing the uncertain non-sample prior information (NSPI) on the value of a set of parameters (cf. Saleh andSen, 1983, andKhan, 2011a), the cdf of a bivariate noncentral chi-square distribution is used to compute the power function of the test. For large sample studies, the cdf of the bivariate noncentral chi-square (BNCC) distribution is used to compute the power function of the test for testing one subset of regression parameters after pre-testing on another subset of parameters of a multivariate simple regression model (MSRM) (cf. Saleh and Sen, 1983, Yunus andKhan, 2011a). For s...