This function computes the nonexceedance probability of a given quantile from the weighted combination of two quantile functions. The distributions are specified by the two parameter object arguments. The left-tail parameter object is the distribution governing the left tail; the right-tail parameter object is the distribution governing the right tail. The quantile function algebra is$$Q(F) = (1-F) \times Q_{lefttail}(F) + F \times Q_{righttail}(F)$$
where $Q(F)$ is the equivalent quantile for nonexceedance probability $F$ computed by the tail weigthing. $Q_{lefttail}(F)$ is the left-tail quantile function; $Q_{righttail}$ is the right-tail quantile function. The par2cdf2 function inverses the above equation for $F$. Parameter objects are discussed in vec2par and lmom2par functions. If the optional weight argument is provided, then the multiplication of 1-F or F is replaced by 1-weight or weight, respectively. If weight=0, then the quantiles for the right tail are returned, and if weight=1, then the quantiles for the left tail are returned.
A word of caution. The resulting weighted quantile function is not checked for monotonic increase with $F$. This is a required property of quantile functions.