I am looking for a monotonically decreasing function to fit a cumulative distribution. The distribution is the number of values of a random variable X, that are greater than Y as a function of Y. In total, there are a few hundred values of X so the distribution does not decrease smoothly. I am thinking of fitting it with a monotonically decreasing polynomial to obtain a smoothly decreasing function of Y. I would like to choose the degree of the polynomial to get a satisfactory fit. Can someone suggest a monotonically decreasing polynomial to use.
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A nonconstant polynomial $p$ is monotonically decreasing if and only if $p'\le 0$ everywhere. Therefore, $-p'$ is the sum of at most two squares of polynomials. (For example, Proposition 1.2.1 here). Therefore, all decreasing polynomials can be generated by the formula $$p(x)=-\int \left((a_0+a_1x+\dots+ a_nx^n)^2 + (b_0+b_1x+\dots +b_nx^n)^2 \right)\,dx \tag1$$ where the coefficients are arbitrary real numbers. The formula (1) generates all such polynomials up to degree $2n+1$.