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It would be useful to provide the loss over time for both training and testing data set. From your description, it seems like you can minimize the training loss, but the testing performance is not going well. If that is the case, try to regularize the model more. One useful approach is doing data argumentation more.


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Although it's unnecessary for a solution, let's connect the elements of this question to familiar statistical concepts. Because $f$ is continuous and decreasing on $[0,\infty)$ it behaves like some multiple of a survival function for a distribution. We can turn it into one by ensuring none of its values exceed $1.$ Since the continuity and decreasing ...


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With the help of the hints I was provided, this is the solution: $\ln⁡(Y_n )=\frac1n (\ln⁡(X_1 )+\ln⁡(X_2 )+\cdots+\ln⁡(X_n ) )$ $\ln⁡(Y_n )=\frac1n\cdot \sum_{i=1}^n \ln⁡(X_i )$ $E[\ln⁡(X_i ) ]=\int_0^1 \ln⁡(x)\cdot 1\, dx=-1$ ($1$ comes from pdf of $X_i$) By Weak Law of Large Numbers, $$\ln⁡(Y_n ) \stackrel{p}\to E[\ln⁡(X_i ) ]=-1$$ Then by exp being ...


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You're missing a small but key point of the definition of weak convergence: $F_n \stackrel{\text w}\to F$ if $\lim_{n\to \infty} F_n(x) = F(x)$ for all $x$ that are continuity points of $F$. This is important because $x\mapsto \lim_n F_n(x)$ is not guaranteed to be a valid CDF, and your example shows this because $\lim_n F_n(0) = 0$ and this would violate ...


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Your issue are the bad starting values. You can easily fit this model using a self-starting model. fit <- nls(TOTAL_CUMULATIVE_AMOUNT ~ SSasympOrig(TENURE, Asym, lrc), data=DF) #C: coef(fit)[["Asym"]] #[1] 2159.835 #k (the log-transformation enforces the boundary condition): -exp(coef(fit)[["lrc"]]) #[1] -0.004412887 plot(...


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You can prove convergence rates for your problem using standard M-estimator analysis. The rate-of-convergence theorem of chapter 3 in van der Vaart, Wellner, 1998 will handle it. It is also given in Lectures 12/13 of https://www.stat.berkeley.edu/~aditya/resources/FullNotes210BSpring2018.pdf


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