Statistical Analysis Stack Exchange
Statistical Analysis Stack Exchange

Questions tagged [concavity]

16 questions
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Median of the sum vs. sum of the median for Gaussian variables

This is a problem I stumbled upon in my research. Consider $n$ Gaussian random variables $x_i \sim \mathcal{N} (\mu_i, \sigma_i^2)$, each with its own mean $\mu_i$ and variance $\sigma_i^2$. Can we say that \begin{equation} \text{Median} \left(…
MrRobot
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What is the shape of the Benini distribution?

The Benini distribution is a continuous univariate distribution that is used in actuarial applications. For all $x \geqslant \sigma$ it has density function: $$\text{Benini}(x| \alpha, \beta, \sigma) = \frac{\alpha + 2 \beta \log (x/\sigma)}{x}…
Ben
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Does the expected value of f(x) go down if f is concave and the variance of X increases?

Let $f(x)$ be concave, $X_1$ a random variable and $X_2$ a mean-preserving spread of $X_1$. The entry on wikipedia defines mean-preserving spread as any $X_2$ such that $$x_{2} \; {\overset {d}{=}}\; (x_{1}+z)$$ where $E[Z | x_1] = 0$ for all…
cd98
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Is there a technique where we keep the proposal in Adaptive Rejection Sampling?

As I understand, the proposal distribution, which I'll call $h(x)$, in adaptive rejection sampling is a linear piece-wise function which converges to the true distribution as the number of iterations goes to infinity. As I understand, in general,…
John Madden
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Stochastic Dominance for convex sum of two random variables with same distribution

This is a very widely used result in finance and economics, and seems fairly intuitive as well (riskier asset is less preferred by risk-averse (concave utility) investors). However, I have not been able to find a proof for this. I see a very similar…
Dayne
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Poisson log-likelihood is concave but not Lipschitz-continuous?

According to He et al. (2016), the log-likelihood of Poisson models, $$ L (\beta) = \sum_{i} - \log(x_{i}^{\mathsf{T}}\beta) + y_{i}x_{i}^{\mathsf{T}}\beta - \log y_{i}! $$ for a random variable, $y_{i} \sim \text{Pois}(x_{i}^\mathsf{T}\beta) $, is…
Durden
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How to prove that the likelihood of a proportional hazards with lognormal baseline model is log concave?

I want to fit a survival model using a proportional hazards assumption $$h(t) = h_0(t)\exp(x^T\beta),$$ where $$h_0(t) = \dfrac{\frac{1}{\sigma t} \phi \left(\frac{\log(t) - \mu}{\sigma}\right)}{\Phi \left(-\frac{\log(t) -…
Posh
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Does log-concavity of a density imply log-concavity of the likelihood?

Many distributions are known to be log-concave: Poisson, Negative Binomial,... However, we are often interested the likelihood function (the density as a function of the parameters, not as a function of the data). Hence my question: Does (or in…
alberto
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Identifiability Versus Convexity

I'm a little unclear on the definitions of "identifiable" and "convex." Consider the case where $X_1, \ldots, X_n \overset{iid}{\sim} \text{Bernoulli}(p)$. Then our likelihood function is $L(p) = p^{\sum_i x_i}(1-p)^{n - \sum x_i}.$ 1. This model is…
Taylor
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Lower bound on expectation of concave function

Let $X$ be a random variable, and let $f$ be a concave function. Are there any known lower bounds for (or methods of lower bounding) $\mathbb{E}[f(X)]$? Jensen's inequality only gives an upper bound on the above.
user19346
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Maximizing a non-parametric Probability Density

Assume we have a set of samples and estimate the underlying distribution with a non-parametric density estimator like the Kernel Density Estimator. Lets assume with a gaussian kernel. In my case it is highly non-convex and multimodal. My goal is to…
Chris
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Is there a measure of association that is not a concave function of mutual information?

I'm reading Mutual Dependence: A Novel Method for Computing Dependencies Between Random Variables and I see in figure 2 and figure 3 that correlation, correlation distance, and mutual dependence are all concave functions of the mutual information.…
DifferentialPleiometry
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Is this objective function convex?

Given that $F(x)$ is the cumulative distribution function (CDF) of continuous random variable $X$, is $$\frac{\int_0^\infty 1-F(x) dx}{\int_{-\infty}^0 F(x) dx}$$ convex? or is it non-convex/concave? How can you derive the answer? And based on the…
develarist
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Convex variant of Bhattacharyya coefficient

For (discrete, finite) probability distributions $P,Q$, the Bhattacharyya coefficient is $B(P,Q) := \sum_x \sqrt{P_x Q_x}$. It can be shown that this is jointly concave in $P$ and $Q$. My question is, is there a monotone increasing function of the…
helloworld
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Concave downward link function for a glm?

I seem to occasionally find datasets where the relationship between X and Y is concave downward. It seems like it should be trivial to find a link function that fits a concave downward curve, but they all seem to fit concave upward curves. Here's a…
dfife
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