Questions tagged [concavity]

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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, ...
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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_{...
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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)}{\...
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Expected value of binomial distribution with concave utility weighting

I am interested in showing that a function $V(m)=\sum_{j=0}^{m}v({j})\binom{m}{j}\left ( 1-p \right )^{j}\left ( p \right )^{m-j}$ is concave in the discrete sense (i.e., $V(m)-V(m-1)>V(m+1)-V(m)$ ...
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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)...
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70 views

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 ...
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Test of concavity for repeated measures data

First question here. I'm trying to figure out what statistical test is appropriate for testing whether a series of data is concave or convex. Specifically, this is coming from a human subjects ...
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61 views

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}{=}}\...
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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 ...
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119 views

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^{\...
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725 views

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 ...
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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 ...
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182 views

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 ...