Erratum to: Parameterizing mixture models with generalized moments

1 Erratum to: Ann Inst Stat Math DOI 10.1007/s10463-014-0490-9

• 1. In Sect. 3.1, the second sentence is “However since the error $\mathrm{e}_r(Q)$ depends on the unknown mixing distribution $Q$, instead we aim to minimize an upper bound of $\mathrm{e}_r(Q)$ instead.” The second “instead” should be deleted.

• 2. In Sect. 4.1, the first sentence below Definition 8 is “According to Carathéodory’s theorem, for each vector ${\varvec{m}} \in \mathrm{conv}(\Gamma _{r+1})$, there exists a convex representation of ${\varvec{m}}$ by $\{u_i(\theta )\}_{i=0}^r$ with $J < r+1$.” The less than sign in “$J < r+1$” should be changed to the less than or equal to sign, i.e., “$J \le r+1$”.

• 3. In Sect. 4.1, the second sentence below the second paragraph in Example 1 is “The boundary vectors of $conv(\Gamma _3)$ are either ${\varvec{u}}(\theta ) \in \mathrm{R}^3$ or $(1-\alpha ){\varvec{u}}(0) + \alpha {\varvec{u}}(1)$, where $0< \alpha < 1$.” Here “$(1- \alpha ){\varvec{u}}(0) + \alpha {\varvec{u}}(1)$” should be changed to “$(1- \alpha ){\varvec{u}}(0) + \alpha {\varvec{u}}(25)$”.

• 4. In Sect. 4.1, the third sentence of the second paragraph in Example 1 is “Therefore, the index of a boundary vector is either $1$ or $3/2$; see Theorem 3.” Here “$3/2$” should be changed to “$1/2$”.

• 5. In Fig. 7, the axis labels in the panel (a) are $u_0 = 1$, $u_1 = \theta -1$, $u_2 = (\theta -1)^2$. The correct axis labels are $u_0(\theta ) = 1$, $u_1=(\theta -12.5)$ and $u_2=(\theta -12.5)^2$; see Fig. 7.

• 6. In Sect. 4.2, the first sentence of the paragraph above Theorem 4 “Since the optimization problem (12) is convex, its solution $\hat{\varvec{m}}_c$ is unique and on the boundary of $\mathcal{K}_r$.” should be changed to “Since the optimization problem (12) is convex, its solution $\hat{\varvec{m}}_c$ is unique and in $\mathcal{K}_r$.”

• 7. In Fig. 8, the legends with $\ell$ should be changed to $\mathcal L$; see Fig. 8.

Fig. 7

Plots of the moment cones induced by a $\{(\theta -12.5)^i\}_{i=0}^2$; and b $\{\phi _i(\theta )\}_{i=0}^2$ for the mixture of Poisson

Fig. 8

Plots of $\mathcal{K}_2$ induced by $\{\phi _i(\theta )\}_{i=0}^2$ and a visual interpretation of Theorems $4$ and $5$