Multivariate Higher-Degree Stochastic Increasing Convexity

Abstract

Building on the seminal work by Shaked and Shanthikumar (Adv Appl Probab 20:427–446, 1988a; Stoch Process Appl 27:1–20, 1988b), Denuit et al. (Eng Inf Sci 13:275–291, 1999; Methodol Comput Appl Probab 2:231–254, 2000; 2001) studied the stochastic s-increasing convexity properties of standard parametric families of distributions. However, the analysis is restricted there to a single parameter. As many standard families of distributions involve several parameters, multivariate higher-order stochastic convexity properties also deserve consideration for applications. This is precisely the topic of the present paper, devoted to stochastic $(s_1,s_2,\ldots ,s_d)$-increasing convexity of distribution families indexed by a vector $(\theta _1,\theta _2,\ldots ,\theta _d)$ of parameters. This approach accounts for possible correlation in multivariate mixture models.

Appendix: Some Results for the Binomial Distribution

Property 6.1

Let $X_{(n,p)}, n \in \mathbb {N}, p \in (0,1)$, denote a random variable distributed according to the Binomial distribution with mean np and variance $np(1-p)$. With $g^{\star }(n,p)=\mathbb {E}[g(X_{(n,p)})]$, define

$$\begin{aligned} \Delta g^{\star }(n,p)=g^{\star }(n+1,p)-g^{\star }(n,p). \end{aligned}$$

Then, we have

$$\begin{aligned} \frac{\partial g^\star }{\partial p} (n,p)= & {} n\mathbb {E} \left[ \Delta g(X_{(n-1,p)})\right] \end{aligned}$$

(6.2)

$$\begin{aligned} \Delta g^\star (n,p)= & {} p\mathbb {E}[\Delta g(X_{(n,p)})] \end{aligned}$$

(6.3)

$$\begin{aligned} \Delta \frac{\partial g^\star }{\partial p}(n,p)= & {} (n+1)p\mathbb {E}[\Delta ^2 g(X_{(n-1,p)})]+\mathbb {E}[\Delta g(X_{(n-1,p)})]. \end{aligned}$$

(6.4)

Proof

We have

$$\begin{aligned} \frac{\partial g^\star }{\partial p}(n,p)= & {} \sum _{i=1}^{n-1} g(i)i{n \atopwithdelims ()i}p^{i-1}(1-p)^{n-i}-\sum _{i=1}^{n-1} g(i)(n-i){n \atopwithdelims ()i}p^{i}(1-p)^{n-i-1}\\&+\, ng(n)p^{n-1}-ng(0)(1-p)^{n-1}. \end{aligned}$$

Using the well-known identities

$$\begin{aligned} i{n \atopwithdelims ()i}=n{n-1\atopwithdelims ()i-1 } \quad \text{ and } \quad (n-i){n \atopwithdelims ()i}=n{n-1\atopwithdelims ()i}, \end{aligned}$$

we can write

$$\begin{aligned} \frac{\partial g^\star }{\partial p} (n,p)= & {} \sum _{i=1}^{n} g(i)n{n-1 \atopwithdelims ()i-1}p^{i-1}(1-p)^{n-i} \\&-\,\sum _{i=0}^{n-1} g(i)n{n-1 \atopwithdelims ()i}p^{i}(1-p)^{n-i-1} \\= & {} \sum _{i=0}^{n-1} g(i+1)n{n-1 \atopwithdelims ()i}p^{i}(1-p)^{n-i-1} \\&-\,\sum _{i=0}^{n-1} g(i)n{n-1 \atopwithdelims ()i}p^{i}(1-p)^{n-i-1}\\= & {} n\sum _{i=0}^{n-1} \Delta g(i){n-1 \atopwithdelims ()i}p^{i}(1-p)^{n-i-1}\\= & {} n\mathbb {E} \left[ \Delta g(X_{(n-1,p)})\right] \end{aligned}$$

which ends the proof of (6.2). Now, let us show that the second equality is valid. Clearly, one has

$$\begin{aligned} \Delta g^\star (n,p)= & {} \sum _{i=0}^{n+1} g(i){n+1 \atopwithdelims ()i}p^{i}(1-p)^{n+1-i}-\sum _{i=0}^{n} g(i){n \atopwithdelims ()i}p^{i}(1-p)^{n-i}. \end{aligned}$$

Since

$$\begin{aligned} {n+1 \atopwithdelims ()i}={n \atopwithdelims ()i}+{n \atopwithdelims ()i-1}, \quad 1 \le i \le n, \end{aligned}$$

we can write

$$\begin{aligned} \Delta g^\star (n,p)= & {} g(0)(1-p)^{n+1}- g(0)(1-p)^{n}+g(n+1)p^{n+1} \\&+\,\sum _{i=1}^{n} g(i)\left( {n \atopwithdelims ()i}+{n \atopwithdelims ()i-1} \right) p^{i}(1-p)^{n+1-i} \\&-\,\sum _{i=1}^{n} g(i){n \atopwithdelims ()i}p^{i}(1-p)^{n-i} \\= & {} g(0)(1-p)^{n+1}-g(0)(1-p)^n+g(n+1)p^{n+1} \\&+\,\sum _{i=1}^{n} g(i){n \atopwithdelims ()i-1}p^{i}(1-p)^{n+1-i} \\&-\,\sum _{i=1}^{n} g(i){n \atopwithdelims ()i}p^{i+1}(1-p)^{n-i} \\= & {} \sum _{i=1}^{n+1} g(i){n \atopwithdelims ()i-1}p^{i}(1-p)^{n+1-i} -\sum _{i=0}^{n} g(i){n \atopwithdelims ()i}p^{i+1}(1-p)^{n-i}\\= & {} \sum _{i=0}^{n} g(i+1){n \atopwithdelims ()i}p^{i+1}(1-p)^{n-i} -\sum _{i=0}^{n} g(i){n \atopwithdelims ()i}p^{i+1}(1-p)^{n-i}\\= & {} p\sum _{i=0}^{n} \big (g(i+1)-g(i)\big ){n \atopwithdelims ()i}p^{i}(1-p)^{n-i}\\= & {} p\sum _{i=0}^{n} \Delta g(i){n \atopwithdelims ()i}p^{i}(1-p)^{n-i}\\= & {} p\mathbb {E}[\Delta g(X_{(n,p)})]. \end{aligned}$$

Finally, let us establish the last formula (6.4). First, recall that for any functions $h_1$ and $h_2:\mathbb {N}\rightarrow \mathbb {R}$, one has

$$\begin{aligned} \Delta \big (h_1(n)h_2(n)\big )=h_1(n+1)\Delta h_2(n)+h_2(n)\Delta h_1(n). \end{aligned}$$

Considering $h_1(n)=n$ and $h_2(n)=\mathbb {E} \left[ \Delta g(X_{(n-1,p)})\right] =(\Delta g)^\star (n-1,p)$, we get

$$\begin{aligned} \Delta \frac{\partial g^\star }{\partial p}(n,p)= & {} \Delta \big (h_1(n)h_2(n)\big )\\= & {} (n+1)\Delta (\Delta g)^\star (n-1,p)+\mathbb {E} \left[ \Delta g(X_{(n-1,p)})\right] \\= & {} (n+1)p\mathbb {E}[\Delta ^2 g(X_{(n-1,p)})]+\mathbb {E} \left[ \Delta g(X_{(n-1,p)})\right] \end{aligned}$$

so that (6.4) is indeed valid. $\square$