Estimation of Rank-Ordered Regret Minimization Models

Abstract

This paper considers the estimation of random regret minimization models using rank-ordered choice data. By analyzing Monte Carlo simulations results, we find that the efficiency increases as we use additional information on the ranking. Compared with the multinomial logit model with utility maximization, the simulation results show that the standard random regret minimization model is slightly worse than the multinomial logit model based on both the mean bias and root mean squared error of the estimator of the model parameter ${{\varvec{\beta }}}$. When using long ranking choice data to estimate the random regret minimization model, based on the mean bias and root mean squared error of the estimator, we find that the rank-ordered random regret minimization model has advantages over the multinomial logit model and the standard random regret minimization model. Analysis of real data shows that our method is very effective in estimating model parameters.

Appendix: Proofs

1.1 A1. Proof of Lemma 3.1

Before proving Lemma 3.1, we first state that for any ${{\varvec{b}}}\in R^{M\times 1}$, we have Lemma A1:

Lemma A1

Under Assumption 3.1, $Q_{\{j_{1},\ldots ,j_{k-1}\}}({{\varvec{x}}}_{n},{{\varvec{b}}})$ is positive semi-definite for any value of ${{\varvec{x}}}_{n}=({{\varvec{x}}}^{'}_{1,n},\ldots ,{{\varvec{x}}}^{'}_{J,n})\in R^{1\times JM},\ k\in \{1,\ldots ,K\}$ and $\{j_{1},\ldots ,j_{k-1}\}\subseteq \mathbb {J}$.

Proof

Let

$$\begin{aligned} \begin{aligned}&Q_{\{j_{1},\ldots ,j_{k-1}\}}({{\varvec{x}}}_{n},{{\varvec{b}}})\\&\quad =\dfrac{\partial ^{2} R_{j_{k},n}}{\partial {{\varvec{b}}}\partial {{\varvec{b}}}^{'}} + \dfrac{\sum \limits _{l\in \mathbb {J}\backslash \{j_{1},\ldots ,j_{k-1}\} }\exp (-R_{l,n}) \left( \dfrac{\partial R_{l,n}}{\partial {{\varvec{b}}}}\dfrac{\partial R_{l,n}}{\partial {{\varvec{b}}}^{'}}-\dfrac{\partial ^{2} R_{l,n}}{\partial {{\varvec{b}}}\partial {{\varvec{b}}}^{'}}\right) }{\sum \limits _{l\in \mathbb {J}\backslash \{j_{1},\ldots ,j_{k-1}\} }\exp (-R_{l,n})}\\&\qquad - \dfrac{\Big (\sum \limits _{l\in \mathbb {J}\backslash \{j_{1},\ldots ,j_{k-1}\} }\exp (-R_{l,n}) \dfrac{\partial R_{l,n}}{\partial {{\varvec{b}}}}\Big )\Big (\sum \limits _{l\in \mathbb {J}\backslash \{j_{1},\ldots ,j_{k-1}\} }\exp (-R_{l,n}) \dfrac{\partial R_{l,n}}{\partial {{\varvec{b}}}^{'}}\Big )}{\Big [\sum \limits _{l\in \mathbb {J}\backslash \{j_{1},\ldots ,j_{k-1}\} }\exp (-R_{l,n})\Big ]^{2}}\\&\quad =Q^{1}_{\{j_{1},\ldots ,j_{k-1}\}}({{\varvec{x}}}_{n},{{\varvec{b}}})+Q^{2}_{\{j_{1},\ldots ,j_{k-1}\}}({{\varvec{x}}}_{n},{{\varvec{b}}}), \end{aligned} \end{aligned}$$

where

$$\begin{aligned} \begin{aligned}&Q^{1}_{\{j_{1},\ldots ,j_{k-1}\}}({{\varvec{x}}}_{n},{{\varvec{b}}})= \dfrac{\sum \limits _{l\in \mathbb {J}\backslash \{j_{1},\ldots ,j_{k-1}\} }\exp (-R_{l,n}) \dfrac{\partial R_{l,n}}{\partial {{\varvec{b}}}}\dfrac{\partial R_{l,n}}{\partial {{\varvec{b}}}^{'}}}{\sum \limits _{l\in \mathbb {J}\backslash \{j_{1},\ldots ,j_{k-1}\} }\exp (-R_{l,n})}\\&\quad - \dfrac{\Big (\sum \limits _{l\in \mathbb {J}\backslash \{j_{1},\ldots ,j_{k-1}\} }\exp (-R_{l,n}) \dfrac{\partial R_{l,n}}{\partial {{\varvec{b}}}}\Big )\Big (\sum \limits _{l\in \mathbb {J}\backslash \{j_{1},\ldots ,j_{k-1}\} }\exp (-R_{l,n}) \dfrac{\partial R_{l,n}}{\partial {{\varvec{b}}}^{'}}\Big )}{\Big [\sum \limits _{l\in \mathbb {J}\backslash \{j_{1},\ldots ,j_{k-1}\} }\exp (-R_{l,n})\Big ]^{2}} \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} Q^{2}_{\{j_{1},\ldots ,j_{k-1}\}}({{\varvec{x}}}_{n},{{\varvec{b}}})= \dfrac{\sum \limits _{l\in \mathbb {J}\backslash \{j_{1},\ldots ,j_{k-1}\} }\exp (-R_{l,n}) \left( \dfrac{\partial ^{2} R_{j_{k},n}}{\partial {{\varvec{b}}}\partial {{\varvec{b}}}^{'}}-\dfrac{\partial ^{2} R_{l,n}}{\partial {{\varvec{b}}}\partial {{\varvec{b}}}^{'}}\right) }{\sum \limits _{l\in \mathbb {J}\backslash \{j_{1},\ldots ,j_{k-1}\} }\exp (-R_{l,n})}. \end{aligned} \end{aligned}$$

$\square$

First, we point out that $Q^{1}_{\{j_{1},\ldots ,j_{k-1}\}}({{\varvec{x}}}_{n},{{\varvec{b}}})$ is positive semi-definite. For any ${{\varvec{x}}}_{n}=({{\varvec{x}}}^{'}_{1,n},\ldots ,{{\varvec{x}}}^{'}_{J,n})$ and $\{j_{1},\ldots ,j_{k-1}\}$, we define the weights:

$$\begin{aligned} w_{l}({{\varvec{x}}}_{n})=\dfrac{\exp (-R_{l,n}) }{\sum \limits _{j\in \mathbb {J}\backslash \{j_{1},\ldots ,j_{k-1}\} }\exp (-R_{j,n})}. \end{aligned}$$

Notably, $w_{l}({{\varvec{x}}}_{n})>0$ and $\sum \limits _{l\in \mathbb {J}\backslash \{j_{1},\ldots ,j_{k-1}\}}w_{l}({{\varvec{x}}}_{n})=1$. Thus, for any constant vector $c\in R^{M\times 1}$, we have

$$\begin{aligned}&c^{'}Q^{1}_{\{j_{1},\ldots ,j_{k}\}}({{\varvec{x}}}_{n},b)c\nonumber \\&\quad = \dfrac{\sum \limits _{l\in \mathbb {J}\backslash \{j_{1},\ldots ,j_{k-1}\} }\exp (-R_{l,n}) (c^{'}\dfrac{\partial R_{l,n}}{\partial {{\varvec{b}}}})^{2}}{\sum \limits _{l\in \mathbb {J}\backslash \{j_{1},\ldots ,j_{k-1}\} }\exp (-R_{l,n})} - \dfrac{\Big (\sum \limits _{l\in \mathbb {J}\backslash \{j_{1},\ldots ,j_{k-1}\} }\exp (-R_{l,n}) (c^{'}\dfrac{\partial R_{l,n}}{\partial {{\varvec{b}}}})\Big )^{2}}{\Big [\sum \limits _{l\in \mathbb {J}\backslash \{j_{1},\ldots ,j_{k-1}\} }\exp (-R_{l,n})\Big ]^{2}}\nonumber \\&\quad =\sum \limits _{l\in \mathbb {J}\backslash \{j_{1},\ldots ,j_{k-1}\} }w_{l}({{\varvec{x}}}_{n})(c^{'}\dfrac{\partial R_{l,n}}{\partial b})^{2} - \Big [\sum \limits _{l\in \mathbb {J}\backslash \{j_{1},\ldots ,j_{k-1}\} }w_{l}({{\varvec{x}}}_{n})(c^{'}\dfrac{\partial R_{l,n}}{\partial b})\Big ]^{2}\nonumber \\&\quad =\sum \limits _{l\in \mathbb {J}\backslash \{j_{1},\ldots ,j_{k-1}\}}[(c^{'}\dfrac{\partial R_{l,n}}{\partial b}) -\lambda _{\{j_{1},\ldots ,j_{k-1}\}}({{\varvec{x}}}_{n};c)]^{2}w_{l}({{\varvec{x}}}_{n})\ \ \end{aligned}$$

(a1)

where $\lambda _{\{j_{1},\ldots ,j_{k-1}\}}({{\varvec{x}}}_{n};c) =\sum \limits _{t\in \mathbb {J}\backslash \{j_{1},\ldots ,j_{k-1}\}}(c^{'}\dfrac{\partial R_{t,n}}{\partial b})w_{t}({{\varvec{x}}}_{n})$. The last expression means that $Q^{1}_{\{ j_{ 1},\ldots ,j_{ k}\}}({{\varvec{x}}}_{ n} , b)$ is positive semi-definite. Next, we prove that $Q^{2}_{\{ j_{ 1},\ldots ,j_{ k}\}}({{\varvec{x}}}_{n} , b)$ is also positive semi-definite. We have

$$\begin{aligned} \begin{aligned} Q^{2}_{\{j_{1},\ldots ,j_{k-1}\}}({{\varvec{x}}}_{n},{{\varvec{b}}})= \dfrac{\sum \limits _{l\in \mathbb {J}\backslash \{j_{1},\ldots ,j_{k-1}\} }\exp (-R_{l,n}) (\dfrac{\partial ^{2} R_{j_{k},n}}{\partial {{\varvec{b}}}\partial {{\varvec{b}}}^{'}}-\dfrac{\partial ^{2} R_{l,n}}{\partial {{\varvec{b}}}\partial {{\varvec{b}}}^{'}})}{\sum \limits _{l\in \mathbb {J}\backslash \{j_{1},\ldots ,j_{k-1}\} }\exp (-R_{l,n})}. \end{aligned} \end{aligned}$$

where $\dfrac{\partial ^{2} R_{l,n}}{\partial {{\varvec{b}}}\partial {{\varvec{b}}}^{'}},\ l=1,\ldots ,J$ are diagonal matrices with dimensions of $M\times M$ and the m-th component on the diagonal is $\dfrac{\partial ^{2} R_{l,n}}{\partial ^{2} {{\varvec{b}}}_{m}}, m=1,\ldots ,M$. For any $l\in \mathbb {J}\backslash \{j_{1},\ldots ,j_{k-1}\}$, Assumption 3.1 means that $\dfrac{\partial ^{2} R_{j_{k},n}}{\partial {{\varvec{b}}}\partial {{\varvec{b}}}^{'}}-\dfrac{\partial ^{2} R_{l,n}}{\partial {{\varvec{b}}}\partial {{\varvec{b}}}^{'}}$ is non-negative definite, so $Q^{2}_{\{j_{1},\ldots ,j_{k}\}}({{\varvec{x}}}_{n},b)$ is positive semi-definite. The proof of Lemma A1 is completed.

Next, we prove part (a) of Lemma 3.1. We remark that $E\Big [d_{y,n}^{K}Q_{\{j_{1},\ldots ,j_{k-1}\}}({{\varvec{x}}}_{n},{{\varvec{b}}})\Big ]$ is positive semi-definite (Lemma A1). Next, we show that $\sum \limits _{y_{n}^{K}\in \mathbb {J}^{K}}E\Big [d_{y,n}^{K}Q_{\{j_{1},\ldots ,j_{k-1}\}}({{\varvec{x}}}_{n},{{\varvec{b}}})\Big ]$ is positive definite, therefore, we only need to verify that $\sum \limits _{y_{n}^{K}\in \mathbb {J}^{K}}E\Big [d_{y,n}^{K}Q^{1}_{\{j_{1},\ldots ,j_{k-1}\}}({{\varvec{x}}}_{n},{{\varvec{b}}})\Big ]$ is positive definite.

By Assumption 3.1, for any $c\in R^{M\times 1}\setminus \{0\}$, there is $y^{K*}=\{j_{1}^{*},\ldots ,j_{K}^{*}\}$ and ${\bar{l}}_{1}, {\bar{l}}_{2}\in \mathbb {J}\backslash \{j_{1}^{*},\ldots ,j_{K-1}^{*}\}$, so that

$$\begin{aligned} \begin{aligned} 0<E\{\Big [c^{'}(\dfrac{\partial R_{{\bar{l}}_{1},n}}{\partial b}-\dfrac{\partial R_{{\bar{l}}_{2},n}}{\partial b})\Big ]^{2}\}<\infty . \end{aligned} \end{aligned}$$

The above equation is rewritten as follows

$$\begin{aligned} \begin{aligned} E\left\{ \Big [c^{'}\left( \dfrac{\partial R_{{\bar{l}}_{1},n}}{\partial b}-\dfrac{\partial R_{{\bar{l}}_{2},n}}{\partial b}\right) \Big ]^{2} \right\} =\int _{\mathscr {D}}\Big [c^{'}\left( \dfrac{\partial R_{{\bar{l}}_{1}}}{\partial b}-\dfrac{\partial R_{{\bar{l}}_{2}}}{\partial b}\right) \Big ]^{2}f({{\varvec{x}}})v(d{{\varvec{x}}}), \end{aligned} \end{aligned}$$

where $f({{\varvec{x}}})$ is the density function of ${{\varvec{x}}}$ with respect to a $\sigma$-finite measure $v(\cdot )$, and $\mathscr {D}= \left\{ {{\varvec{x}}}=({{\varvec{x}}}_{1}^{'},\ldots ,{{\varvec{x}}}_{J}^{'})\in R^{1\times JM}:\left| c^{'}\left( \dfrac{\partial R_{{\bar{l}}_{1}}}{\partial b}-\dfrac{\partial R_{{\bar{l}}_{2}}}{\partial b}\right) \right| >0 \right\}$. We denote

$$\begin{aligned} \mathscr {D}_{s}=\{{{\varvec{x}}}\in R^{1\times JM}:\left| c^{'}\left( \dfrac{\partial R_{{\bar{l}}_{1}}}{\partial b}-\dfrac{\partial R_{{\bar{l}}_{2}}}{\partial b}\right) \right| \ge \frac{1}{s}\}, \end{aligned}$$

where $s\in \mathbb {N}$, $\mathscr {D}=\cup _{s\in \mathbb {N}}\mathscr {D}_{s}$. Since

$$\begin{aligned} \lim \limits _{s \rightarrow +\infty }\int _{\mathscr {D}_{s}}\left[ c^{'}\left( \dfrac{\partial R_{{\bar{l}}_{1}}}{\partial b}-\dfrac{\partial R_{{\bar{l}}_{2}}}{\partial b}\right) \right] ^{2}f({{\varvec{x}}})v(d{{\varvec{x}}}) =\int _{\mathscr {D}}\left[ c^{'}\left( \dfrac{\partial R_{{\bar{l}}_{1}}}{\partial b}-\dfrac{\partial R_{{\bar{l}}_{2}}}{\partial b}\right) \right] ^{2}f({{\varvec{x}}})v(d{{\varvec{x}}})>0 \end{aligned}$$

by the Lebesgue’s dominated convergence theorem, there exists $s^{*}\in \mathbb {N}$ such that

$$\begin{aligned} \int _{\mathscr {D}_{s^{*}}}\left[ c^{'}\left( \dfrac{\partial R_{{\bar{l}}_{1}}}{\partial b}-\dfrac{\partial R_{{\bar{l}}_{2}}}{\partial b}\right) \right] ^{2}f({{\varvec{x}}})v(d{{\varvec{x}}})>0, \end{aligned}$$

The above equation means that $\int _{\mathscr {D}_{s^{*}}}f({{\varvec{x}}})v(d{{\varvec{x}}})>0$. We define

$$\begin{aligned} \mathscr {D}_{s^{*},l}=\left\{ {{\varvec{x}}}\in R^{1\times JM}:\left| \left( c^{'}\dfrac{\partial R_{{\bar{l}}_{l}}}{\partial b}\right) -\lambda _{\{j_{1}^{*},\ldots ,j_{K-1}^{*}\}}({{\varvec{x}}};c)\right| \ge \frac{1}{2s^{*}} \right\} \end{aligned}$$

where $l=1,2$. Because

$$\begin{aligned} \left| c^{'}\left( \dfrac{\partial R_{{\bar{l}}_{1}}}{\partial b}-\dfrac{\partial R_{{\bar{l}}_{2}}}{\partial b}\right) \right| \le \left| c^{'}\dfrac{\partial R_{{\bar{l}}_{1}}}{\partial b}-\lambda _{\{j_{1}^{*},\ldots ,j_{K-1}^{*}\}}({{\varvec{x}}};c)\right| +\left| c^{'}\dfrac{\partial R_{{\bar{l}}_{2}}}{\partial b}-\lambda _{\{j_{1}^{*},\ldots ,j_{K-1}^{*}\}}({{\varvec{x}}};c)\right| , \end{aligned}$$

So $\mathscr {D}_{s^{*}}\subseteq \mathscr {D}_{s^{*},1}\cup \mathscr {D}_{s^{*},2}$, then, we have

$$\begin{aligned} 0<\int _{\mathscr {D}_{s^{*}}}f({{\varvec{x}}})v(d{{\varvec{x}}}) \le \int _{\mathscr {D}_{s^{*},1}}f({{\varvec{x}}})v(d{{\varvec{x}}}) +\int _{\mathscr {D}_{s^{*},2}}f({{\varvec{x}}})v(d{{\varvec{x}}}). \end{aligned}$$

Without loss of generality, assuming that $\int _{\mathscr {D}_{s^{*},1}}f({{\varvec{x}}})v(d{{\varvec{x}}})>0$, we have

$$\begin{aligned} E\Big [\sum \limits _{y_{n}^{K}\in \mathbb {J}^{K}}d_{y,n}^{K}Q^{1}_{\{j_{1},\ldots ,j_{k-1}\}}({{\varvec{x}}}_{n},{{\varvec{b}}})\Big ]= & {} E\Big [\sum \limits _{y_{n}^{K}\in \mathbb {J}^{K}\setminus \{y_{n}^{*K}\}}d_{y,n}^{K}Q^{1}_{\{j_{1},\ldots ,j_{k-1}\}}({{\varvec{x}}}_{n},{{\varvec{b}}})\Big ]\\&+E\Big [d_{y^{*},n}^{K}Q^{1}_{\{j_{1},\ldots ,j_{k-1}\}}({{\varvec{x}}}_{n},{{\varvec{b}}})\Big ]. \end{aligned}$$

According to Lemma 3.1, we know that the first term at the right end of the equation is positive semi-definite. Next, we prove that the second term is positive definite. According to the law of iterated expectation, we have

$$\begin{aligned}&c^{'}E[d_{y^{*},n}^{K}Q^{1}_{\{j_{1^{*}},\ldots ,j_{K-1^{*}}\}}({{\varvec{x}}}_{n},b)]c\\&\quad =\int Pr(d_{y^{*},n}^{K}=1|{{\varvec{x}}};{{\varvec{\beta }}})[c^{'}Q^{1}_{\{j_{1^{*}},\ldots ,j_{K-1^{*}}\}}({{\varvec{x}}},b)c]f({{\varvec{x}}})v(d{{\varvec{x}}})\\&\quad =\int Pr(d_{y^{*},n}^{K}=1|{{\varvec{x}}};{{\varvec{\beta }}})\\&\qquad \left\{ \sum \limits _{l\in \mathscr {J}\backslash \{j_{1^{*}},\ldots ,j_{K-1^{*}}\}} \left[ \left( c^{'}\dfrac{\partial R_{l}}{\partial b}\right) -\lambda _{\{j_{1^{*}},\ldots ,j_{K-1^{*}}\}}({{\varvec{x}}};c)\right] ^{2}w_{l}({{\varvec{x}}})\right\} f({{\varvec{x}}})v(d{{\varvec{x}}})\\&\quad \ge \left( \frac{1}{2s^{*}}\right) ^{2}\int _{\mathscr {D}_{s^*,1}} Pr(d_{y^{*},n}^{K}=1|{{\varvec{x}}};{{\varvec{\beta }}})w_{{\bar{l}}_{1}}({{\varvec{x}}})f({{\varvec{x}}})v(d{{\varvec{x}}})\\&\quad \ge \left( \frac{1}{2s^{*}}\right) ^{2}\Big [\min \limits _{{{\varvec{x}}}\in \mathscr {D}_{s^*,1}} Pr(d_{y^{*},n}^{K}=1|{{\varvec{x}}};{{\varvec{\beta }}}) w_{{\bar{l}}_{1}}({{\varvec{x}}})\Big ]\int _{\mathscr {D}_{s^*,1}}f({{\varvec{x}}})v(d{{\varvec{x}}})>0. \end{aligned}$$

The second equality follows from expression (a1). We highlight that $\min \limits _{{{\varvec{x}}}\in \mathscr {D}_{s^*,1}}Pr(d_{y^{*},n}^{K}=1|{{\varvec{x}}};{{\varvec{\beta }}}) w_{{\bar{l}}_{1}}({{\varvec{x}}})>0$ is positive because $Pr(d_{y^{*},n}^{K}=1|{{\varvec{x}}};{{\varvec{\beta }}}) w_{{\bar{l}}_{1}}({{\varvec{x}}})$ is continuous and strictly positive. Therefore, $E[d_{y^{*},n}^{K}Q^{1}_{\{j_{1^{*}},\ldots ,j_{K-1^{*}}\}}({{\varvec{x}}}_{i},b)]$ is positive definite, and part (a) of Lemma 3.1 holds.

The following derivation proves that part (b) of Lemma 3.1 holds, we have

$$\begin{aligned} \begin{aligned} \mathop {I^{K}({{\varvec{\beta }}})}&=\sum \limits _{y_{n}^{K}\in \mathbb {J}^{K}}\sum \limits _{k=1}^{K-1} E\Big [d_{y,n}^{K}Q^{1}_{\{j_{1},\ldots ,j_{k-1}\}}({{\varvec{x}}}_{n},{{\varvec{\beta }}})\Big ]+\sum \limits _{y_{n}^{K}\in \mathbb {J}^{K}} E\Big [d_{y,n}^{K}Q^{1}_{\{j_{1},\ldots ,j_{k-1}\}}({{\varvec{x}}}_{n},{{\varvec{\beta }}})\Big ]\\&\quad +\sum \limits _{y_{n}^{K}\in \mathbb {J}^{K}}\sum \limits _{k=1}^{K} E\Big [d_{y,n}^{K}Q^{2}_{\{j_{1},\ldots ,j_{k-1}\}}({{\varvec{x}}}_{n},{{\varvec{\beta }}})\Big ] \end{aligned} \end{aligned}$$

In the above equation, the first and third terms are positive semi-definite according to Lemma A1, and the second term is positive definite based on the previous result. Therefore, $\mathop {I^{K}(\beta )}$ is positive definite, and part (b) of Lemma 3.1 holds.

1.2 A2. Proof of Theorem 3.1

Without the loss of generality, we first consider the case of ${\widetilde{K}}=K-1$ for the general case of $1\le {\widetilde{K}}<K$, and obtain similar conclusions by induction. We have

$$\begin{aligned} \begin{aligned} \mathop {I^{K}({{\varvec{\beta }}})}&=\sum \limits _{y_{n}^{K}\in \mathbb {J}^{K}}\sum \limits _{k=1}^{K} E\Big [d_{y,n}^{K}Q_{\{j_{1},\ldots ,j_{k-1}\}}({{\varvec{x}}}_{n},{{\varvec{\beta }}})\Big ]\\&=\sum \limits _{y_{n}^{K}\in \mathbb {J}^{K}}\sum \limits _{k=1}^{K-1}E\Big [d_{y,n}^{K}Q_{\{j_{1},\ldots ,j_{k-1}\}}({{\varvec{x}}}_{n},{{\varvec{\beta }}})\Big ] +\sum \limits _{y_{n}^{K}\in \mathbb {J}^{K}}E\Big [d_{y,n}^{K}Q_{\{j_{1},\ldots ,j_{K-1}\}}({{\varvec{x}}}_{n},{{\varvec{\beta }}})\Big ]. \end{aligned} \end{aligned}$$

Furthermore,

$$\begin{aligned} \begin{aligned}&\sum \limits _{y_{n}^{K}\in \mathbb {J}^{K}}\sum \limits _{k=1}^{K-1}E\Big [d_{y,n}^{K}Q_{\{j_{1},\ldots ,j_{k-1}\}}({{\varvec{x}}}_{n},{{\varvec{\beta }}})\Big ]\\&\quad =\sum \limits _{k=1}^{K-1}\sum \limits _{y_{n}^{K-1}\in \mathbb {J}^{K-1}}E\Big [ \Big (\sum \limits _{l\in \mathbb {J}\backslash \{j_{1},\ldots ,j_{K-1}\}} d_{\{j_{1},\ldots ,j_{K-1},l\},n}^{K}\Big )Q_{\{j_{1},\ldots ,j_{k-1}\}}({{\varvec{x}}}_{n},{{\varvec{\beta }}})\Big ]\\&\quad =\sum \limits _{y_{n}^{K}\in \mathbb {J}_{K}}\sum \limits _{k=1}^{K-1}E\Big [d_{y,n}^{K-1}Q_{\{j_{1},\ldots ,j_{k-1}\}}({{\varvec{x}}}_{n},{{\varvec{\beta }}})\Big ]\\&\quad =\mathop {I^{K-1}({{\varvec{\beta }}})}. \end{aligned} \end{aligned}$$

The first equation holds because $\sum \limits _{y_{n}^{K-1}\in \mathbb {J}^{K-1}}\sum \limits _{l\in \mathbb {J}\backslash \{j_{1},\ldots ,j_{K-1}\}}=\sum \limits _{y_{n}^{K}\in \mathbb {J}^{K}}$ and the second equation holds based on $d_{\{j_{1},\ldots ,j_{K-1}\},n}^{K-1}=\sum \limits _{l\in \mathbb {J}\backslash \{j_{1},\ldots ,j_{K-1}\}}d_{\{j_{1},\ldots ,j_{K-1},l\},n}^{K}$.

So

$$\begin{aligned} \begin{aligned} \mathop {I^{K}({{\varvec{\beta }}})}-\mathop {I^{K-1}({{\varvec{\beta }}})}= \sum \limits _{y_{n}^{K}\in \mathbb {J}_{K}}E\Big [d_{y,n}^{K}Q_{\{j_{1},\ldots ,j_{K-1}\}}({{\varvec{x}}}_{n},{{\varvec{\beta }}})\Big ]. \end{aligned} \end{aligned}$$

According to Lemma 3.1(a), $\mathop {I^{K}({{\varvec{\beta }}})}-\mathop {I^{K-1}({{\varvec{\beta }}})}$ is obviously positive definite.