Incomplete paired comparisons in case of multiple choice and general log-concave probability density functions

Abstract

A scoring method based on paired comparison allowing multiple choice is investigated. We allow general log-concave probability density functions for the random variables describing the difference of the objects. This case involves Bradley–Terry models and Thurstone models as well. A sufficient condition is proved for the existence and uniqueness of the maximum likelihood estimation of the parameters in case of incomplete comparisons. The axiomatic properties of the method are also investigated.

Appendix

The proof of Theorem 1 follows the proof of the main theorem in Orbán-Mihálykó et al. (2017). The existence of a maximal value can be concluded from the fact that the log-likelihood function is continuous, has finite limits and is bounded in all variables $m_{i,j}=m_{i}-m_{j}$$i=1,\ldots ,n-1,j=i+1,\ldots ,n.$ The lemmas and the theorems in the “Appendix” in Orbán-Mihálykó et al. (2017) remain true in case of general strictly log-concave p.d.f.-s. Nevertheless, Theorem 4 in Orbán-Mihálykó et al. (2017) utilizes the special form of the Gaussian p.d.f., therefore it can not be used in case of general p.d.f.-s. The statement is true for general cases but because of the general c.d.f.-s $F_{i,j}$, a more general proof is needed. Now, we prove the following

Theorem 6

The function $\quad H_{i,j}(x,y)=\log ( F_{i,j}(x)-F_{i,j}(y))$ is strictly concave on the set $y<x\in {\mathbb {R}}$.

Proof

Let $f_{i,j}$ denote the probability density function of $F_{i,j}$. The set $C=\{ (x,y)\mid y<x\in {\mathbb {R}}\} \subset {\mathbb {R}}^{2}$ is convex. We prove that the matrix

$$\begin{aligned} D^{i,j}=\left[ \begin{array} [c]{cc} d_{1,1}^{i,j} &{} d_{1,2}^{i,j}\\ d_{2,1}^{i,j} &{} d_{2,2}^{i,j} \end{array} \right] \end{aligned}$$

(44)

with entries

$$\begin{aligned}&d_{1,1}^{i,j}=f_{i,j}(x)\dfrac{\dfrac{\left( f_{i,j}\right) ^{\prime } (x)}{f_{i,j}(x)}\cdot \left( F_{i,j}(x)-F_{i,j}(y)\right) -f_{i,j} (x)}{\left( F_{i,ji,j}(x)-F_{i,j}(y)\right) ^{2}}, \end{aligned}$$

(45)

$$\begin{aligned}&d_{1,2}^{i,j}=d_{2,1}^{i,j}=\dfrac{f_{i,j}(x)\cdot f_{i,j}(y)}{\left( F_{i,j}(x)-F_{i,j}(y)\right) ^{2}}, \end{aligned}$$

(46)

$$\begin{aligned}&d_{2,2}^{i,j}=f_{i,j}(y)\dfrac{-\dfrac{\left( f_{i,j}\right) ^{\prime } (y)}{f_{i,j}(y)}\cdot \left( F_{i,j}(x)-F_{i,j}(y)\right) -f_{i,j} (y)}{\left( F_{i,j}(x)-F_{i,j}(y)\right) ^{2}} \end{aligned}$$

(47)

is negative definite on C.

First we show that

$$\begin{aligned} d_{1,1}^{i,j}<0. \end{aligned}$$

(48)

Let $y\in {\mathbb {R}}$ and let us define

$$\begin{aligned} f_{i,j,y}(x)=\frac{\left( f_{i,j}\right) ^{\prime }(x)}{f_{i,j}(x)} \cdot \left( F_{i,j}(x)-F_{i,j}(y)\right) -f_{i,j}(x),\quad y\le x\in {\mathbb {R}} \end{aligned}$$

(49)

Due to the strictly log-concave property $( \log f_{i,j}(x)) ^{\prime \prime }=( \frac{( f_{i,j}) ^{\prime }(x)}{f_{i,j}(x)}) ^{\prime }\le 0,$ moreover in every interval the strict inequality holds for some points, we get

$$\begin{aligned} \left( f_{i,j,y}^{\quad }\right) ^{\prime }(x)=\left( \frac{\left( f_{i,j}\right) ^{\prime }(x)}{f_{i,j}(x)}\right) ^{\prime }\left( F_{i,j} (x)-F_{i,j}(y)\right) \le 0,\quad y<x\in {\mathbb {R}}, \end{aligned}$$

(50)

We see that the function $f_{i,j,y}(x)$ is monotone decreasing. If we take a point z from the interval (y, x) for which $f_{i,j,y}^{\quad ^{\prime } }(z)<0$ holds, then

$$\begin{aligned} f_{i,j,y}(x)<f_{i,j,y}(z)<f_{i,j,y}(y)=-f_{i,j}(y)<0. \end{aligned}$$

(51)

It implies

$$\begin{aligned} \frac{\partial ^{2}H_{i,j}(x,y)}{\partial x^{2}}=f_{i,j}(x)\cdot \frac{f_{i,j,y}(x)}{\left( F_{i,j}(x)-F_{i,j}(y)\right) ^{2}}<0. \end{aligned}$$

(52)

Now we investigate the determinant of the matrix containing the second order partial derivatives. Let $x\in {\mathbb {R}}$ and consider

$$\begin{aligned} g_{i,j,x}(y)=-\left( \dfrac{\left( f_{i,j}^{\prime }\right) ^{\prime } (y)}{f_{i,j}(y)}\right) ^{\prime }(F_{i,j}(x)-F_{i,j}(y)-f_{i,j}(y),\quad x\ge y\in {\mathbb {R}} \end{aligned}$$

(53)

Now

$$\begin{aligned} \det (D^{_{i,j}})=\dfrac{f_{i,j}(x)f_{i,j}(y)}{\left( F_{i,j}(x)-F_{i,j} (y)\right) ^{4}}\cdot (f_{i,j,y}(x)g_{i,j,x}(y)-f_{i,j}(x)f_{i,j}(y)),\nonumber \\ \quad y<x\in {\mathbb {R}}. \end{aligned}$$

(54)

As

$$\begin{aligned} g_{i,j,x}^{\prime }(y)=-\left( \dfrac{f_{i.j}(y)}{f_{i,j}(y)}\right) ^{\prime }\left( F_{i,j}(x)-F_{i,j}(y)\right) \ge 0,\quad y\le x\in {\mathbb {R}} \end{aligned}$$

(55)

therefore $g_{i,j,x}(y)$ is increasing. Take a point $z\in (y,x)$ for which $0<g_{i,j,x}^{\prime }(z),$

$$\begin{aligned} g_{i,j,x}(y)<g_{i,j,x}(z)<g_{i,j,x}(x)=-f_{i,j}(x)<0,\quad y<x\in {\mathbb {R}}. \end{aligned}$$

(56)

(51) together with (56) imply the inequality

$$\begin{aligned} f_{i,j}\left( x\right) f_{i,j}\left( y\right)<f_{i,j,y}\left( x\right) g_{i,j,x}\left( y\right) ,\quad y<x\in {\mathbb {R}} \end{aligned}$$

(57)

consequently 0 < $\det (D^{_{i,j}})$ holds. $\square$

The following Lemma is the pair of Lemma 6 in Orbán-Mihálykó et al. (2017).

Lemma 1

If $0<c_{1}$ and $0<c_{2}$, then $G:{\mathbb {R}}^{2}\rightarrow {\mathbb {R}}^{2},$

$$\begin{aligned} G_{i,j}(x,y)=c_{1}\log F_{i,j}\left( x\right) +c_{2}\log \left( 1-F_{i,j}(y)\right) \end{aligned}$$

(58)

is strictly concave.

Proof

The proof is technical, it is left to the reader. $\square$

The remaining parts of the proof do not rely on the special form of the cumulative distribution functions, therefore the lemmas can be applied in the general case, as well, hence the statement is proved.