2. Preliminaries and some definitions

Throughout this paper, ‘increasing’ and ‘decreasing’ are taken as ‘non-decreasing’ and ‘non-increasing’, respectively, and the generalised inverse of an increasing function v is denoted as $v^{-1}(u)=\sup\{x\in\mathbb{R}\colon v(x)\leq u\}$ . Moreover, the beta function with parameters $a,b>0$ is denoted by $\mathcal{B}(a,b)=\int_0^1 t^{a-1}(1-t)^{b-1}\,{\mathrm{d}} t$ . Finally, given an absolutely continuous CDF, its density function is denoted by the corresponding lowercase letter.

We shall consider two general families of stochastic orders, characterised either by integration or shape assumptions, which are shown to be crucial for establishing comparisons between expectations of order statistics.

Setting particular choices for the generator class, we obtain some well-known stochastic orders. We recall some relevant classes of functions before presenting the translation of the previous definition into specific ordering relations.

We will focus on the following integral stochastic orders, obtained from Definition 2.1 for particular generator classes.

The ICV, ICX, and SS orders are well known (see, for instance, [Reference Shaked and Shanthikumar20]). Differently, the IAS order seems not to have been studied. As we will discuss in Section 6.2, the IAS order has the disadvantage, unlike the others, that an easy-to-check characterisation is not available. The relationships among classes of functions yield the following implications (see [Reference Shaked and Shanthikumar20, Theorem 4.A.55] for the first line, while the second is proved later in Proposition 6.1):

\begin{align*}\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c}X\geq_\mathrm{st}Y & \Longrightarrow & X\geq_\mathrm{ss}Y & \Longrightarrow & X\geq_\mathrm{icx}Y, \\X\geq_\mathrm{st}Y & \Longrightarrow & X\geq_\mathrm{ias}Y & \Longrightarrow & X\geq_\mathrm{icv}Y.\end{array}\end{align*}

All these orders imply inequality of the means, $\mathbb{E} X\geq \mathbb{E} Y$ , since the identity function belongs to each of the above classes.

We now introduce a second general family of stochastic orders.

Similarly to the integral stochastic orders defined earlier, the following transform orders may be obtained from Definition 2.4 by taking $\mathcal{H}$ as the class of convex and star-shaped functions, respectively.

We should note that the standard stochastic order may be seen both as an integral and a transform order. In fact, $X\geq_\mathrm{st}Y$ if $F^{-1}\circ G(x)\leq x$ , for every x.

In this article, we show that a useful approach to obtaining interesting stochastic inequalities consists in a suitable combination of integral and transform orderings based on a common generator class.

3. Main result

We now address the comparison of order statistics with respect to integral stochastic orders. We shall be taking F as the CDF of interest, and G some suitably chosen reference CDF. It is well known that the CDF of $X_{i:n}$ is given by $F_{B_{i:n}} \circ F$ , where $F_{B_{i:n}}$ is the CDF of a beta random variable with parameters i and $n-i+1$ , that is, $B_{i:n}\sim \mathrm{beta}(i,n-i+1)$ (see [Reference Jones6]). Equivalently, one can write $X_{i:n}\; {\stackrel{\mathrm{d}}{=}}\; F^{-1}\circ B_{i:n}$ . This representation renders it difficult to establish conditions for a stochastic comparison between two different order statistics, say $X_{i:n}$ and $X_{j:m}$ , since the result depends on the four parameters i,j,n,m and on the analytical form of F. In a parametric framework, F is assumed to be known up to defining several real parameters, so the problem boils down to a mathematical exercise, which may still be analytically complicated. However, if F is in some nonparametric class then the problem is more complicated, and, as we show in the sequel, it can be solved just by adding some shape constraints on F. In this nonparametric framework, results may still be obtained by applying a simple decomposition trick: write $X_{i:n}\stackrel{\mathrm{d}}{=} F^{-1}\circ G\circ G^{-1}\circ B_{i,n}$ and assume that F is related to some known G by a suitable transform order. Indeed, in this case, the analytical form of G being known, the problem reduces to a simpler comparison between known RVs, namely $G^{-1}\circ B_{i:n}$ and $G^{-1}\circ B_{j:m}$ .

For the sake of convenience and flexibility in the applications of our main result, we introduce the following notation.

We may now state our main result, which establishes sufficient conditions for comparing expected order statistics.

Note that, if $\mathcal{H}$ is closed under the composition of functions, the assumption of preservation of the $\geq_\mathcal{H}^\mathrm{I}$ order is automatically fulfilled. Despite the simplicity of Theorem 3.1, its applications are remarkably interesting, showcasing the profound implications of the interplay between integral and transform orders.

4. Types of class generators

Definitions 2.1 and 2.4 become particularly interesting when the generator classes are chosen as well-known and popular families. We will now show that some of the already mentioned classes are encompassed within this framework, and add a number of further interesting families of distributions that can also be addressed. Indeed, different choices of the class $\mathcal{H}$ and of the reference CDF G in Theorem 3.1 yield different families of the type $\mathcal{F}_\mathcal{H}^G$ , which, we recall, are defined via a transform order. As will be shown, when $\mathcal{H}$ is the class of increasing convex or concave functions, $\mathcal{F}_\mathcal{H}^G$ may be characterised using the convex transform order. Hence, we will refer to these choices of $\mathcal{H}$ as convex-ordered families. Similarly, when $\mathcal{H}$ is the class of star-shaped or (increasing) anti-star-shaped functions, $\mathcal{F}_\mathcal{H}^G$ may be characterised via the star transform order, so we will refer to these choices as star-ordered families. For the sake of simplicity, besides the already defined classes $\mathcal{C}$ of convex functions and $\mathcal{S}$ of functions that are star-shaped at the origin, we shall define $\mathcal{V}$ as the class of concave functions and $\mathcal{A}$ as the class of increasing anti-star-shaped functions. Bear in mind that a function is convex if and only if its inverse is concave, so that $F\geq_\mathcal{C}^\mathrm{T}G$ is equivalent to $F\leq_\mathcal{V}^\mathrm{T}G$ . The same relation holds between star-shaped and increasing anti-star-shaped functions: $F\geq_\mathcal{S}^\mathrm{T}G$ is equivalent to $F\leq_\mathcal{A}^\mathrm{T}G$ . This is stated as follows.

The results that follow from Theorem 3.1 obviously depend on the choice of G. In particular, we will consider the uniform distribution on the unit interval, with CDF $U(x)=x$ , $x\in[0,1]$ , the exponential distribution, with CDF $\mathcal{E}(x)=1-{\mathrm{e}}^{-x}$ , $x\geq0$ , the standard logistic distribution with CDF ${L}(x)={1}/({\text{e}^{-x}+1})$ , $x\in\mathbb{R}$ , and the log-logistic distribution with shape parameter equal to 1, hereafter LL1, with CDF $LL(x)= x/({1+x})$ , $x\geq0$ . These reference distributions, as described below, lead to several well-known families of distributions. We will also consider the corresponding negative versions: in general, if $Y\sim G$ then $-Y\sim G_-$ , where $G_-(x)=1-G(\!-x)$ . Note that, due to symmetry, for the logistic distribution we have $L=L_-$ .

Combining the classes $\mathcal{C}$ , $\mathcal{V}$ , $\mathcal{S}$ , or $\mathcal{A}$ with the choices of G discussed above, we may generate several different families of distributions, some of them well known in the literature. An application of Theorem 3.1 will derive inequalities that hold for each of the constructed classes of distributions. Naturally, some of these are more interesting than others. Hereafter we will focus on the following.

• • The class of concave CDFs, also known as the decreasing density (DD) class, as it requires the existence of a decreasing PDF (except, possibly, at the right-endpoint of its support). This may be obtained by $\mathcal{F}_{\mathcal{C}}^{U}=\{F\colon F\geq^\mathrm{T}_{\mathcal{C}}U\} = \{F\colon F\geq_\mathrm{c}U\}=\{F^{-1}\in\mathcal{C}\}={\mathcal{V}}$ . This class has received much attention in the literature; for instance, it is a typical assumption for shape-constrained statistical inference [Reference Groeneboom and Jongbloed5]. Among known parametric models, the gamma, the log-logistic, and the Weibull distributions, with shape parameters less than or equal to 1, belong to this class.

• • The class of convex CDFs, also known as the increasing density (ID) class, as it requires the existence of an increasing PDF (except, possibly, at the right-endpoint of its support). This may be obtained by $\mathcal{F}_{\mathcal{V}}^{U}=\{F\colon F\geq^\mathrm{T}_{\mathcal{V}}U\} = \{F\colon U\geq_\mathrm{c}F\}=\{F^{-1}\in\mathcal{V}\}={\mathcal{C}}$ . This class is generally less applicable than the DD one, as it requires bounded support and contains few known parametric models.

• • The class of star-shaped CDFs. In the case of absolutely continuous distributions, this is also known as the class of distributions with increasing density on average (IDA). This may be obtained as $\mathcal{F}_{\mathcal{A}}^{U}=\{F\colon F\geq^\mathrm{T}_{\mathcal{A}}U\}=\{F\colon U\geq_\ast F\} = {\mathcal{S}}$ . This class extends the applicability of the ID class.

• • The class of anti-star-shaped CDFs. In the case of absolutely continuous distributions, this is also known as the class of distributions with decreasing density on average (DDA), as it requires ${F(x)}/{x}=(1/x)\int_0^xf(t)\,{\mathrm{d}} t$ to be decreasing. This may be obtained by $\mathcal{F}_{\mathcal{S}}^{U}=\{F\colon F\geq^\mathrm{T}_{\mathcal{S}}U\}=\{F\colon F\geq_\ast U\} = \{F^{-1}\in\mathcal{S}\}={\mathcal{A}}$ . This is an interesting class, as it extends the applicability of the popular DD class, allowing for nonmonotonicity of the PDF and jumps in the CDF.

• • The class of distributions with a convex hazard function, $H=-\ln (1-F)$ , that is, the well-known increasing hazard rate (IHR) class [Reference Marshall and Olkin16], as it requires the existence of an increasing hazard rate function $h=f/({1-F})$ (except, possibly, at the right-endpoint of the support). This may be obtained by $\mathcal{F}_{\mathcal{V}}^{\mathcal{E}} = \{F\colon F\geq^\mathrm{T}_{\mathcal{V}}\mathcal{E}\}=\{F\colon F^{-1}\circ\mathcal{E}\in\mathcal{V}\} = \{F\colon \mathcal{E}\geq_\mathrm{c}F\}$ . The properties and applicability of IHR models are well known.

• • The class of distributions with a star-shaped hazard function. This is known as the IHR on average (IHRA) class, as it requires ${H(x)}/{x}=({1}/{x})\int_0^xh(t)\,{\mathrm{d}} t$ , in the absolutely continuous case, to be increasing. This class may be obtained as $\mathcal{F}_{\mathcal{A}}^{\mathcal{E}} = \{F\colon F\geq^\mathrm{T}_{\mathcal{A}}\mathcal{E}\}=\{F\colon F^{-1}\circ\mathcal{E}\in\mathcal{A}\} = \{F\colon\mathcal{E}\geq_\ast F\}$ . This is a relevant class (see [Reference Marshall and Olkin16, Reference Shaked and Shanthikumar20]) that extends the applicability of the IHR class (in the nonnegative case).

• • The class of distributions with a concave hazard function, that is, the decreasing hazard rate (DHR) class [Reference Marshall and Olkin16], as it requires the existence of a decreasing hazard rate function $h=f/({1-F})$ . Analogously to the previous example, this class may be obtained by $\mathcal{F}_{\mathcal{C}}^{\mathcal{E}}=\{F\colon F\geq^\mathrm{T}_{\mathcal{C}}\mathcal{E}\} = \{F\colon F^{-1}\circ\mathcal{E}\in\mathcal{C}\}=\{F\colon \mathcal{E}\leq_\mathrm{c}F\}$ .

• • The class of distributions with an anti-star-shaped hazard function. This is known as the DHR on average (DHRA) class, as it requires ${H(x)}/{x}=({1}/{x})\int_0^xh(t)\,{\mathrm{d}} t$ , in the absolutely continuous case, to be decreasing. This class may be obtained as $\mathcal{F}_{\mathcal{S}}^{\mathcal{E}} = \{F\colon F\geq^\mathrm{T}_{\mathcal{S}}\mathcal{E}\}=\{F\colon F^{-1}\circ\mathcal{E}\in\mathcal{S}\} = \{F\colon\mathcal{E}\leq_\ast F\}$ . It extends the applicability of the DHR class (in the nonnegative case).

• • The class of CDFs such that $\log F$ is concave, also characterised by $f/F$ being decreasing, known as the decreasing reversed hazard rate (DRHR) class. This is a rather broad class of distributions. We can obtain this class by taking $\mathcal{H}=\mathcal{C}$ and $\mathcal{F}_{\mathcal{C}}^{\mathcal{E}_-}=\{F\colon F \geq_\mathrm{c} \mathcal{E}_-\}$ , the class of functions that dominate $\mathcal{E}_-$ with respect to the convex transform order.

• • The class of distributions with a convex odds function, ${F}/({1-F})$ , that is, the increasing odds rate (IOR) class [Reference Lando, Arab and Oliveira13], as it requires the existence of an increasing odds rate function $f/{(1-F)^2}$ (except, possibly, at the right-endpoint of the support). This may be obtained by $\mathcal{F}_{\mathcal{V}}^{{LL}} = \{F\colon F\geq^\mathrm{T}_{\mathcal{V}}{LL}\}=\{F\colon F^{-1}\circ {LL}\in\mathcal{V}\} = \{F\colon{LL}\geq_\mathrm{c}F\}$ . The properties and applicability of IOR models are discussed in [Reference Lando, Arab and Oliveira13, Reference Lando, Arab and Oliveira14].

• • The class with a concave odds function may be similarly defined as the decreasing odds rate (DOR) class, which may be obtained as $\mathcal{F}_{\mathcal{C}}^{LL}$ .

• • The class of distributions with a convex log-odds function, $\log({F}/({1-F}))$ , that is, the increasing log-odds rate (ILOR) class [Reference Zimmer, Wang and Pathak23], as it requires the existence of an increasing log-odds rate function $f/({F(1-F)})$ . This may be obtained by $\mathcal{F}_{\mathcal{V}}^{L}=\{F\colon L\geq_\mathrm{c}F\}$ .

• • The class with a concave log-odds function may be similarly defined as the decreasing log-odds rate (DLOR) class, which may be obtained as $\mathcal{F}_{\mathcal{C}}^{L}$ .

5. Convex-ordered families

In this section, we apply Theorem 3.1 to families of distributions that may be obtained through the convex transform order, extending some recent results of [Reference Lando, Arab and Oliveira12]. All the results are summarised in the following corollaries. Although some cases are already proved in [Reference Lando, Arab and Oliveira12], we report them here for the sake of completeness.

The flexibility concerning the choice of the $\mathcal{H}$ family in Theorem 3.1 allows for the following extension. Note that the first four cases of this corollary follow trivially from the previous result, and the fact that $X \leq_\mathrm{icx} Y$ if and only if $-X \geq_\mathrm{icv}-Y$ [Reference Shaked and Shanthikumar20, Theorem 4.A.1].

Proof. Note that if $G^{-1}\circ F$ is increasing concave, then $F^{-1}\circ G$ is increasing convex, and that the ICX order is obviously preserved under increasing convex transformations. As follows from [Reference Shaked and Shanthikumar20, Theorem 4.A.63] (note that the ICX order is, in [Reference Shaked and Shanthikumar20], referred as 2-icx) in conjunction with [Reference Lando, Arab and Oliveira12, Lemma 2.6], a sufficient condition for $G^{-1}\circ B_{i:n}\geq_{icx}G^{-1}\circ B_{j:m}$ is that $i\leq j$ and $\mathbb{E}G^{-1}\circ B_{i:n}\geq\mathbb{E} G^{-1}\circ B_{j:m}$ . Then, setting G as the uniform, unit exponential, LL1, standard logistic, negative exponential, and negative LL1, we obtain conditions (i)–(vi), respectively. We verify only case (v), the less obvious one, corresponding to $G=\mathcal{E}_{-}$ , where we need to compute

\begin{align*} \mathbb{E}\log B_{i:n} = \int_{0}^1\frac{t^{i-1}(1-t)^{n-i}\log t}{\mathcal{B}(i,n-i+1)}\,{\mathrm{d}} t = \psi(i)-\psi(n+1) = -\sum_{k=i}^{n} \frac{1}{k}, \end{align*}

using repeatedly [Reference Viola21, (6.44)], where $\psi(x)={\Gamma^\prime(x)}/{\Gamma(x)}$ , $x\geq 0$ , with $\Gamma$ representing the Euler gamma function, is the digamma function (we refer the reader to [Reference Viola21] for the properties of $\psi$ ).

Note that, since both the ICV and the ICX orders imply inequality between the means, Corollaries 5.1 and 5.2 provide assumptions implying that $\mathbb{E}X_{i:n}\geq \mathbb{E} X_{j:m}$ . Furthermore, we may derive conditions for the comparison with the mean of their parent distribution by setting $j=m=1$ or $i=n=1$ , respectively.

6. Star-ordered families

In this section we deal with families of distributions of the form $\{F\colon F\geq_\ast G\}$ , which include the family of anti-star-shaped CDFs and the DHRA family, using the SS order. Then, we move to families of the form $\{F\colon F\leq_\ast G\}$ , which include the family of star-shaped distributions and the IHRA family, using the new IAS order.

6.1. SS order of order statistics

Let us start with some preliminary discussion. As starshapedness refers only to functions with domain $[0,+\infty)$ , in this section we consider only nonnegative RVs. First, a simple preservation property.

Lemma 6.1. ([Reference Shaked and Shanthikumar20, Theorem 4.A.56].) $X\geq_\mathrm{ss}Y$ if and only if $h(X)\geq_\mathrm{ss}h(Y)$ , for every star-shaped function.

It is also useful to remark that a function $\phi$ is star-shaped if and only if its generalised inverse $\phi^{-1}$ is increasing anti-star-shaped, as proved in Lemma 4.1. We now recall the following characterisation of the SS order.

Theorem 6.1 ([Reference Shaked and Shanthikumar20, Theorem 4.A.54].) $X\geq_\mathrm{ss}Y$ if and only if, for every $x\geq 0$ , $\int_x^\infty t\,{\mathrm{d}} F(t) \geq \int_x^\infty t\,{\mathrm{d}} G(t)$ .

In the following subsections, we will frequently deal with transformations of beta RVs using the result stated next. The proof is omitted since it follows straightforwardly, requiring a simple observation of the shape of the graphical representation of the function considered in each case.

Lemma 6.2. Let $T_{a,b}(x)=x^a(1-x)^b$ , where $a,b\in\mathbb{R}$ , $c>0$ . Define R(a,b,c) to be the set of roots of the equation $T_{a,b}(x)=c$ that are in [0,1], and represent by $\#R(a,b,c)$ its cardinality. Then (i) if $ab<0$ , $\#R(a,b,c)=1$ ; (ii) if $ab>0$ , $\#R(a,b,c)\leq2$ ; (iii) if $ab=0$ , $\#R(a,b,c)\leq 1$ .

This lemma means that when $c>0$ , R(a,b,c) has at most two elements.

Using the above lemmas, it is not difficult to apply Theorem 3.1 to wide families of distributions, as discussed in the following subsections.

The next theorem deals with the case of anti-star-shaped CDFs, denoted as DDA distributions.

Theorem 6.2. Assume that F is anti-star-shaped. We write

(6.1) \begin{equation} Z(x) = \frac{i}{n+1}\big(1-F_{B_{i+1:n+1}}(x)\big) - \frac{j}{m+1}\big(1-F_{B_{j+1:m+1}}(x)\big). \end{equation}

If ${i}/({n+1})\geq{j}/({m+1})$ and, for every

\begin{align*}r\in R\bigg(i-j,n-i-(m-j),\frac{\mathcal{B}(i,n-i+1)}{\mathcal{B}(j,m-j+1)}\bigg)\end{align*}

it holds that $Z(r)\geq 0$ , then $X_{i:n}\geq_\mathrm{ss} X_{j:m}$ .

Proof. Since $F^{-1}$ is star-shaped, the result holds by Theorem 3.1 and Lemma 6.1, provided that $B_{i:n}\geq_\mathrm{ss} B_{j:m}$ , which, taking into account Theorem 6.1 and the distribution of the beta order statistics mentioned before ( $B_{i:n}\sim \mathrm{beta}(i,n-i+1)$ and $B_{j:m}\sim \mathrm{beta}(j,m-j+1)$ [Reference Jones6]), is equivalent to

(6.2) \begin{equation} \int_x^1\frac{t^{i}(1-t)^{n-i}}{\mathcal{B}(i,n-i+1)}\,{\mathrm{d}} t \geq \int_x^1\frac{t^{j}(1-t)^{m-j}}{\mathcal{B}(j,m-j+1)}\,{\mathrm{d}} t \quad \text{for all } x\in[0,1]. \end{equation}

It is easily seen that (6.2) is equivalent to $Z(x)\geq 0$ for every $x\in[0,1]$ . Now, the extreme points of Z are at 0, 1, or among the solutions of

\begin{align*}T_{i-j,(n-i)-(m-j)}(x)=\frac{\mathcal{B}(i,n-i+1)}{\mathcal{B}(j,m-j+1)},\end{align*}

hence the result follows immediately from the assumptions.

The results of Theorem 6.2 can be compared with Corollary 5.2(i). Assume that ${i}/({n+1})\geq{j}/({m+1})$ or, equivalently, that $Z(0)\geq0$ . If F is concave (DD class), then $X_{i:n}\geq_\mathrm{icx}X_{j:m}$ for $i\leq j$ . If F is increasing anti-star-shaped (yielding the wider DDA class), then the stronger order $X_{i:n}\geq_\mathrm{ss}X_{j:m}$ holds if $Z(r)\geq0$ for r in the described set. Recall that the ICX order is necessary for the SS order, and $i\leq j$ is necessary for the ICX order. So, the condition $Z(r)\geq0$ , for r in the set defined in Theorem 6.2, is stronger than just $i\leq j$ .

We may use Theorem 6.2 to get a complete geometric description of the $\geq_\mathrm{ss}$ -comparability of order statistics when F is DDA. Assume the sample sizes $n\leq m$ are given. Based on [Reference Arab, Oliveira and Wiklund2, Theorem 1], we know that $B_{i:n}\geq_\mathrm{st}B_{j:m}$ , which implies $B_{i:n}\geq_\mathrm{ss}B_{j:m}$ , whenever $i>j$ and $n-i<m-j$ , that is, whenever $i>j$ . Likewise, this result also implies that $B_{i:n}\leq_\mathrm{st}B_{j:m}$ , implying $B_{i:n}\leq_\mathrm{ss}B_{j:m}$ , whenever $i<j$ and $n-i>m-j$ , which is equivalent to $n-i>m-j$ (see Figure 1). For the region $i<j<i+m-n$ we have no $\geq_\mathrm{st}$ -comparability. The line $j=(m+1)i/(n+1)$ corresponds to points such that $Z(0)=0$ , where Z is given by (6.1). Above this line we have $Z(0)<0$ , hence, according to Theorem 6.1, there is no $\geq_\mathrm{ss}$ -comparability. Finally, we are left with the region where $i<j$ and ${i}/({n+1})\geq{j}/({m+1})$ , the region not shaded in Figure 1, where actual verification of (6.2) is needed.

Figure 1. $\geq_\mathrm{ss}$ -comparability for distributions in the DDA class.

For (i,j) in the unshaded region it is easily seen that

\begin{align*}Z^\prime(x)=-\frac{x^i(1-x)^{n-i}}{\mathcal{B}(i+1,n-i+1)}+\frac{x^j(1-x)^{m-j}}{\mathcal{B}(j+1,m-j+1)} < 0\end{align*}

whenever x is close to 0 or 1. Moreover, as Lemma 6.2 implies that Z has two extreme points in (0,1), the monotonicity of Z is ‘ $\searrow\nearrow\searrow$ ’. Numerical verification shows that the initial interval where Z is decreasing is rather small, so Z will remain nonnegative whenever $Z(0)={i}/(n+1)-{j}/(m+1)>0$ is large enough. Therefore, we expect that points (i,j) not satisfying the assumption in Theorem 6.2 will be close to the top border of the unshaded region. A few examples illustrating this behaviour are shown in Figure 2.

Figure 2. Points fulfilling (marked with $\circ$ ) and not fulfilling (marked with $\bullet$ ) the assumptions of Theorem 6.2. Left: $n=20$ , $m=30$ . Right: $n=30$ , $m=80$ .

We now extend our approach to the family of DHRA distributions.

Theorem 6.3. Assume that F is DHRA. Let

\begin{align*} Z(x) & = \left(\begin{array}{c}n\\i\end{array}\right){i}\sum_{k=0}^{i-1}\left(\begin{array}{c}{i-1}\\k\end{array}\right)(\!-1)^{i-1-k} \frac{({\mathrm{e}}^{-x})^{n-k}(\!-kx+nx+1)}{(k-n)^2} \\ & \quad - \left(\begin{array}{c}{m}\\j\end{array}\right){j}\sum_{k=0}^{j-1} \left(\begin{array}{c}{j-1}\\k\end{array}\right)(\!-1)^{j-1-k} \frac{({\mathrm{e}}^{-x})^{m-k}(\!-kx+mx+1)}{(k-m)^2}. \end{align*}

If $Z(0)\geq0$ and $Z(\!-\log(1-r))\geq 0$ for every

\begin{align*}r\in R\bigg(i-j,(n-i)-(m-j),\frac{\mathcal{B}(i,n-i+1)}{\mathcal{B}(j,m-j+1)}\bigg),\end{align*}

then $X_{i:n}\geq_\mathrm{ss} X_{j:m}$ .

Proof. Since $F^{-1}\circ \mathcal{E}$ is star-shaped, the result holds by Theorem 3.1 and Lemma 6.1, provided that $\mathcal{E}^{-1}\circ B_{i:n}\geq_\mathrm{ss} \mathcal{E}^{-1}\circ B_{j:m}$ , or, equivalently, $-\log(1-B_{i:n})\geq_\mathrm{ss}-\log(1-B_{j:m})$ . This may be expressed as

(6.3) \begin{multline} \frac{1}{\mathcal{B}(i,n-i+1)}\int_x^\infty t(1-{\mathrm{e}}^{-t})^{i-1}{\mathrm{e}}^{-(n-i+1)t}\,{\mathrm{d}} t \\ \geq \frac1{\mathcal{B}(j,m-j+1)}\int_x^\infty t(1-{\mathrm{e}}^{-t})^{j-1}{\mathrm{e}}^{-(m-j+1)t}\,{\mathrm{d}} t \quad \text{for all } x\geq0. \end{multline}

Using the binomial theorem, we obtain

\begin{align*} & \frac{1}{\mathcal{B}(i,n-i+1)}\int_x^\infty t(1-{\mathrm{e}}^{-t})^{i-1}{\mathrm{e}}^{-(n-i+1)t}\,{\mathrm{d}} t \\ & \qquad = \frac{1}{\mathcal{B}(i,n-i+1)}\sum_{k=0}^{i-1}\int_x^\infty(\!-1)^{i-1-k} \left(\begin{array}{c}{i-1}\\k\end{array}\right)t{\mathrm{e}}^{-(n-k)t}\,{\mathrm{d}} t \\ & \qquad = \left(\begin{array}{c}{n}\\i\end{array}\right){i}\sum_{k=0}^{i-1}\left(\begin{array}{c}{i-1}\\k\end{array}\right)(\!-1)^{i-1-k}\frac{{\mathrm{e}}^{(n-k)t}(\!-kx+nx+1)}{(k-n)^2}, \end{align*}

and similarly for the second term, hence (6.3) is equivalent to $Z(x)\geq 0$ for every $x\geq0$ . Note that $Z(+\infty)=\lim_{x\to+\infty}Z(x)=0$ , due to the exponential terms. Now, the function $Z\circ \mathcal{E}^{-1}$ is continuous on [0, 1], so it is nonnegative if and only if its minimal value in [0, 1] is nonnegative. The extreme points of $Z\circ\mathcal{E}^{-1}$ are easily seen to be among the solutions of

\begin{align*}T_{i-j,(n-i)-(m-j)}(1-{\mathrm{e}}^{-x})=\frac{\mathcal{B}(i,n-i+1)}{\mathcal{B}(j,m-j+1)},\end{align*}

hence the result follows immediately from the assumption that ${Z\circ\mathcal{E}^{-1}(r)}\geq 0$ for every

\begin{equation*} r\in R\bigg(i-j,(n-i)-(m-j),\frac{\mathcal{B}(i,n-i+1)}{\mathcal{B}(j,m-j+1)}\bigg). \end{equation*}

A complete geometric picture of the $\geq_\mathrm{ss}$ -comparability for DHRA distributions produces a plot similar to the one in Figure 1. The shaded regions where comparability exists are the same, but the directions of the $\geq_\mathrm{ss}$ -comparability are reversed, taking into account that $1-B_{i:n}$ and $1-B_{j:m}$ still have beta distributions with the parameters swapped. Moreover, the dash-dotted line in Figure 1 is now replaced by setting to zero the two terms appearing in Corollary 5.2(ii), that is, for each $i\leq n$ going through the coordinates j and $j+1$ such that $\sum_{k=n-i+1}^n1/k-\sum_{k=m-j+1}^m1/k$ and $\sum_{k=n-i+1}^n1/k-\sum_{k=m-(j+1)+1}^m1/k$ have opposite signs. The region below this curve, corresponding to $Z(0)>0$ , and above the diagonal is seen to be where we have no $\geq_\mathrm{ss}$ -comparability. The remaining region needs numerical verification. Hence, with respect to Figure 1, we reverse the direction of the comparisons, swap the unshaded and shaded areas between the two straight lines, and the separating dash-dotted line is no longer straight.

6.2. Properties of the IAS order

Our method can be applied to classes of the form $\{F\colon G\geq_* F\}$ using the IAS order. This includes the important IHRA class, obtained for $G=\mathcal{E}$ , and also the IDA class, where we take $G=U$ , the uniform distribution. In some sense the IAS order behaves like the SS order, with the disadvantage that it does not seem to have a simple characterisation based on a transformation of the CDFs, analogous to Theorem 6.1, which makes it difficult to check.

The IAS order satisfies the following properties.

Proposition 6.1. Let X and Y be nonnegative random variables with CDFs F and G, respectively.

1. (i) $X\geq_\mathrm{st}Y\ \Longrightarrow\ X\geq_\mathrm{ias}Y\ \Longrightarrow\ X\geq_\mathrm{icv}Y$ .

2. (ii) $X\geq_\mathrm{ias}Y$ implies $h(X)\geq_\mathrm{ias}h(Y)$ for every increasing anti-star-shaped h.

3. (iii) Let $\Theta$ be a random variable, and let $F(\,\cdot\mid\theta)$ and $G(\,\cdot\mid\theta)$ be the conditional CDFs of X and Y with respect to the event $\Theta=\theta$ . If $F(\,\cdot\mid\theta)\geq_\mathrm{ias}G(\,\cdot\mid\theta)$ for every possible realisation $\theta$ of $\Theta$ , then $X\geq_\mathrm{ias}Y$ .

Proof.

1. (i) The first implication follows from the fact that all increasing anti-star-shaped functions are increasing. Let $\phi$ be an increasing concave function. If $\phi(0)=0$ , $\phi$ is also anti-star-shaped and $X\geq_\mathrm{ias}Y$ implies that $\mathbb{E}\phi(X)\geq\mathbb{E}\phi(Y)$ . If $\phi(0)\neq0$ , define $\phi_0(x)=\phi(x)-\phi(0)$ , which is still increasing concave, and proceed similarly.

2. (ii) Let $\phi$ be any increasing anti-star-shaped function. Taking into account that h is increasing, the quotient

   \begin{align*}\frac{\phi(h(x))}{x} = \frac{\phi(h(x))}{h(x)} \frac{h(x)}{x}\end{align*}
   is the product of two decreasing functions, hence it is decreasing itself. That is, $\phi\circ h$ is increasing anti-star-shaped. Therefore, $X\geq_\mathrm{ias}Y$ implies that $\mathbb{E}(\phi \circ h(X))\geq \mathbb{E}(\phi \circ h (Y))$ or, equivalently, $h(X)\geq_\mathrm{ias}h(Y)$ .

3. (iii) This follows directly from the tower law of conditional expectations.

Properties (ii) and (iii) mean that the IAS order is closed under composition and mixtures, respectively. Property (i) shows why the IAS order can be useful. In fact, it measures size and dispersion at the same time. The IAS order implies inequality of the means; moreover, if $\mathbb{E} X=\mathbb{E} Y$ , then $X\geq_\mathrm{ias}Y$ implies that $\textrm{Var}(X)\leq\textrm{Var}(Y)$ . We now provide some examples showing that neither of the implications in Proposition 6.1(i) is an equivalence. The above properties suggest that the IAS order can be used as a valid (and stronger) alternative to the commonly used ICV order whenever we deal with star-ordered families. However, for technical reasons, the verification of the IAS order is complicated, as discussed in the next subsection.

Example 6.1. Let $p\in(0,1)$ , $p^\prime>p$ , and $s\in (0,1)$ be given, and consider the random variables X and Y with distributions $\mathbb{P}(X=p)=1-\mathbb{P}(X=0)=s$ and $\mathbb{P}(Y=p^\prime)=1-\mathbb{P}(Y=0)=r={sp}/{p^\prime}$ . As $r<s$ , the CDFs of X and Y cross, so $X\not\geq_\mathrm{st}Y$ . However, given any increasing anti-star-shaped function $\phi$ , so that $\phi(0)=0$ and ${\phi(x)}/{x}$ is decreasing, we have $\mathbb{E}\phi(X)=s\phi(p)=sp{\phi(p)}/{p} \geq sp{\phi(p^\prime)}/{p^\prime}=r\phi(p^\prime)=\mathbb{E}\phi(Y)$ . Hence $X\geq_\mathrm{ias}Y$ , showing that the first implication in Proposition 6.1(i) is indeed not an equivalence.

Example 6.2. To show that the second implication is also not an equivalence, take X exponentially distributed, with CDF $\mathcal{E}$ , and Y with Weibull distribution, with shape parameter 2 and scale parameter $\Gamma\big(\frac32\big)$ . Therefore, $\mathbb{E} X=\mathbb{E} Y$ and their CDFs cross once. According to [Reference Shaked and Shanthikumar20, Theorem 4.A.22], $Y\geq_\mathrm{icv}X$ . On the other hand, considering the increasing anti-star-shaped function

\begin{align*} \phi(x)=\begin{cases} 10x, & x<\frac{1}{10}, \\ 1, & \frac{1}{10}\leq x < 1, \\ x, & x\geq 1, \\ \end{cases} \end{align*}

it is easy to verify that $\mathbb{E}\phi(X)>\mathbb{E}\phi(Y)$ , so $Y\not\geq_\mathrm{ias}X$ .

6.3. IAS order of order statistics: A heuristic approach

From a practical point of view, a simple characterisation of the IAS order, described in distributions terms, seems unavailable and remains an open problem. An alternative approach may be based on Theorem 6.4. To state our result, we need some additional notation. Given $a>0$ , define

\begin{align*}u_a(x)= \begin{cases} 0, &x < a,\\ x, &x\geq a.\\\end{cases}\end{align*}

Moreover, given sequences $a_1,\ldots,a_n>0$ and $b_1,\ldots,b_n>0$ , define

(6.4) \begin{equation} s_n(x)=\sum_{\ell=1}^n b_\ell u_{a_\ell}(x).\end{equation}

Theorem 6.4. Let $X\sim F$ and $Y\sim G$ be absolutely continuous nonnegative random variables with density functions f and g, respectively. $X\geq_\mathrm{ias}Y$ if and only if, for every positive integer n, and for every $0=a_0\leq a_1\leq\cdots\leq a_n$ and $b_1,\ldots,b_n>0$ , and every $s_n$ defined according to (6.4), we have

(6.5) \begin{multline} \int_0^\infty s_n(t)f\circ s_n(t)\,{\mathrm{d}} t + \sum_{\ell=1}^n a_\ell(F(a_\ell)-F(a_{\ell-1})) \\ \geq \int_0^\infty s_n(t)g\circ s_n(t)\,{\mathrm{d}} t + \sum_{\ell=1}^n a_\ell(G(a_\ell)-G(a_{\ell-1})). \end{multline}

Proof. As proved in Lemma 4.1, h is increasing anti-star-shaped if and only if $h=s^{-1}$ , where s is star-shaped. By the change of variable $h(x)=t$ , $\int_0^\infty h(x)\,{\mathrm{d}} F(x) = \int_0^\infty t\,{\mathrm{d}} F\circ s(t)$ , and equivalently for the integration with respect to G. Hence, we need to prove that $\int_0^\infty t\,{\mathrm{d}} F\circ s(t)\geq \int_0^\infty t\,{\mathrm{d}} G\circ s(t)$ for every star-shaped function s. Now, every star-shaped function s can be approximated by a sequence $s_n$ , where the sequences $a_\ell$ and $b_\ell$ satisfy the given assumptions. Indeed, this follows directly from approximating the increasing function ${s(x)}/{x}$ by an increasing step function. Therefore, by monotonous approximation, $X\geq_\mathrm{ias}Y$ if and only if $\int_0^\infty t\,{\mathrm{d}} F\circ s_n(t)\geq \int_0^\infty t\,{\mathrm{d}} G\circ s_n(t)$ for every integer n, and every $a_\ell$ and $b_\ell$ as given. As $F\circ s_n$ and $G\circ S_n$ have discontinuities at $a_1,\ldots,a_n$ , and F and G have densities f and g, respectively, we obtain

\begin{align*} \int_0^\infty t\,{\mathrm{d}} F\circ s_n(t) = \int_0^\infty s_n(t)f\circ s_n(t)\,{\mathrm{d}} t + \sum_{\ell=1}^n a_\ell(F(a_\ell)-F(a_{\ell-1})), \end{align*}

and similarly for $\int_0^\infty t\,{\mathrm{d}} G\circ s_n(t)$ .

Although Theorem 6.4 provides a necessary and sufficient condition, it requires the verification of infinitely many inequalities, thus reducing its usability for a direct verification of the IAS order. To address this difficulty we propose the following simulation algorithm to check if $X\geq_\mathrm{ias}Y$ . Let K be the total number of repetitions; for every $k=1,\ldots,K$ :

1. (i) randomly generate n, say n(k), from a discrete distribution with infinite support;

2. (ii) randomly generate the sequences $0=a_0\leq a_1\leq\cdots\leq a_{n(k)}$ and $b_1,\ldots,b_{n(k)}$ , and define $s_n$ according to (6.4);

3. (iii) compute

   \begin{align*} I_F(k) & = \int_0^\infty s_n(t)f\circ s_n(t)\,{\mathrm{d}} t + \sum_{\ell=1}^n a_\ell(F(a_\ell)-F(a_{\ell-1})), \\ I_G(k) & = \int_0^\infty s_n(t)g\circ s_n(t)\,{\mathrm{d}} t + \sum_{\ell=1}^n a_\ell(G(a_\ell)-g(a_{\ell-1})); \end{align*}

4. (iv) if $R(K)=(1/K)\sum_{k=1}^K\mathbb{I}(I_F(k)-I_G(k))<1$ then $X\not\geq_\mathrm{ias}Y$ ; otherwise, if $R(K)=1$ we have an indication that X may dominate Y in the IAS order.

This procedure can be used to check whether $X_{i:n}\geq_\mathrm{ias}X_{j:m}$ .

Corollary 6.1. Assume that $G\geq_\ast F$ . If, for every positive integer n and for every choice of $0=a_0\leq a_1\leq\cdots\leq a_n\leq 1$ and $b_1,\ldots,b_n>0$ ,

(6.6) \begin{multline} \int_0^\infty s_n(t)f_{B_{i:n}}\circ G^{-1}\circ s_n(t)\,{\mathrm{d}} t + \sum_{\ell=1}^n a_\ell\big(F_{B_{i:n}}\circ G^{-1}(a_\ell)-F_{B_{i:n}}\circ G^{-1}(a_{\ell-1})\big) \\ \geq \int_0^\infty s_n(t)f_{B_{j:m}}\circ G^{-1}\circ s_n(t)\,{\mathrm{d}} t + \sum_{\ell=1}^n a_\ell\big(F_{B_{j:m}}\circ G^{-1}(a_\ell)-F_{B_{j_m}}\circ G^{-1}(a_{\ell-1})\big), \end{multline}

then $X_{i:n}\geq_\mathrm{ias}X_{j:m}$ .

Proof. According to Theorem 6.4, (6.6) is equivalent to $G^{-1}\circ B_{i:n}\geq_\mathrm{ias} G^{-1}\circ B_{j:m}$ . Then, the result follows from the fact that $F^{-1}\circ G $ is increasing anti-star-shaped and this class is closed under composition, noting that, as we are integrating with respect to a beta distribution, we only need to consider the approximation in [0, 1].

Similarly to the previous applications, choosing a particular G leads to conditions for $X_{i:n}\geq_\mathrm{ias}X_{j:m}$ when F belongs to the appropriate family of distributions. For example, taking $G=U$ we find conditions that apply when F is star-shaped (or in the IDA class, referring to the families described in Section 4), while the choice $G=\mathcal{E}$ gives conditions when F is IHRA.

Example 6.3. Take $n = 200$ , $i=70$ , $m = 70$ , $j=8$ . In this case we have $B_{70:200}\not\geq_\mathrm{st} B_{8:70}$ , which means that $X_{70:200}\not\geq_\mathrm{st} X_{8:70}$ . Hence, as the strongest of the stochastic order fails to hold, we may be interested in checking whether some weaker order, such as ICV or IAS, holds. If F is convex, these values satisfy $i\geq j$ and ${i}/({n+1})\geq{j}/({m+1})$ , so $X_{70:200}\geq_\mathrm{icv}X_{8:70}$ . However, if F is not convex but only star-shaped, we can check the condition of Corollary 6.1 for $G=U$ , using the proposed algorithm. Taking $K=1000$ and randomly generating n from a Poisson distribution with parameter $\lambda=20$ , we obtain $R(1000)=1$ , suggesting, although not actually proving, that $X_{70:200}\geq_\mathrm{ias}X_{8:70}$ . This would imply that $\mathbb{E} X_{70:200}\geq\mathbb{E} X_{8:70}$ , although $X_{70:200}\not\geq_\mathrm{st}X_{8:70}$ , as the identity is an increasing anti-star-shaped function. Note that $X_{70:200}\geq_\mathrm{icv}X_{8:70}$ means that $\mathbb{E} u(X)\geq\mathbb{E} u(Y)$ for every increasing concave function u, a large subset of the class of increasing anti-star-shaped functions. Nevertheless, the proposed algorithm was not able to identify an increasing anti-star-shaped function violating (6.6), although, of course, it is not guaranteed that such a function does not exist. Now, take $n=200$ , $i = 65$ , $m = 40$ , $j = 10$ . Even in this case, $X_{65:200}\not\geq_\mathrm{st} X_{10:40}$ . If F is convex we still have $X_{65:200}\geq_\mathrm{icv}X_{10:40}$ . However, applying our algorithm again (with the same settings), we obtain $R(1000)=0.89$ , so in this case we know that $B_{65:200}\not\geq_\mathrm{ias}B_{10:40}$ . Accordingly, we cannot conclude that $X_{65:200}$ dominates $X_{10:40}$ in the IAS order, although we assume that F is star-shaped. Similar examples can be provided for the IHRA case.

7. Bounds for probabilities of exceedance

Consider a scenario where we represent the lifetime of a k-out-of-n system as $X_{k:n}$ . A notable challenge in reliability analysis involves determining the probability that an individual component’s lifetime falls below or exceeds the expected lifetime of the entire system, denoted as $\mathbb{E}X_{k:n}$ . In a parametric setting, this probability can be precisely computed using the mathematical formula of the parent CDF F. However, when the exact form of F is unknown, we can leverage information about its overall shape to establish upper or lower bounds for this probability. This would follow from the application of Jensen’s inequality, under the assumption that F belongs to a convex-ordered family $\mathcal{F}_{\mathcal{V}}^G$ or $\mathcal{F}_{\mathcal{C}}^G$ . We remark that the case in which G is the uniform distribution has already been discussed in [Reference Ali and Chan1].

Put otherwise, this result means that, if $F\in\mathcal{F}_{\mathcal{V}}^G$ , the expected order statistic $\mathbb{E} X_{i:n}$ is always less than or equal to the $ p_{i:n}^G$ -quantile of X. Similarly, if $F\in\mathcal{F}_{\mathcal{C}}^G$ , the expected order statistic $\mathbb{E} X_{i:n}$ is always greater than or equal to the $p_{i:n}^G$ -quantile of X, that is, $\mathbb{E} X_{i:n}\geq F^{-1}(p_{i:n}^G)$ . This result also enables a useful characterisation of the LL1 distribution. Indeed, we can generally approximate $\mathbb{E} X_{i:n}$ with $F^{-1}({i}/n)$ : for $n\to\infty$ and ${i}/n\to p$ (constant), $F^{-1}({i}/n)\to\mathbb{E} X_{i:n}$ . This result is exact for finite n if and only if X has an LL1 distribution.

Common choices of G yield the following explicit expressions for $p_{i:n}^G$ :

• • If $G=U$ , $p_{i:n}^U={i}/({n+1})$ .

• • If $G=\mathcal{E}$ , $p_{i:n}^\mathcal{E}=1-\exp\big({-}\sum_{k=n-i+1}^n 1/k\big)$ .

• • If $G={LL}$ , $p_{i:n}^{LL}={i}/n$ .

• • If $G=\mathcal{E_-}$ , $p_{i:n}^\mathcal{E_-}=\exp\big({-}\sum_{k=i}^n 1/k\big)$ .

Table 1 shows the $p_{i:n}^G$ bounds for $n=10$ and some choices of G. The application of our results is quite straightforward. For instance, if we know that the CDF of interest, F, is IHR and has a decreasing density, as is the case, for example, for the Gompertz distributions (for suitably chosen parameters), then the probability of having $X\leq \mathbb{E} X_{i:n}$ is always between $p_{i:n}^{U}$ and $p_{i:n}^{\mathcal{E}}$ , that is,

\begin{align*}\frac{i}{n+1}\leq \mathbb{P}(X\leq \mathbb{E} X_{i:n})\leq 1-\exp\Bigg({-}\sum_{k=n-i+1}^n \frac1k\Bigg).\end{align*}

If $i=3$ and $n=10$ , this means that $\mathbb{P}(X\leq \mathbb{E} X_{i:n})\in{[0.273,0.285]}$ . Similarly, if F is IOR and DRHR, then

\begin{align*}\exp\Bigg({-}\sum_{k=i}^n \frac1k\Bigg)\leq \mathbb{P}(X\leq \mathbb{E} X_{i:n})\leq \frac{i}{n}.\end{align*}

As these classes are wider than those previously considered, these bounds are generally weaker, so, for $i=3$ and $n=10$ , we now find $\mathbb{P}(X\leq \mathbb{E} X_{i:n})\in{[0.240,0.300]}$ .

Table 1. $p_{i:10}^G$ for different choices of G.

The bounds with respect to the families of distributions are sharp, as illustrated in Figure 3, where we have plotted true probabilities for two distributions that are both IRH and DRHR (Weibull with shape parameter larger than 1, and power distribution), and the inverted power distribution with exponent $\frac12$ , which is not IHR and hence violates the lower bound.

Figure 3. Upper bounds (solid) and true values for $\mathbb{P}(X\leq \mathbb{E} X_{i:n})$ with respect to the lower bounds (the horizontal line): Weibull(3,1) (dashed), $F(x)=x^3$ (dotted), $F(x)=1-\sqrt{1-x}$ (dash-dotted).