Bounds of expectations of order statistics for distributions possessing monotone reversed failure rates

Abstract

In the literature, the sharp positive upper mean-variance bounds on the expectations of order statistics based on independent identically distributed random variables with the decreasing and increasing failure rates, have been recently presented. In this paper we determine analogous evaluations in the dual cases when the parent distributions have monotone reversed failure rates.

1 Introduction

The failure (hazard) rate function is one of the crucial tools in the random lifetime analysis. For an absolutely continuous random variable X with a distribution function F and the respective density function f, it is defined as

$$\begin{aligned} h_X (x)= & {} \frac{f(x)}{1-F(x)}= \lim _{\vartriangle x \searrow 0} \frac{\mathbb {P}(x < X \le x + \vartriangle \! x)}{\vartriangle \! x\,\,\mathbb {P}(X> x)} \\= & {} \lim _{\vartriangle x \searrow 0} \frac{\mathbb {P}( X \le x +\vartriangle \!x |X > x)}{\vartriangle \!x}, \end{aligned}$$

and it is interpreted as the infinitesimal probability of failure of an item just after time x under the condition that it survived till time x. High values of the failure rate suggest that the item has a tendency of failing fast. Indeed, if X has the failure rate uniformly greater than Y, the relation is inherited by the respective distribution functions which means that X precedes Y in the classic stochastic order. It is also important to study variability of the failure rate. The increasing (decreasing) failure rate property of a random variable means that that it has an increasing (decreasing, respectively) tendency of failing as the time runs. We adhere to a popular convention of writing that the failure rate is increasing (decreasing), and denoting the property by IFR (DFR, respectively) if the function is actually nondecreasing (nonincreasing, respectively).

A dual notion to the failure rate, called the reversed failure (hazard) rate

$$\begin{aligned} r_X (x)= & {} \frac{f(x)}{F(x)}= \lim _{\vartriangle x \searrow 0} \frac{\mathbb {P}(x- \vartriangle \! x \le X \le x )}{\vartriangle \! x\,\,\mathbb {P}(X \le x)} \\= & {} \lim _{\vartriangle x \searrow 0} \frac{\mathbb {P}( X \ge x -\vartriangle \!x |X \le x)}{\vartriangle \!x}, \end{aligned}$$

represents the infinitesimal probability that X failed just before x under the condition that it is already failed at x. Conversely to the standard failure rate case, the large values of the reversed failure rate assure a long life of the object. Indeed, if the reversed failure rate of X is always greater than that of Y then X succeeds Y in the stochastic order. The fact that $r_X$ decreases (increases) means that X has a decreasing (increasing) tendency of failure at large values of time, and in consequence, the greater probability of failing earlier (later, respectively). If X has a nonincreasing (nondecreasing, respectively) reversed failure rate function, we say that it has the decreasing (increasing, respectively) reversed failure rate property, and denote it shortly by DRFR (IRFR, respectively).

Combining the simple relations

$$\begin{aligned} \frac{f(x)}{F(x)} = \frac{f(x)}{1-F(x)} \, [1-F(x)]\, \frac{1}{F(x)} \end{aligned}$$

with trivial observations that the last two factors of the right-hand side are nonincreasing we immediately conclude that the DFR property implies a nonincreasing density function (DD, for short), and the latter implies the DRFR. Similarly we note that every IRFR distributions has a nondecreasing density (shortly ID), and every element of the ID family has an increasing failure rate. Since all $\frac{f(x)}{1-F(x)}$, f(x) and $\frac{f(x)}{F(x)}$ represent possibilities of failure in a close neighborhood of x, all the properties of decreasing failure rate, density and reversed failure rate functions reflect the decreasing tendency of failing in time. The DRFR property is the most general one while the DFR is the most stringent. If either of the functions is increasing, the failure is more probable rather earlier than later. Then IRFR is the most severe condition, and IFR is the mildest one. These and many other results concerning the distributions with monotone standard and reversed failure rates can be found in (Shaked and Shanthikumar 2007, Sect. 1.1.B).

Order statistics and their linear combinations are widely used in the statistical inference (see, e.g., David and Nagaraja 2003, Chapters 7–9). The mixtures of their distributions are applied for analysis of reliability system lifetimes (see, e.g., Samaniego 2007). They frequently appear in the survival analysis, especially when some censoring schemes of observation are incorporated. We focus here on the standard model of order statistics based on independent identically distributed samples (iid for brevity). The classic general bounds for the expectations of sample maxima centered about the population mean were determined independently by Gumbel (1954) and Hartley and David (1954). They were expressed in the population standard deviation scale units. Their counterparts for nonextreme order statistics were presented in Moriguti (1953).

More subtle evaluations under the restriction that the parent distribution is either DFR or DD were established in Danielak (2003). The bounds were positive and valid for order statistics with sufficiently large ranks. Order statistics with low ranks based on DFR and DD populations were treated in Rychlik (2009a, 2009b), respectively. Evaluations of expected extreme order statistics based on iid IFR samples were established by Rychlik (2014) and Rychlik and Szymkowiak (2021), whereas the bounds for the remaining order statistics from the IFR populations were presented in Goroncy and Rychlik (2016). Similar results under the assumption that the parent distribution has an increasing density function can be found in Goroncy and Rychlik (2015). We finally mention the optimal bounds on the expectations of generalized order statistics with the general, DFR, and DD baseline distributions determined in Goroncy (2014), Bieniek (2006), Goroncy (2020), Bieniek (2008) and Goroncy (2017), respectively, as well as with the decreasing generalized failure rate distributions by Bieniek and Goroncy (2020).

In this paper we determine bounds on the expectations of order statistics coming from iid samples with monotone reversed failure rate. We assume that iid random variables $X_1,\ldots ,X_n$ have a common DRFR (IRFR) distribution function F, say, with an expectation $\mu \in \mathbb {R}$ and a positive finite variance $\sigma ^2$, and denote the respective order statistics by $X_{1:n} \le \ldots \le X_{n:n}$. Under these assumptions, we provide nonnegative sharp upper bounds on the expectations of standardized order statistics $\mathbb {E}\frac{X_{j:n}-\mu }{\sigma }$. Note that simple bounds for the sample minima are equal to 0, because $\mathbb {E}X_{1:n} \le \mathbb {E}X_1$ with no assumptions on the joint distributions. The optimality of $\mathbb {E}\frac{X_{1:n}-\mu }{\sigma } \le 0$ for the DRFR family follows from Rychlik (2009a, 2009b), where attainability of the zero bound was proven for narrower DD and DFR families. We also show that 0 is the optimal bound for other order statistics with small ranks coming from the DRFR populations. On the contrary, in the IRFR case all the bounds for $2 \le j \le n$ are positive, and the zero bound is sharp for the sample minima then. In Sect. 2 we present some auxiliary tools necessary for establishing our main results. The bounds for the DRFR populations are present in Sect. 3. Section 4 contains analogous results for the IRFR case. Our theoretical results are illustrated by exemplary numerical bounds for various sample sizes and numbers of order statistics in Sect. 5. The proofs are presented in the Appendix.

2 Auxiliary results

Our results are mostly derived with use of so called projection method. The method was proposed in Gajek and Rychlik (1996), and widely developed in Rychlik (2001). We start from the observation that nonincrease (nondecrease) of the reversed failure rate $\frac{f(x)}{F(x)}$ implies that $\ln F(x)$ is concave (convex, respectively) on the support of F. The family of distributions with concave (convex) logarithm of the distribution function is slightly larger than the family of DRFR (IRFR, respectively) distributions. It contains the distributions which have a version of density functions such that $\frac{f(x)}{F(x)}$ is nonincreasing (nondecreasing, respectively) on their interval supports, and possibly an atom at the left (right, respectively) end-point of the support. The family with concave (convex) logarithms of distribution functions is closed in the weak convergence topology, and the family with nonincreasing (nondecreasing, respectively) ratios $\frac{f(x)}{F(x)}$ is a dense subset of it. It follows that sharp bounds on the expectations of order statistics can be attained by possibly discontinuous elements of the class with concave (convex) $\ln F(x)$, but these bounds can be also attained in the limit by sequences of continuous parent DRFR (IRFR, respectively) distributions. Therefore for convenience of our further investigations, we identify the family of distributions with concave (convex) $\ln F(x)$ with the family of DRFR (IRFR, respectively) distributions.

Moreover, we focus rather on the inverses $F^{-1}(e^x)$, $x<0$, of the original functions $\ln F(x)$ which are convex (concave) in the DRFR (IRFR, respectively) case. This means that F succeeds (precedes, respectively) the negative exponential distribution function $V(x) = e^x$, $x<0$, in the convex transform order introduced by van Zwet (1964). We treat the compositions $F^{-1}(e^x)$, $x<0$, as the elements of the Hilbert space $L^2(\mathbb {R}_-, e^xdx)$. The compositions are nondecreasing functions on the negative half-axis, and so are their translations $F^{-1}(e^x)-\mu$, $x<0$. Note that $(F^{-1}\circ V - \mu , 1)_V =\int _{-\infty }^{0}[F^{-1}(e^x)-\mu ] e^x \,dx = \int _{0}^{1} [F^{-1}(x)-\mu ]\,dx =0,$ and

$$\begin{aligned} \Vert F^{-1} \circ V - \mu \Vert _V^2 = \int _{-\infty }^{0}[F^{-1}(e^x)-\mu ]^2 e^x \,dx = \int _{0}^{1} [F^{-1}(x)-\mu ]^2\,dx = \sigma ^2,\nonumber \\ \end{aligned}$$

(2.1)

where $( \cdot , \cdot )_V$ and $\Vert \cdot \Vert _V$ denote the inner product and norm of $L^2(\mathbb {R}_-, e^x\,dx)$, respectively, and $\mu$ and 1 are simply the constant functions equal to $\mu$ and 1, respectively, on the whole interval $(-\infty ,0)$. Summing up, the family of functions $F^{-1}(e^x)-\mu$, $x<0$, constitute the convex cone

$$\begin{aligned} \breve{\mathcal {C}}^0_{V} = \{ g \in L^2(\mathbb {R}_-, e^x\,dx): g\; \textrm{nondecreasing, convex, } (g,1)_V =0\} \end{aligned}$$

(2.2)

in the DRFR case, and

(2.3)

for the IRFR family. Let

$$\begin{aligned} f_{j:n}(x){} & {} = n B_{j-1,n-1}(x) =n \left( {\begin{array}{c}n-1\\ j-1\end{array}}\right) x^{j-1} (1-x)^{n-j}, \qquad 0<x<1,\nonumber \\{} & {} \quad j=1,\ldots ,n, \end{aligned}$$

(2.4)

denote the density function of the jth order statistic $U_{j:n}$ based on iid standard uniform variables $U_1,\ldots , U_n$, where

$$\begin{aligned} B_{k,m}(x) = \left( {\begin{array}{c}m\\ k\end{array}}\right) x^k(1-x)^{m-k}, \qquad 0<x<1,\;k=1,\ldots ,m, \end{aligned}$$

(2.5)

are the Bernstein polynomials of degree m. Also, we shall further use the distribution functions of (2.4) defined as $F_{j:n}(x) = \sum _{i=j}^{n} B_{i,n}(x) = \sum _{i=j}^{n}\left( {\begin{array}{c}n\\ i\end{array}}\right) x^i(1-x)^{n-i}$, $0<x<1,\;j=1,\ldots ,n.$ Put

$$\begin{aligned} h_{j:n}(x) = f_{j:n}(e^x) = n \left( {\begin{array}{c}n-1\\ j-1\end{array}}\right) e^{(j-1)x} (1-e^x)^{n-j}, \qquad x<0,\;j=1,\ldots ,n, \end{aligned}$$

(2.6)

The following relations are crucial for our analysis. In the DRFR case we have

$$\begin{aligned} \mathbb {E}X_{j:n}-\mu= & {} \int _{-\infty }^{0} [F^{-1}(e^x)-\mu ] [h_{j:n}(x)-1]e^x\,dx \le \int _{-\infty }^{0} [F^{-1}(e^x)-\mu ] \breve{P}^0_{V}(h_{j:n}-1)(x)e^x\,dx \nonumber \\\le & {} \left[ \int _{-\infty }^{0} [F^{-1}(e^x)-\mu ]^2e^x\,dx \right] ^{1/2} \left[ \int _{-\infty }^{0} [ \breve{P}^0_{V}(h_{j:n}-1)(x)]^2 e^x\,dx \right] ^{1/2} \nonumber \\= & {} \sigma || \breve{P}^0_{V}(h_{j:n}-1) ||_V \nonumber \\ \end{aligned}$$

(2.7)

(see, e.g., (Balakrishnan 1981, Corollary 1.4.2), (Rychlik 2001, Theorem 1)), where the function $\breve{P}^0_{V}(h_{j:n}-1)$ denotes the projection of $h_{j:n}-1$ onto the cone (2.2). This means that the bound on $\mathbb {E}\frac{X_{j:n}-\mu }{\sigma }$ in the class of iid DRFR samples amounts to $|| \breve{P}^0_{V}(h_{j:n}-1) ||_V$. Moreover, if it is non-zero, the equalities in () are attained iff $F^{-1}(e^x)-\mu$ and $\breve{P}^0_{V}(h_{j:n}-1)(x)$ are proportional. Under the condition (), the precise attainability condition of the bound is $\frac{F^{-1}(e^x)-\mu }{\sigma }= \frac{\breve{P}^0_{V}(h_{j:n}-1)(x)}{|| \breve{P}^0_{V}(h_{j:n}-1) ||_V}.$ If the norm of projection is equal to 0, the attainability condition is $\sigma = 0$, which contradicts our assumptions. However, in such cases we present the families of nondegenerate parent distributions which attain the zero bounds in the limit.

It actually suffices to consider the projection onto the convex cone

$$\begin{aligned} \breve{\mathcal {C}}_{V} = \{ g \in L^2(\mathbb {R}_-, e^x\,dx): g\; \textrm{nondecreasing, convex} \}, \end{aligned}$$

(2.8)

since () is translation invariant which assures that $\breve{P}^0_{V}(h_{j:n}-1)= \breve{P}_{V}(h_{j:n}-1) = \breve{P}_{V}h_{j:n}-1$. (cf. Danielak 2003, Lemma 1). In conclusion we obtain

$$\begin{aligned} \mathbb {E}\frac{X_{j:n}-\mu }{\sigma } \le \left( \Vert \breve{P}_{V}h_{j:n}\Vert _V^2 -1 \right) ^{1/2} \end{aligned}$$

(2.9)

with the equality condition

$$\begin{aligned} \frac{F^{-1}(e^x)-\mu }{\sigma } = \frac{\breve{P}_{V}h_{j:n}(x)-1}{\left( \Vert \breve{P}_{V}h_{j:n}\Vert _V^2 -1 \right) ^{1/2}}, \end{aligned}$$

(2.10)

when the right-hand side of () is positive. Similar arguments and results are valid for the IRFR distributions if we replace $\breve{\mathcal {C}}^0_{V}$, $\breve{P}^0_{V}$, $\breve{\mathcal {C}}_{V}$, and $\breve{P}_{V}$ by , the projection operator on ,

(2.11)

and denoting the projection onto (), respectively.

Accordingly, our crucial problems consist in finding the projections of $h_{j:n}(x) = f_{j:n}(e^x)$, $j=1,\ldots , n$, on the cones () and (). They are solved in two steps. We first describe possible shapes of the candidates for the projections. This allows us to confine our attention to some parametric subfamilies of () and (). Then we determine the optimal parameters characterizing the projection functions.

We first analyze the shapes of functions $h_{j:n}(x)$, $x<0$ in the lemma below.

Lemma 1

Consider $h_{j:n}(x)$, $x<0$ given by (). If $j=1$, then $h_{1:n}$ is decreasing. If $j=n$, then $h_{n:n}$ is increasing and strictly convex. If $j=n-1$, then $h_{n-1:n}$ is convex increasing, concave increasing, and concave decreasing. For every $j=2, \ldots , n-2$, the function $h_{j:n}$ is strictly convex and increasing, strictly concave and increasing, strictly concave and decreasing and finally strictly convex and decreasing.

For the first step of the procedure of establishing the shape of the projection we do not need precise forms of projected functions, but only their basic shape properties. We also admit more general weight function than the exponential one. To this end we assume the following.

$(\breve{\textbf{A}})$:

Suppose that for some $a= - \infty<b <c \le d \le +\infty$, a function h is nonnegative, bounded and twice differentiable on (a, d), satisfies $\lim _{x \rightarrow a} h(x)=0$, and h is strictly increasing and strictly convex on (a, b), strictly increasing and concave on (b, c), and strictly decreasing on (c, d). Moreover, we assume that for a probability distribution function W supported on (a, d) with a positive density function w there we have $\int _a^d h(x)w(x)\,dx =1$ and $\int _a^d h^2(x)w(x)\,dx <+\infty$.

Note that $h(x) = h_{j:n}(x) = f_{j:n}(e^x)$, $j=2,\ldots ,n-1$, and $W(x)=V(x)= e^x$, $x<0$, satisfy $(\breve{\textbf{A}})$ with $a=-\infty$, $b=x_1(j,n)$, $c=x_0(j,n)$ and $d=0$. The precise form of the projection $\breve{P}_{W}h$ of h satisfying conditions $(\breve{\textbf{A}})$ on

$$\begin{aligned} \breve{\mathcal {C}}_{W} = \{ g \in L^2((-\infty , d), w(x)\,dx): g\; \textrm{nondecreasing, convex} \}, \end{aligned}$$

(2.12)

is presented in Theorem 1 below.

Theorem 1

For h and W satisfying the assumptions $(\breve{\textbf{A}})$ we define the functions

$$\begin{aligned} \breve{\lambda }(\beta )= & {} \frac{\int \limits ^d_\beta (x-\beta )[h(x) - h(\beta )]w(x)dx}{\int \limits ^d_\beta (x-\beta )^2w(x)dx}\,\\ \breve{K}(\beta )= & {} \breve{\lambda }(\beta ) - h'(\beta ), \\ \breve{L}(\beta )= & {} \int ^d_\beta h(x)w(x)dx - \int ^d_\beta [\breve{\lambda }(\beta )(x-\beta ) + h(\beta )]w(x)dx, \end{aligned}$$

for $-\infty \le \beta \le b$, and the set

$$\begin{aligned} \breve{\mathcal {K}}=\{-\infty < \beta \le b: \breve{K}(\beta ) \ge 0 \text{ and } \breve{L}(\beta ) = 0\}. \end{aligned}$$

If $\breve{\mathcal {K}}$ is nonempty and $\beta _* = \sup \breve{\mathcal {K}}$, then

$$\begin{aligned} \breve{P}_{W}h(x) = \left\{ \begin{array}{ll} h(x), &{} a=-\infty \le x \le \beta _*,\\ \breve{\lambda }(\beta _*)(x-\beta _*) + h(\beta _*), &{} \beta _* \le x < d. \end{array} \right. \end{aligned}$$

(2.13)

Otherwise, if $\breve{\mathcal {K}} = \emptyset$ then

$$\begin{aligned} \breve{P}_{W}h(x) = 1. \end{aligned}$$

Now we present tools allowing us to calculate the bounds on the expectations of order statistics based on samples with IRFR marginal distributions. We first modify the assumptions $(\breve{\textbf{A}})$ in order to apply Lemma 1 of Goroncy and Rychlik (2016).

Let h be a bounded, twice differentiable function on $(-\infty ,d)$ such that

$$\begin{aligned} \int \limits _{-\infty }^dh(x)w(x)dx =0, \quad \int \limits _{-\infty }^dh^2(x)w(x)dx < \infty . \end{aligned}$$

(2.14)

We further assume that h is strictly decreasing on $(-\infty ,a)$, strictly convex increasing on (a, b), strictly concave increasing on (b, c) with $h(c)> 0 \ge h(-\infty )$, and strictly decreasing on (c, d) with $h(d) =h(-\infty )$ for some $-\infty \le a<b<c< d$.

The assumptions are satisfied by the functions $f_{j:n}(e^x)-1$, $j=2,\ldots ,n-1$. In Goroncy and Rychlik (2016), they were formulated so that they could be applied for evaluations of spacings of order statistics as well. The main difference between our assumptions and those of Goroncy and Rychlik (2016) is that the left end point of the domain of h being equal to 0 there was replaced by $-\infty$. Accordingly, the forms of projections onto

(2.15)

are presented in Theorem 2 below. To this aim we consider auxiliary functions

(2.16)

(2.17)

(2.18)

(2.19)

defined on $(-\infty ,d)$.

Theorem 2

Assume that the zero $-\infty< \beta _* <c$ of () belongs to (b, c), the set is nonempty, and $\alpha _*=\inf \{\alpha \in \mathcal {Y}\}$. Then

(2.20)

is the projection of h on (). Otherwise we define

$$\begin{aligned} P_{\alpha }h(x)= \frac{\int _\alpha ^d h(y)w(y)dy}{1-W(\alpha )} \left[ \frac{ (x-\alpha ) \textbf{1}_{( -\infty ,\alpha )}(x)}{-\int _{-\infty }^\alpha (y-\alpha )w(y)dy} +1 \right] , \quad \beta _* \le \alpha < d, \end{aligned}$$

(2.21)

with

$$ ||P_{\alpha }h||^2 = \frac{\left( \int _\alpha ^d h(x)w(x)dx\right) ^2 \left[ \int \limits _{-\infty }^\alpha (x-\alpha )^2w(x)dx-\left( \int \limits _{-\infty }^\alpha (x-\alpha ) w(x)dx\right) ^2 \right] }{(1-W(\alpha ))^2\left[ \int \limits _{-\infty }^\alpha (x-\alpha )w(x)dx\right] ^2}. $$

Let $\mathcal {Z}$ denote the set of arguments $\alpha \ge \beta _*$ satisfying

$$\begin{aligned} \frac{\int \limits _\alpha ^d h(x)w(x)dx}{1-W(\alpha )} = - \frac{\int \limits _{-\infty }^\alpha (x-\alpha )h(x)w(x)dx \int \limits _{-\infty }^\alpha (x-\alpha )w(x)dx}{\int \limits _{-\infty }^\alpha (x-\alpha )^2w(x)dx-\left( \int \limits _{-\infty }^\alpha (x-\alpha )w(x)dx\right) ^2} >0. \end{aligned}$$

Then $\mathcal {Z}$ is nonempty, and for unique $\alpha _* = \arg \max _{\alpha \in \mathcal {Z}} ||P_{\alpha }h||^2$.

In our detailed considerations we also use the variation diminishing property (VDP, for short) of the Bernstein polynomials () (cf., e.g., Rychlik 2001, Lemma 14 and Schoenberg 1959). Note that the VDP holds also for compositions $B_{i,m}\circ F$ of the Bernstein polynomials with any nondecreasing function F with values in [0, 1], e.g., a distribution function.

3 Decreasing reversed failure rate

We start with considering the sample maxima from DRFR samples, and recall the classic result of Gumbel (1954) and Hartley and David (1954). Applying the Schwarz inequality they concluded that the following bound

$$\begin{aligned} \mathbb {E}\frac{X_{n:n}-\mu }{\sigma }\le & {} \left[ \int _{0}^{1}\frac{[F^{-1}(x)-\mu ]^2}{\sigma ^2} \,dx \right] ^\frac{1}{2}\left[ \int _{0}^{1} [nx^{n-1}-1]^2 \,dx \right] ^\frac{1}{2} = \frac{n-1}{\sqrt{2n-1}},\nonumber \\ \end{aligned}$$

(3.1)

is valid for all iid samples with arbitrary parent distribution function F, mean $\mu$ and variance $0< \sigma ^2 <\infty$. Moreover, the equality above is attained for the following location-scale family of parent distribution functions

$$\begin{aligned} F(x){} & {} =\left[ \frac{1}{n}\left( \frac{n-1}{\sqrt{2n-1}}\frac{x-\mu }{\sigma }+1\right) \right] ^\frac{1}{n-1}, \qquad \mu - \sigma \frac{\sqrt{2n-1}}{n-1} \le x \le \mu \nonumber \\{} & {} \quad + \sigma \sqrt{2n-1}. \end{aligned}$$

(3.2)

Note that these have the decreasing density functions

$$\begin{aligned} f(x)=\frac{1}{n\sqrt{2n-1}\sigma }\left[ \frac{1}{n} \left( \frac{n-1}{\sqrt{2n-1}}\frac{x-\mu }{\sigma }+1\right) \right] ^{-\frac{n-2}{n-1}} \end{aligned}$$

on their supports. Since each distribution with a decreasing density function is a member of the DRFR family, we conclude that the bound () is optimal for the decreasing reversed failure rate distributions, and it becomes equality for the DRFR distribution functions (). Therefore the case $j=n$ is excluded from our further investigations.

Theorem 3

Consider the iid sequence $X_1,\ldots ,X_n$ of random variables based on the decreasing reversed failure rate distribution with the expectation $\mu$ and standard deviation $\sigma$, and let $X_{1:n},\ldots ,X_{n:n}$ be respective order statistics. For fixed $2 \le j \le n-1$ we define

$$\begin{aligned} \breve{\lambda }_{j:n}(\beta )= & {} \frac{1}{\beta ^2+2\beta -2e^\beta +2} \nonumber \\{} & {} \left[ f_{j:n}(e^\beta )(1-e^\beta +\beta )-\beta -\sum \limits _{i=j}^n\frac{1}{i}(1-F_{i:n}(e^\beta ))\right] , \end{aligned}$$

(3.3)

$$\begin{aligned} \breve{K}_{j:n}(\beta )= & {} j[f_{j+1:n}(e^\beta )-f_{j:n}(e^\beta )]\nonumber \\{} & {} +\frac{1}{\beta ^2 +2\beta -2e^\beta +2}\Bigg [(\beta ^2+3\beta -3e^\beta +3)f_{j:n}(e^\beta )\nonumber \\{} & {} -\beta -\sum \limits _{i=j}^n\frac{1}{i}(1-F_{i:n}(e^\beta ))\Bigg ], \end{aligned}$$

(3.4)

$$\begin{aligned} \breve{L}_{j:n}(\beta )= & {} 1-F_{j:n}(e^\beta )-f_{j:n}(e^\beta )(1-e^\beta )\nonumber \\{} & {} - \frac{e^\beta -\beta -1}{\beta ^2+2\beta -2e^\beta +2}\Bigg [(1-e^\beta +\beta )f_{j:n}(e^\beta )\nonumber \\{} & {} -\beta -\sum \limits _{i=j}^n\frac{1}{i}(1-F_{i:n}(e^\beta ))\Bigg ]. \end{aligned}$$

(3.5)

If either $j=1$ or $2 \le j\le n-1$ and the set

$$\begin{aligned} \breve{\mathcal {K}}_{j:n}=\{-\infty < \beta \le b=x_1(j,n): \breve{K}_{j:n}(\beta )\ge 0 \text{ and } \breve{L}_{j:n}(\beta ) = 0\} \end{aligned}$$

(3.6)

is empty, then we have

$$\begin{aligned} \mathbb {E}\frac{X_{j:n}-\mu }{\sigma }\le 0. \end{aligned}$$

The equality is attained in the limit by the family of baseline distributions being the mixtures of atoms at $\mu +\sigma (e^\beta -1-\beta )$ and linearly transformed left-truncated negative exponential distributions $\frac{1}{1-e^\beta } \exp \left( \frac{ \frac{x-\mu }{\sigma }-e^\beta +1+\beta }{\sqrt{1+2\beta e^\beta - e^{2\beta }}}+\beta \right)$, $\mu + \sigma (e^\beta -1-\beta )< x < \mu +\sigma \big ( e^\beta -1-\beta - \beta \sqrt{ 1+2\beta e^\beta - e^{2\beta }}\big )$ with respective probabilities $e^\beta$, and $1-e^\beta$, as $0> \beta \rightarrow 0$.

If the set () is nonempty and $\beta _* = \sup \breve{{\mathcal K}}_{j:n}$ then we have the following bound

$$\begin{aligned} \mathbb {E}\frac{X_{j:n}-\mu }{\sigma }\le \breve{B}= \breve{B}(j,n), \end{aligned}$$

(3.7)

where

$$\begin{aligned} \breve{B}^2= & {} 2n\frac{{2j-2\atopwithdelims ()j-1}{2n-2j\atopwithdelims ()n-j}}{{2n\atopwithdelims ()n}}F_{2j-1:2n-1}(e^{\beta _*}) \nonumber \\{} & {} + \frac{1}{\beta _*^2+2\beta _*-2\textrm{e}^{\beta _*}\!+2}\cdot \! \left[ \left[ (1-\textrm{e}^{\beta _*})^2-\beta _*^2\textrm{e}^{\beta _*}\right] \! f^2_{j:n}(\textrm{e}^{\beta _*})\right. \nonumber \\{} & {} \left. +\left( \beta +\sum \limits _{i=j}^n\frac{1}{i}(1-F_{i:n}(\textrm{e}^{\beta _*}))\right) ^2\right] -1. \end{aligned}$$

(3.8)

The equality in () holds for the absolutely continuous DRFR distribution function

$$\begin{aligned} F(x)=\left\{ \begin{array}{ll} f^{-1}_{j:n}(1+\frac{x-\mu }{\sigma }\breve{B}),&{} x<\mu +\frac{\sigma }{\breve{B}}[f_{j:n}(e^{\beta _*})-1],\\ &{}\\ \textrm{exp}\left\{ \frac{\frac{x-\mu }{\sigma }\breve{B}+1-f_{j:n}(e^{\beta _*})}{\breve{\lambda }_{h_{j:n},V}(\beta _*)} +\beta _*\right\} ,&{}\mu +\frac{\sigma }{\breve{B}}[f_{j:n}(e^{\beta _*})-1]\le x<\\ &{}<\mu +\frac{\sigma }{\breve{B}}[f_{j:n}(e^{\beta _*})-\breve{\lambda }_{h_{j:n},V}(\beta _*)\beta _*-1],\\ &{}\\ 1,&{}x\ge \mu +\frac{\sigma }{\breve{B}}[f_{j:n}(e^{\beta _*})-\breve{\lambda }_{h_{j:n},V}(\beta _*)\beta _*-1], \end{array} \right. \end{aligned}$$

(3.9)

Although it does not directly follows from the statement of Theorem 3, the set () is empty for small values of j, and then we obtain the zero bound, whereas it is nonempty for large j, and then we obtain a strictly positive bound defined in (). We do not have precise descriptions of $j=j(n)$ at which the values of bounds switch from 0 to positive bounds. We can only say that such indices satisfy $j(n) \le \frac{n+1}{2}$. This follows from the fact that the bounds in the DD case amount to 0 for $j \le \frac{n+1}{2}$, and they are positive otherwise (see, e.g., Rychlik 2009b). Since the DRFR family is wider than DD, the bounds for the expectations of given order statistics from decreasing reversed failure rate distributions cannot be less than the corresponding bounds in the decreasing density case.

We also note that the bounds for the extreme order statistics can be determined with use of the projection method as well. For $j=1$ the function () is decreasing. It is easy to verify that among the nondecreasing functions the one lying closest to $h_{1:n}$ is the constant function equal to 1. This is obviously the member of the family of nondecreasing convex functions (). Applying () we immediately check that the bound is equal to 0. For $j=n$ () is belongs to (), and so $\breve{P}_{V}h_{n:n}= h_{n:n}$ It is easy to check that

$$ \Vert \breve{P}_{V}h_{n:n}\Vert ^2_V= \Vert h_{n:n}\Vert ^2_V = n^2 \int _0^1 x^{2n-2}dx = \frac{n^2}{2n-1}, $$

which in view of () gives the bound of ().

4 Increasing reversed failure rate

In the IRFR case we take

(4.1)

The counterparts of ()–() for are given by

(4.2)

(4.3)

(4.4)

(4.5)

The variation diminishing property of Bernstein polynomials allows us to deduce the following properties of the above functions. The first one () is first negative and then positive. Its zero is necessarily less than $x_0$. For $\beta > x_0$ it has to be positive, because it represents the difference of $f_{j:n}$ at $e^\beta$ and its integral over $(e^\beta ,1)$. Then the function is decreasing over the interval, and the integral is less than the value of function at the left end-point of the integral. Function () is first positive and then negative, whereas () is either negative on the whole $\mathbb {R}_-$ or it has one interval of positivity separated from $-\infty$ and 0. The last one () has two zeros at most. If j is relatively small with respect to n so that $\sum _{r=j}^{n} \frac{1}{r} >2$, then it is first negative and then positive. Otherwise it has either 0 or 2 zeros.

Theorem 4

We consider iid random variables $X_1,\ldots , X_n$ with an IRFR marginal distribution function, and $2 \le j \le n-1$. Suppose that the zero $-\infty< \beta _* <x_0$ of () belongs to $(x_1,x_0)$, and there exists a unique $x_1< \alpha _* < \beta _*$ satisfying and . Then

(4.6)

where

(4.7)

The equality in () holds for the following distribution function

(4.8)

Otherwise, if the above conditions are not satisfied, the set of points $\beta _* \le \alpha <0$ being the solutions to the equation

$$\begin{aligned} \textrm{e}^\alpha -F_{j:n}(\textrm{e}^\alpha )=\frac{1-\textrm{e}^\alpha }{2-\textrm{e}^\alpha }\left( \textrm{e}^\alpha - \sum \limits _{i=j}^n\frac{1}{i}F_{i:n}(\textrm{e}^\alpha )\right) , \end{aligned}$$

say $\mathcal {Z}$, is nonempty, and for , with

(4.9)

we have the following bound

(4.10)

The equality above is attained for the distribution function given by

$$\begin{aligned} F(x)=\left\{ \begin{array}{ll} \exp \left\{ \left( \frac{x-\mu }{\sigma }\sqrt{\frac{(1-\textrm{e}^{\alpha _*})(2-\textrm{e}^{\alpha _*})}{\textrm{e}^{\alpha _*} -F_{j:n}(\textrm{e}^{\alpha _*})}}-1\right) \textrm{e}^{\alpha _*}+\alpha _*\right\} ,&{} x< \mu +\sigma \sqrt{\frac{\textrm{e}^{\alpha _*}-F_{j:n}(\textrm{e}^{\alpha _*})}{(1-\textrm{e}^{\alpha _*})(2-\textrm{e}^{\alpha _*})}},\\ &{}\\ 1,&{} x\ge \mu +\sigma \sqrt{\frac{\textrm{e}^{\alpha _*}-F_{j:n}(\textrm{e}^{\alpha _*})}{(1-\textrm{e}^{\alpha _*})(2-\textrm{e}^{\alpha _*})}}. \end{array} \right. \end{aligned}$$

(4.11)

Formula () represents a linearly transformed negative exponential distribution. Distribution function () has a jump of size $1- e^{\beta _*}$ at the right end-point of its support. Note that both () and () have interval supports unbounded on the left. This is a common property of all IRFR distributions (see Block et al. 1998).

We applied the above result for numerical evaluations of expectations of order statistics for various sample sizes n and order statistics ranks $2 \le j \le n-1$. In our numerical calculations it neither happened that conditions of the first part of Theorem 4 were satisfied, and that any bounds were calculated with use of (). The reason was that the intervals of concave increase of belong to very short left neighborhood of the function maximum, right to $\beta _*$. However, we could not prove formally that these conditions do not hold for any j and n. Therefore for every pair j and n we should first check validity of the assumptions allowing use of bound (). If they do not hold, then we apply the ().

In the remaining cases of extreme order statistics the bounds and their proofs are substantially simpler.

Theorem 5

If F has the increasing reversed failure rate, then we have the following optimal bound for the maximum of an iid sample

$$\begin{aligned} \mathbb {E}\frac{X_{n:n}-\mu }{\sigma }\le 1-\frac{1}{n}. \end{aligned}$$

(4.12)

The equality is attained for the location-scale family of negative exponential distributions

$$\begin{aligned} F(x)=\left\{ \begin{array}{ll} \exp \left( \frac{x-\mu }{\sigma }-1\right) ,&{}x<\mu +\sigma ,\\ 1,&{}x\ge \mu +\sigma . \end{array} \right. \end{aligned}$$

(4.13)

In the case of sample minimum we have the trivial bound

$$\begin{aligned} \mathbb {E}\frac{X_{1:n}-\mu }{\sigma }\le 0, \end{aligned}$$

which is attained in the limit e.g., by the family of right truncated at $\mu + \frac{\sigma }{\sqrt{2e^{-\alpha }-1}}$ negative exponential distributions with location $\mu + \frac{\sqrt{2 e^{-\alpha }-1}}{\sigma e^\alpha }+ e^{-\alpha } -\alpha$ and scale $\frac{\sigma e^\alpha }{\sqrt{2 e^{-\alpha }-1}}$, $\alpha <0$, as $\alpha \rightarrow - \infty$.

Theorem 5 shows that the bounds for the sample maxima in the IRFR case are less than 1, but they approach 1 if the sample size increases. It follows that all the bounds of Theorem 4 do not exceed 1 as well.

5 Numerical results

In Tables 1 and 2 we present numerical values for the mean-standard deviation bounds on the expectations of order statistics with ranks $2 \le j \le n$ coming from samples of sizes 10, and 20, respectively, based on iid random variables with monotone reversed failure rates. Columns 2 and 3 are connected with the DRFR families, and columns 4 and 5 correspond to the IRFR case. The bounds are presented in columns 3 and 5, respectively, and they are based on formulae () and (), respectively, except for the sample maxima, where () and () are used. Note that formula () is nowhere used. Columns 2 and 4 contain a partial information about the marginal distributions which attain the respective bounds. The values $e^{\beta _*}$ in the second columns represent the contribution of the part of the optimal distribution function with the shape of the inverse of the density function $f_{j:n}$ of the corresponding order statistic in the standard uniform case (see ()). Here $1- e^{\beta _*}$ is the mass of the linearly transformed left-truncated negative exponential distribution. The values $e^{\alpha _*}$ of columns 4 provide the probability masses of the negative exponential part of the distribution attaining the bound (see ()), whereas $1- e^{\alpha _*}$ is the value of atom located at the right end-point of the support of the distribution. For $j=1$ the bounds in both the DRFR and IRFR cases are equal to 0, and they are not placed in the tables.

Table 1 Upper bounds on the expectations of standardized order statistics $\mathbb {E}\frac{X_{j:n}-\mu }{\sigma }$ based on samples of size $n=10$ from iid populations with monotone reversed failure rate distributions

Table 2 Upper bounds on the expectations of standardized order statistics $\mathbb {E}\frac{X_{j:n}-\mu }{\sigma }$ based on samples of size $n=20$ from iid populations with monotone reversed failure rate distributions

The bounds for the DRFR and IRFR families increase with j (which is obvious), but those for DRFR change faster than in IRFR case. In the former case for a significant proportion of j’s less than $\frac{1}{2}$ the bounds are equal to 0., and then they increase relatively fast. For the sample maximum, the DRFR bound $\frac{n-1}{\sqrt{2n-1}}$ coincides with one valid for the iid samples with an arbitrary parent distribution. As n increases, they tend to infinity at the rate $\sqrt{\frac{n}{2}}$. In the IRFR case all the bounds except for the sample minimum are positive, but they increase much slower with j. The maximal value $1 - \frac{1}{n}$ attained by the sample maxima does not exceed 1. Our numerical examples show that the bounds in the DRFR and IRFR cases are approximately equal when j is close to $\frac{2}{3} n$.

We finally note that due to relations DD$\subset$DRFR and IRFR$\subset$ID the bounds for the DRFR populations are greater than for the populations with decreasing density functions (cf. Danielak 2003) and the reversed relations concern the bounds in the IRFR and ID cases (cf. Goroncy and Rychlik 2016).

Appendix

Proof of Lemma 1

It is easy to check that $h_{1:n}(x) = n(1-e^x)^{n-1}$ is decreasing, and $h_{n:n}(x) = n e^{(n-1)x}$ is increasing and strictly convex. This information is sufficient for our purposes. The remaining cases need more elaborate analysis. Since $f_{j:n}$, $j=2,\ldots , n-1$, are increasing on $\left( 0, \frac{j-1}{n-1} \right)$, and decreasing on $\left( \frac{j-1}{n-1}, 1\right)$, then $h_{j:n}(x) = f_{j:n}(e^x)$ are increasing-decreasing as well, and they are maximized at $x_0= x_0(j,n) = \ln \frac{j-1}{n-1}$. They vanish at $-\infty$ and 0. The second derivatives are given by

$$\begin{aligned} h''_{j:n}(x) = n[(j-1)^2 B_{j-1,n-1}(e^x)-j(2j-1) B_{j,n-1}(e^x) +j(j+1)B_{j+1,n-1}(e^x)]. \end{aligned}$$

(A.1)

If $j=n-1$, then the last summand disappears. It follows that the second derivative is first positive, it changes the sign at $x_1=x_1(n-1,n) = 2 \ln \frac{n-2}{n-1} < x_0(n-1,n) = \ln \frac{n-2}{n-1}$, and it is ultimately negative. This means that $h_{n-1:n}$ is convex increasing on $(-\infty , x_1)$ concave increasing on $(x_1,x_0)$, and concave decreasing on $(x_0,0)$.

Once for $j=2, \ldots , n-2$ we change the variable taking $y= e^x \in (0,1)$, we modify () to the form

$$\begin{aligned} \tilde{h}''_{j:n}(y) = c(y) [ (n-1)^2y^2 -(2nj-n-3j+2)y + (j-1)^2 ], \end{aligned}$$

where c(y) is a strictly positive function. Since the quadratic function in the square brackets has a positive determinant $\Delta = (n-j)^2 + 4 (j-1)(n-1) (n-j)$, and negative $-2(j-1)(n-1) -(n-j)$ and positive $(2n-1)(n-j)$ values of the derivative at 0 and 1, respectively, it has two zeros in (0, 1). They amount to

$$\begin{aligned} y_1(j,n)= & {} \frac{j-1}{n-1} + \frac{n-j-\sqrt{(n-j)^2 + 4 (j-1)(n-1) (n-j)}}{2(n-1)^2},\\ y_2(j,n)= & {} \frac{j-1}{n-1} + \frac{n-j+\sqrt{(n-j)^2 + 4 (j-1)(n-1) (n-j)}}{2(n-1)^2}. \end{aligned}$$

The above representations easily show that $y_1(j,n)< \frac{j-1}{n-1} < y_2(j,n)$. This implies that for $x_i=x_i(j,n) = \ln y_i(j,n)$, $i=1,2$, we have the following. For every $j=2, \ldots , n-2$, the function $h_{j:n}(x)$, $x<0$, is strictly convex and increasing on $(-\infty , x_1)$, strictly concave and increasing on $(x_1,x_0)$, strictly concave and decreasing on $(x_0,x_2)$ and finally strictly convex and decreasing on $(x_2,0)$. In fact, the relations $x_1< x_0 < x_2$ can be also deduced from a simple observation that every local maximum of a smooth function lies in an interval of its concavity. $\square$

Sketch of the proof of Theorem 1

The proof is analogous to the proof of the first statement of Proposition 1 in Danielak (2003). The second statement of this proposition describes the form of linear increasing projection which does not need to be considered here. Therefore, the first step in proving Theorem 1 consists of determining the possible shapes of projection of functions h satisfying assumptions $(\breve{\textbf{A}})$. Let $\breve{\mathcal {C}}^*_{W}\subset \breve{\mathcal {C}}_{W}$ with () be the class of functions such that either

$$\begin{aligned} g^*(x)=\left\{ \begin{array}{ll} h(x),&{} x\le \beta ,\\ h(\beta )+\alpha (x-\beta ),&{} x>\beta , \end{array} \right. \end{aligned}$$

for some $-\infty <\beta \le b$ and $\alpha \ge h'(\beta )$, or $g^*(x)=C$, for some $C>0$. Then for any $g\in \breve{\mathcal {C}}_{W}$ there exists a function $g^*\in \breve{\mathcal {C}}^*_{W}$ such that

$$\begin{aligned} ||h-g||_W \ge ||h-g^*||_W. \end{aligned}$$

The shape of $g^*$ is similar to the one of Lemma 3 of Danielak (2003). The main difference is that she assumed that the left end-point a of the joint domain of h and W is finite, whereas we take $a=-\infty$. The conclusion of Lemma 3 in Danielak (2003) admits that $\breve{\mathcal {C}}^*_{W}$ contains linear functions with positive slopes as well. These are excluded in our case, because every such function $\ell$, say, defined on $(-\infty , d)$ takes on negative values on the left. However, $\ell _+= \max \{\ell , 0 \} \in \breve{\mathcal {C}}_{W}$ and lies closer to nonnegative h than $\ell$.

Having the shapes $g^*$ of projections of h satisfying assumptions $(\breve{\textbf{A}})$, we observe that they depend on some parameters $\alpha , \beta$ and C. Hence in the next stage it is enough to determine the optimal parameters using analytical methods, and this implies the results of the theorem. $\square$

Sketch of the proof of Theorem 2

The procedure is analogous to the one described in the sketch of the proof of Theorem 1. First we determine the possible shapes of projection of h satisfying onto the convex cone () (in fact, it coincides with the projection onto (), because () holds). Let with () be the class of functions described by either of the formulae

$$\begin{aligned} g_*(x)=\left\{ \begin{array}{ll} \lambda (x-\alpha )+h(\alpha ),&{}-\infty \le x<\alpha ,\\ h(x),&{}\alpha \le x<\beta ,\\ h(\beta ),&{}\beta \le x<d, \end{array} \right. \end{aligned}$$

for $b\le \alpha <\beta \le c$, $\lambda \ge h'(\alpha )$, or

$$\begin{aligned} g_*(x)=\left\{ \begin{array}{ll} \lambda (x-\alpha )+\gamma ,&{} -\infty \le x<\alpha ,\\ \gamma ,&{}\alpha \le x<d, \end{array} \right. \end{aligned}$$

for $\lambda > 0$ and $0< \alpha <d$. Then for every function there exists a function such that

$$\begin{aligned} ||h-g_*||_W\le ||h-g||_W. \end{aligned}$$

Functions $g_*$ presented above are similar to those in Lemma 1 of Goroncy and Rychlik (2016). The main difference is that 0 appearing there is replaced by $-\infty$. We traced all the steps of the rather long proof of former lemma and verified that the change of the left end-point does not disturb arguments nor the resulting claim. Since the shapes $g_*$ of projections of h satisfying assumptions depend on some $\lambda , \alpha ,\beta , \gamma$, we further determine the optimal parameters using analytical methods. As a result, we obtain the thesis of Theorem 2. $\square$

Proof of Theorem 3

We start from a trivial observation that $\mathbb {E}X_{1:n} \le \mu =\mathbb {E}X_1$, $n \ge 2$, with the equality for the degenerate parent distribution. For $2 \le j \le n-1$ we combine formulae () and () with the statement of Theorem 1 for fixed $h(x) = h_{j:n}(x) = f_{j:n}(e^x)$ and $W(x)= V(x)= e^x$, $x<0$.

After some tedious calculations we obtain

$$\begin{aligned} \int \limits _\beta ^0(x-\beta )[h_{j:n}(x)-h_{j:n}(\beta )]e^x\,dx = f_{j:n}(\textrm{e}^\beta )(1-e^\beta +\beta )- \beta -\sum \limits _{i=j}^n\frac{1}{i}(1-F_{i:n}(e^\beta )), \end{aligned}$$

which provides (). Subtracting $h'_{j:n}(x)$ from () we obtain ().

Referring to Theorem 1 we conclude that if () is not empty, then $\breve{P}_{V}h_{j:n}$ has the form () with specially chosen $h=h_{j:n}$ and $W=V$, $\breve{\lambda }_*=\breve{\lambda }_{j:n}$ defined in () and $\beta _*$ maximizing (). Its squared norm amounts to

$$\begin{aligned} \Vert \breve{P}_{V}h_{j:n}\Vert _V^2= & {} \int _{-\infty }^{0} [\breve{P}_{V}h_{j:n}(x)]^2e^x\,dx \\= & {} 2n\frac{{2j-2\atopwithdelims ()j-1}{2n-2j\atopwithdelims ()n-j}}{{2n\atopwithdelims ()n}}F_{2j-1:2n-1}(e^{\beta _*}) +\breve{\lambda }_{h_{j:n},V}^2(\beta _*)(\beta _*^2+2\beta _*-2e^{\beta _*}+2)\\{} & {} +\, 2\breve{\lambda }_{h_{j:n},V}(\beta _*)f_{j:n}(e^{\beta _*})(e^{\beta _*}\!-\!1\!-\!\beta _*)+f_{j:n}^2(e^{\beta _*})(1 -e^{\beta _*})= \breve{B}^2+1. \end{aligned}$$

(cf.()). Due to (), the inequality () with B defined in () holds. The equality in () occurs when () is satisfied. The respective distribution function has the explicit form ().

Suppose that $2 \le j \le n-1$, and () is an empty set. Then the projection $\breve{P}_{V}h_{j:n}$ is a constant function equal to 1. By (), the bound for $\mathbb {E} \frac{X_{j:n}-\mu }{\sigma }$ is equal to 0. We prove that this bound is the best possible by constructing a family of parent DRFR distribution functions $\{ F_\beta \}_{\beta <0}$ such that for $\beta \rightarrow 0$ the expressions $\mathbb {E}_\beta \frac{X_{j:n}-\mu }{\sigma }$ tend to 0 as well. The construction shall give the same conclusion for $j=1$. Consider the family of distribution functions

$$\begin{aligned} F_\beta (x) = \left\{ \begin{array}{ll} 0, &{} x<0, \\ e^{x+\beta }, &{} 0 \le x \le - \beta , \\ 1, &{} x \ge -\beta , \end{array} \right. \qquad \beta <0. \end{aligned}$$

(A.2)

They have quantile functions

$$\begin{aligned} F^{-1}_\beta (x) = \left\{ \begin{array}{ll} 0, &{} 0< x \le e^\beta , \\ \ln x - \beta , &{} e^\beta \le x<1, \\ \end{array} \right. \qquad \beta <0, \end{aligned}$$

satisfying

$$\begin{aligned} F^{-1}_\beta (e^x) = \left\{ \begin{array}{ll} 0, &{} -\infty< x \le \beta , \\ x - \beta , &{} \beta \le x<0, \\ \end{array} \right. \qquad \beta <0. \end{aligned}$$

(A.3)

Convexity of () implies that all () belong to the DRFR family.

We have

$$\begin{aligned} \mathbb {E}_\beta X_1= & {} \int _{-\infty }^{0} F^{-1}_\beta (e^x) e^x \,dx = \int _{\beta }^{0}(x-\beta )e^x\,dx = e^\beta -1- \beta , \\ \mathbb {E}_\beta X_1^2= & {} \int _{-\infty }^{0} [F^{-1}_\beta (e^x)]^2 e^x\, dx = \int _{\beta }^{0}(x-\beta )^2e^x\,dx = \beta ^2 +2 \beta +2 - 2 e^\beta . \end{aligned}$$

Simple calculations show that

$$\begin{aligned} \mathbb {V}ar_\beta \,X_1= 1+2\beta e^\beta - e^{2\beta }. \end{aligned}$$

Furthermore, we have

$$\begin{aligned} \mathbb {E}_\beta X_{j:n}= & {} \int _{-\infty }^{0} F^{-1}_\beta (e^x) f_{j:n}(e^x) e^x dx =- \beta - \sum _{i=j}^{n} \frac{1}{i} [1-F_{i:n}(e^\beta )], \end{aligned}$$

and we obtain

$$\begin{aligned} \mathbb {E}_\beta \frac{X_{j:n}-\mu }{\sigma } = \frac{1-e^\beta - \sum _{i=j}^{n} \frac{1}{i} [1-F_{i:n}(e^\beta )] }{\sqrt{ 1+2\beta e^\beta - e^{2\beta }}}. \end{aligned}$$

We show that

$$\begin{aligned} \lim _{\beta \rightarrow 0}\mathbb {E}_\beta \frac{X_{j:n}-\mu }{\sigma }= & {} \lim _{\beta \rightarrow 0} \frac{1-e^\beta - \sum _{i=j}^{n} \frac{1}{i} [1-F_{i:n}(e^x)]}{(1-e^\beta )^2}\, \frac{(1-e^\beta )^2}{\sqrt{ 1+2\beta e^\beta - e^{2\beta }}} \nonumber \\= & {} \lim _{\beta \rightarrow 0} Q(e^\beta ) R(\beta ), \end{aligned}$$

(A.4)

say, is equal to 0. Applying the de l’Hospital rule twice we calculate

$$\begin{aligned} \lim _{\beta \rightarrow 0} Q(e^\beta )&= \lim _{x \rightarrow 1}Q(x) = \lim _{x \rightarrow 1}\frac{1-x - \sum _{i=j}^{n} \frac{1}{i} [1-F_{i:n}(x)]}{(1-x)^2} \\&= \lim _{x \rightarrow 1} \frac{\sum _{i=j}^n \frac{1}{i} f'_{j:n}(x)}{2} = - \frac{1}{2}, \end{aligned}$$

because

$$\begin{aligned} f'_{i:n}(1) = \left\{ \begin{array}{ll} 0, &{} i=1, \ldots ,n-2, \\ -n(n-1), &{} i=n-1, \\ n(n-1), &{} i=n. \end{array} \right. \end{aligned}$$

Triple application of the de l’Hospital rule gives

$$\begin{aligned} \lim _{\beta \rightarrow 0} R^2(\beta )= & {} \lim _{\beta \rightarrow 0} \frac{(1-e^\beta )^4}{ 1+2\beta e^\beta - e^{2\beta }} =\lim _{\beta \rightarrow 0} \frac{-4(1-e^\beta )e^\beta (16e^{2\beta }-11e^{\beta } + 1)}{2\beta e^\beta + 6e^\beta - 8 e^{2\beta }} =0. \end{aligned}$$

This proves that () actually amounts to 0. To get the same conclusions for the parent DRFR distributions with arbitrarily selected $\mu$ and $\sigma$, it suffices to transform the family () as follows

$$\begin{aligned} G_\beta (x)= & {} F_\beta \left( \frac{\frac{x-\mu }{\sigma }-e^\beta +1+\beta }{\sqrt{1+2\beta e^\beta -e^{2\beta }}}\right) \\= & {} \left\{ \begin{array}{ll} 0, &{} x < \mu + \sigma (e^\beta -1-\beta ), \\ \exp \left( \frac{ \frac{x-\mu }{\sigma }-e^\beta +1+\beta }{\sqrt{1+2\beta e^\beta - e^{2\beta }}}+\beta \right) , &{} \mu + \sigma (e^\beta -1-\beta ) \le x \\ &{} \le \mu +\sigma \big ( e^\beta -1-\beta - \beta \sqrt{ 1+2\beta e^\beta - e^{2\beta }}\big ),\\ 1, &{} x \ge \mu +\sigma \big ( e^\beta -1-\beta - \beta \sqrt{ 1+2\beta e^\beta - e^{2\beta }}\big ), \end{array} \right. \end{aligned}$$

which represents the mixture distribution described in the statement of Theorem 3. This completes the proof. $\square$

Proof of Theorem 4

In the IRFR case, mimicking () we obtain

(A.5)

where denotes the projection of onto the cones (2.3) and (2.11) (cf. (2.7) and Lemma 1 of Danielak (2003) mentioned above). Therefore we apply the results of Theorem 2 in the particular case , $j=2,\ldots ,n-1$, for calculating the bounds on the expectations of order statistics coming from IRFR family.

Assume first that satisfies the conditions of the first statement of Theorem 2, i.e., $\beta _* > x_1$ and is nonempty. In fact, it consists of only one point. It follows from the facts that the second derivative of the squared distance with respect to $\alpha$ is positive (cf. Gajek and Rychlik 1998, Lemma 4(b)). This means that every subinterval of $\mathcal {Y}$ contains only one local minimum of the distance. Since the set where is positive consists only of a single interval (if any), $\alpha _*$ is determined uniquely.

The particular form of the projection () in our case is represented as

(A.6)

Since

$$\begin{aligned} \int \limits _\alpha ^\beta f^2_{j:n}(\textrm{e}^x)\textrm{e}^xdx= & {} \int \limits _{\textrm{e}^\alpha }^{\textrm{e}^\beta } n^2 \left( {\begin{array}{c}n-1\\ j-1\end{array}}\right) ^2x^{2j-2}(1-x)^{2n-2j}dx\\= & {} \frac{n^2\left( {\begin{array}{c}n-1\\ j-1\end{array}}\right) ^2}{(2n-1)\left( {\begin{array}{c}2n-2\\ 2j-2\end{array}}\right) }\int \limits _{\textrm{e}_\alpha }^{\textrm{e}^\beta } f_{2j-1:2n-1}(x)dx\\= & {} n\frac{\left( {\begin{array}{c}2j-2\\ j-1\end{array}}\right) \left( {\begin{array}{c}2n-2j\\ n-j\end{array}}\right) }{\left( {\begin{array}{c}2n-1\\ n\end{array}}\right) }\left[ F_{2j-1:2n-1}(\textrm{e}^\beta )- F_{2j-1:2n-1}(\textrm{e}^\alpha )\right] , \end{aligned}$$

we obtain the squared norm of (), which is equivalent to

After plugging () into the above formula we obtain ().

Otherwise, if the above conditions do not hold the projection of has a simpler form

(A.7)

Its squared norm equals to (). The projection of has the greatest value of the norm among all functions of the form (). Therefore the maximal norm coincides with the desired bound.

The equality condition for both inequalities () and () follow from the equality condition analogous to (), that is

(A.8)

which applied to appropriately adjusted (), () and (), () results in distribution functions given by () and (), respectively. $\square$

Proof of Theorem 5

For $j=n$ we already noticed that the function $h_{n:n}(x)= f_{j:n}(e^x)$ is strictly increasing and convex on $\mathbb {R}_-$, and so is the function . This means that the projection of () onto () is a linear and increasing function, say for some $a>0$ and $b \in \mathbb {R}$. Indeed, the graphs of every pair of strictly increasing convex function and nondecreasing concave functions with the same integrals over $\mathbb {R}_-$ with the weight $e^x$ have exactly two common points. The increasing linear function passing through these points runs between the concave and concave function and it is a better approximation of the convex function than the original concave one. The parameters a, b of the optimal linear approximation can be derived by minimizing the following squared distance $\int _{-\infty }^{0}[n e^{(n-1)x}- 1 - ax - b]^2 e^x\,dx =\frac{n^2}{2n-1} +2a^2 +c^2 -2ac + \frac{2a}{n} -2c.$ This is a quadratic convex function in each coordinate, and it is maximized by the pair $a= 1- \frac{1}{n}>0$ and $c= 2- \frac{1}{n}$, i.e. for $a=b=1- \frac{1}{n}$. We have and so provides the optimal bound in (). The marginal distribution for which the equality is attained is determined by the equation $\frac{F^{-1}(\textrm{e}^x)-\mu }{\sigma }=x+1,$ which is a particular form of () with $j=n$. The explicit form of the distribution function is presented in ().

In the sample minimum case the bound of () is . Since is decreasing, its projection onto the family of nondecreasing functions is obvious a constant function. This is equal to 0, because This is obviously concave as well which implies that and consequently . We prove that this zero bound is attained in the limit by a family of IRFR distributions. Define the family of distribution functions

$$\begin{aligned} F_\alpha (x) = \left\{ \begin{array}{ll} \exp (e^{\alpha } (x-1)+ \alpha ), &{} x<1, \\ 1, &{} x \ge 1, \end{array} \right. \qquad \alpha < 0. \end{aligned}$$

(A.9)

The respective quantile functions composed with the exponential functions

$$\begin{aligned} F_\alpha ^{-1}(e^x) = \left\{ \begin{array}{ll} e^{-\alpha } (x-\alpha )+1, &{} -\infty<x \le \alpha , \\ 1, &{} \alpha \le x<0, \end{array} \right. \qquad \alpha < 0, \end{aligned}$$

are concave which means that actually all $F_\alpha$, $\alpha < 0$, have the IRFR property. Simple calculations imply

$$\begin{aligned} \mathbb {E}_\alpha X_1= & {} \int _{-\infty }^{0} F^{-1}_\alpha (e^x) e^x\,dx = e^{-\alpha } \int _{-\infty }^{\alpha } (x-\alpha ) e^x \,dx + \int _{-\infty }^{0} e^x\,dx =0, \\ \mathbb {V}ar_\alpha X_1= & {} \mathbb {E}_\alpha X_1^2\\= & {} \int _{-\infty }^{0}[ F^{-1}_\alpha (e^x)]^2 e^x\,dx =e^{-2\alpha } \int _{-\infty }^{\alpha } (x-\alpha )^2 e^x \,dx \\{} & {} + 2 e^{-\alpha } \int _{-\infty }^{\alpha } (x-\alpha ) e^x \,dx + \int _{-\infty }^{0} e^x\,dx \\= & {} 2 e^{-\alpha }-1. \end{aligned}$$

Note that $\sigma = \sqrt{\mathbb {V}ar_\alpha X_1} \rightarrow \infty$ as $\alpha \rightarrow - \infty$. Moreover,

$$\begin{aligned} \mathbb {E}_\alpha X_{1:n}= & {} \int _{-\infty }^{0} F^{-1}_\alpha (e^x)f_{1:n}(e^x) e^x\,dx \\= & {} n e^{-\alpha } \int _{-\infty }^{\alpha } (x-\alpha ) (1- e^x)^{n-1} e^x \,dx + n \int _{-\infty }^{0} (1- e^x)^{n-1} e^x \,dx \\= & {} n e^{-\alpha } \sum _{i=0}^{n-1} \left( {\begin{array}{c}n-1\\ i\end{array}}\right) (-1)^{n-1-i} \int _{-\infty }^{\alpha } (x-\alpha ) e^{(n-i)x}\,dx +1\\= & {} \sum _{i=0}^{n-1} \left( {\begin{array}{c}n\\ i\end{array}}\right) (-1)^{n-i} e^{(n-1-i)\alpha } +1, \end{aligned}$$

because the integrals in the penultimate line amount to $- \frac{e^{(n-i)\alpha }}{n-i}$, $i=0, \ldots , n-1$. We easily notice that $\lim \nolimits _{\alpha \rightarrow - \infty } \mathbb {E}_\alpha X_{1:n} = -n+1,$ and in consequence,

$$\begin{aligned} \lim _{\alpha \rightarrow - \infty }\frac{ \mathbb {E}_\alpha X_{1:n}- \mathbb {E}_\alpha X_{1}}{\sqrt{\mathbb {V}ar_\alpha X_{1}}} = 0. \end{aligned}$$

(A.10)

A simple location and scale transformation of the distributions functions () generates the family of distributions with arbitrarily fixed mean $\mu$ and variance $\sigma ^2$ that preserves the relation (). This family is presented in the statement of Theorem 5. $\square$