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## Über dieses Buch

About 10 years ago I began studying evaluations of distributions of or­ der statistics from samples with general dependence structure. Analyzing in [78] deterministic inequalities for arbitrary linear combinations of order statistics expressed in terms of sample moments, I observed that we obtain the optimal bounds once we replace the vectors of original coefficients of the linear combinations by the respective Euclidean norm projections onto the convex cone of vectors with nondecreasing coordinates. I further veri­ fied that various optimal evaluations of order and record statistics, derived earlier by use of diverse techniques, may be expressed by means of projec­ tions. In Gajek and Rychlik [32], we formulated for the first time an idea of applying projections onto convex cones for determining accurate moment bounds on the expectations of order statistics. Also for the first time, we presented such evaluations for non parametric families of distributions dif­ ferent from families of arbitrary, symmetric, and nonnegative distributions. We realized that this approach makes it possible to evaluate various func­ tionals of great importance in applied probability and statistics in different restricted families of distributions. The purpose of this monograph is to present the method of using pro­ jections of elements of functional Hilbert spaces onto convex cones for es­ tablishing optimal mean-variance bounds of statistical functionals, and its wide range of applications. This is intended for students, researchers, and practitioners in probability, statistics, and reliability.

## Inhaltsverzeichnis

### 1. Introduction and Notation

Abstract
This work presents a method of using projections of functions onto convex cones in Hilbert spaces for determining sharp bounds on values of statistical functionals over general and restricted families of distributions, expressed in terms of moment parameters of the distributions. The method is based on representing the statistical functionals and families of distributions as fixed elements and convex cones, respectively, in a common real Hilbert space. Then the norm of the projection of the element onto the cone provides the optimal bound. The distribution for which the bound is attained is derived by a simple transformation of the projection.
Tomasz Rychlik

### 2. Basic Notions

Abstract
We recall here some basic facts about the Hilbert spaces that are used in the sequel. They can be found in textbooks on functional analysis (see, e.g., Balakrishnan [9]). A pair (H, (· , ·)) is called a real inner product space if H is a real linear space and the function (· , ·) : H × H ↦ ℜ, referred to further as the inner product, is linear in each argument when the other is fixed, symmetric under rearrangement of arguments, and positive if both arguments are identical and nonzero. These properties imply the Schwarzinequality
$$\forall g,h \in H (g,h) \leqslant {[(g,g)(h,h)]^{1/2}}$$
(2.1)
.
Tomasz Rychlik

### 3. Quantiles

Abstract
The main results of this chapter come from Rychlik [90]. The bounds for quantiles of general distributions were obtained by Moriguti [58]. Those for symmetric and symmetric unimodal distributions may also be concluded from the Chebyshev and Gauss inequalities, respectively. Vysochanskii and Petunin [102] presented a refinement of the Gauss inequality for unimodal distributions. Further generalizations can be found in Dharmadhikari and Joag-dev [25, Section 1.5]. We also notice that the Markov inequality yields
$${F^{ - 1}}(p) \leqslant \frac{{\mu F}}{{1 - p}}$$
for quantiles of nonnegative random variables. Another implication of the Markov inequality is the second moment bound
$${F^{ - 1}}(p) \leqslant \frac{{mF}}{{{{(1 - p)}^{1/2}}}}$$
.
Tomasz Rychlik

### 4. Order Statistics of Independent Samples

Abstract
Bounds for expectations of order statistics from general i.i.d. samples, due to Moriguti [58], are presented in Section 4.1. Bounds of Sections 4.2 and 4.4 for populations with decreasing density and failure rate, and symmetric uni-modal ones, were obtained by Gajek and Rychlik [33]. Results of Section 4.3 for restricted families determined by the star order come from Rychlik [89]. Section 4.5, partially based on Okolewski and Rychlik [65], is devoted to the study of quantile estimation bias in various nonparametric families of distributions.
Tomasz Rychlik

### 5. Order Statistics of Dependent Observations

Abstract
Assume that Y1,…, Y n are possibly dependent and identically distributed. Recalling arguments of Rychlik [79], in Section 5.1 we conclude sharp bounds (2.25) and (2.27) on expectations of general L-statistics and single order statistics, respectively, depending on the common marginal distribution of the observations. Next we apply the projections of functionals defined in (2.25) and (2.27) for establishing respective moment bounds over general and restricted families of marginals. In Section 5.2 general, symmetric, and nonnegative observations are treated, and respective deterministic bounds for arbitrary samples are concluded. The results cited are from Rychlik [81], but some earlier partial solutions are also mentioned. In the remainder of Chapter 5 we confine ourselves to single order statistics. In Section 5.3 we present mean-variance and second moment bounds on the expectations of order statistics for families of parent distributions related to a given one in the convex order. The mean-variance bounds for F $${\underline \succ _c}$$W and F $${\underline \prec _c}$$ W are not published elsewhere, except for DFR and IFR distributions, given in Rychlik [87]. The results for general W and W = U (i.e., for the decreasing density distributions) are presented here as well. Rychlik [84] established second moment bounds for F $${\underline \succ _c}$$ ($${\underline \prec _c}$$)W with general W. The decreasing density and failure rate distributions were studied in Gajek and Rychlik [32].
Tomasz Rychlik

### 6. Records and kth Records

Abstract
In comparison with evaluations of other statistical functionals discussed here, investigations for record values are still at a preliminary stage, and only a few results are presented now. Examples of Section 6.1 show that the range of record values can be arbitrarily large when all types of interdependence among the original variables are admitted. For the case of i.i.d. sequences with general and symmetric distributions, mean-variance bounds on standard and kth records, due to Nagaraja [59] and Raqab [74], respectively, are presented in Section 6.2. In Section 6.3 second moment bounds for distributions with decreasing density and failure rate are cited from Gajek and Okolewski [31]. Finally, we discuss evaluations of Rychlik [83] for increments of first records coming from various families of parent distributions. Note that Raqab [75] derived pth absolute moment bounds on expectations of first records in general and symmetric populations based on the Hölder inequality. Nagaraja [59] used the Jensen inequality for deriving some quantile bounds on expectations of records (see also Arnold and Balakrishnan [5, Section 6.2]). Some bounds and approximations can be found in Arnold et al. [8, Sections 3.8 and 3.9].
Tomasz Rychlik

### 7. Predictions of Order and Record Statistics

Abstract
In this chapter we evaluate expected increments of future order and record statistics in the i.i.d. samples under conditions that some previous values are known. These are important for predicting prospective failures in reliability systems and shock models on the grounds of former data. The results are based on representations of conditional expectations of order statistics and records in terms of unconditional expectations of other ones presented in Section 2.2, and evaluations of the latter derived in Sections 4.1, 4.2, 6.2, and 6.3.
Tomasz Rychlik

### 8. Further Research Directions

Abstract
In the monograph we mostly focused on the optimal upper bounds, but certainly the lower ones are needed for evaluating the actual ranges of the functionals over given classes of distributions. Generally, the best upper and lower bounds are not symmetric about zero, but the latter can also be derived by means of our projection method. To this end we should analyze the negatives of the functionals under study. Only the functional corresponding to the order statistics and L-statistics (see (2.27) and (2.25)) in the dependent case needs a more subtle transformation. Changing the signs of coefficients c j , 1 ≤ jn, in (2.26) results in constructing a functional g−c different from −gc. Also, one should realize that generally projecting a functional and its negative are different problems that should be solved separately by use of specific arguments. With few exceptions they cannot be derived one from the other in a simple way.
Tomasz Rychlik

### Backmatter

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