Let us consider the infinite sequence , , of independent and identically distributed random variables with the common cummulative distribution function (cdf) and finite mean
and variance. By denote the order statistics of . Further, we are interested in the increasing subsequences of of the th greatest order statistics, for a fixed . Formally, we define the (upper) th records , , by introducing first the th record times as
Then the th record values are given by
Note that classic upper records are defined by , and we say that such a record occurs at time if is greater than the maximum of previous observations .
Records are widely used, not only in the statistical applications. The most obvious one that arises at the first glance is the prediction of sport achievements and natural disasters. The first mention of the classic records comes from Chandler (1952), while the th record values were introduced by Dziubdziela and Kopociński (1976). For the comprehensive overview of the results on the record values the reader is referred to Arnold, Balakrishnan and Nagaraja (1998) and Nevzorov (2001).
The distribution function of the th record value is given by the following formula
If the cdf
is absolutely continuous with a probability density function (pdf), then the distribution function (1.1) also has the pdf given by
In particular case of the standard unform underlying cdf , the corresponding distribution of the uniform th record is given by the following equations
Now, recall the cdf of the generalized Pareto distribution
Next, we say that the cdf precedes the cdf in the convex transform order, and we write , if the composition is concave on the support of . Following the reasoning of Goroncy and Rychlik (2015) and Bieniek and Szpak (2017), we consider the following family of distributions with the increasing generalized failure rate defined as with respect to
Indeed, if the distribution function is continuous with the density function , then the generalized failure rate defined as
is increasing. Note that the expression in is just the product of the conventional failure rate and a power of the survival function .
we obtain the standard uniform distribution functionand the family IGFR(1)=ID of the increasing density distributions, respectively. On the other hand for , the cdf
is the cdf of the standard exponential distribution and in the result we get the family IGFR(0)=IFR of the increasing failure rate distributions.
The aim of this paper is to establish the optimal upper bounds on
where the cdf is restricted to the IGFR() class of distributions, for arbitrarily chosen and . In the special case , which reduces to the order statistics , the readers are referred to Rychlik (2014), who established the optimal bounds for ID and IFR distributions.
The bounds on the th records, in particular classic record values have been widely considered in the literature, beginning with Nagaraja (1978). He used the Schwarz inequality to obtain the upper bounds on the expectations of the classic records, which were expressed in terms of the mean and standard deviation of the underlying distribution. The Hölder inequality was used by Raqab (2000), who presented more general bounds expressed in the scale units generated by the
th central absolute moments, . Also, he considered records from the symmetric populations. Differences of the consecutive record values (called record spacings) based on the general populations and from distributions with the increasing density and increasing failure rate were considered by Rychlik (1997). His results were generalized by Danielak (2005) into the arbitrary record increments.
The th record values were considered by Grudzień and Szynal (1985), who by use of the Schwarz inequality obtained non-sharp upper bounds expressed in terms of the population mean and standard deviation. Respective optimal bounds were derived by Raqab (1997), who applied the Moriguti (1953) approach. Further, the Hölder along with the Moriguti inequality were used by Raqab and Rychlik (2002) in order to get more general bounds. Gajek and Okolewski (2003) dealt with the expected th record values based on the non-negative decreasing density and decreasing failure rate populations evaluated in terms of the population second raw moments. Results for the adjacent and non-adjacent th records were obtained by Raqab (2004) and Danielak and Raqab (2004a). Evaluations for the second records from the symmetric populations were considered by Raqab and Rychlik (2004). Danielak and Raqab (2004b) presented the mean-variance bounds on the expectations of th record spacings from the decreasing density and decreasing failure rate families of distributions. Further, Raqab (2007) considered second record increments from decreasing density families. Bounds for the th records from decreasing generalized failure rate populations were evaluated by Bieniek (2007). Expected th record values, as well as their differences from bounded populations were determined by Klimczak (2007), who expressed the bounds in terms of the lengths of the support intervals.
Regarding the lower bounds on the record values, there are not many papers concerning the problem, in opposite to the literature on the lower bounds for the order statistics and their linear combinations (see e.g. Goroncy and Rychlik (2006a), Goroncy and Rychlik (2006b, 2008), Rychlik (2007), Goroncy (2009)). The lower bounds on the expected th record values expressed in units generated by the central absolute moments of various orders, in the general case of the arbitrary parent distributions were presented by Goroncy and Rychlik (2011). There are also a few papers concerning the lower bounds on records indirectly, namely in the more general case of the generalized order statistics (Goroncy (2014), Bieniek and Goroncy (2017)).
Below we present a procedure which provides the basis of obtaining the optimal upper bounds on in the case of our interest. It is well known that
Due to the further application, we subtract 1 from in the formula above, but one could replace it with an arbitrary constant. Changing the variables in , for a fixed, absolutely continuous cdf with the pdf on the support , , we obtain
Further assume that satisfies
Let us consider the Hilbert space of the square integrable functions with respect to on , and denote the norm of an arbitrary function as
Moreover, let stand for the projection operator onto the following convex cone
In order to find the upper optimal bounds on , we will use the Schwarz inequality combined with the well-known projection method (see Rychlik (2001), for details). It is clear that can be bounded by the -norm of the projection of the function , as follows
with the equality attained for cdf satisfying
In our case we fix and the problem of establishing the optimal upper bounds on easily boils down to determining the -norm of the projection of the function onto . Note that in order to apply the projection method, we need the condition to be fulfilled by the distribution function . Bieniek (2008) showed, that in that case we need to confine ourselves to parameters , what we do in our further considerations.
2 Auxiliary results
In this section we recall the results of Goroncy and Rychlik (2015, 2016), who determined the projection of the function satisfying particular conditions, onto the cone of nondecreasing and concave functions. These conditions are presented below.
(A) Let be bounded, twice differentiable function on , such that
Moreover, assume that is strictly decreasing on , strictly convex increasing on , strictly concave increasing on with , and strictly decreasing on with for some .
The projection of the function satisfying conditions (A) onto the convex cone is either first linear, then coinciding with and ultimately constant, or just linear and then constant, depending on the behaviour on some particular auxiliary functions, which are introduced below.
which is decreasing on , increasing on and decreasing on , having the unique zero in . Moreover, let
for . The precise form of the projection of the function satisfying (A) onto the cone is described in the proposition below (cf. Goroncy and Rychlik (2016), Proposition 1).
If the zero of belongs to the interval and the set is nonempty, then
where is the projection of h onto . Otherwise we define
Let denote the set of arguments satisfying the following condition
Then is nonempty and for unique .
Note that there are only two possible shapes of projection functions of the function onto . The first one requires compliance with certain conditions and can be briefly described as: linear - identical with - constant (l-h-c, for short). The second possible shape does not have a part which corresponds to the function , and we will refer to it as l-c (linear and constant) from now on. The original version of this proposition can be found in Goroncy and Rychlik (2015), however there was no clarification about the parameter in case of the l-c type of the projection, therefore we refer to Goroncy and Rychlik (2016).
We will also need some results on the projection of the functions satisfying conditions (), which are a slight modification of conditions (A). We state that the function satisfies () if conditions (A) are modified so that and . This in general means that the function does not have the decreasing part at the right end of the support and in particular does not have to be bounded from above. The proposition below (cf. Goroncy and Rychlik (2016), Proposition 6) describes the shape of the projection in this case.
If the function satisfies conditions , then the set is nonempty and for we have
3 Main results
Let us focus now on the case and denote
We also denote .
The substantial matter in determining the bounds on is to learn the shapes of the functions for arbitrary and , which correspond with the shapes of compositions , and are presented in the lemma below (comp. with Bieniek (2007), Lemma 3.2).
If , then the shape of is as follows:
If , then , , is convex increasing.
If , then , , is convex increasing, concave increasing and concave decreasing.
If , then is concave increasing-decreasing, and , , is convex increasing, concave increasing and concave decreasing.
If , then is concave increasing, concave decreasing and convex decreasing, and , , is convex increasing, concave increasing, concave decreasing and convex decreasing.
If , then the shape of is as follows:
If , then is linear increasing and , , is convex increasing.
If , then is concave increasing and then decreasing, , , is convex increasing, concave increasing, and decreasing.
If , then the shape of is as follows:
If , then is concave increasing, , , is convex increasing and concave increasing.
If , then is concave increasing, concave decreasing and convex decreasing and , , is convex increasing, concave increasing, concave decreasing and convex decreasing.
It is worth mentioning that slight differences between the lemma above and Lemma 3.2 in Bieniek (2007) are the result of different notations of record values.
Note that the case is covered by the above lemma, except the setting (ii) for , which is not possible in this case (cf. Rychlik (2001), p.136). Case comes from Rychlik (2001, p.136). In order to determine the shape of for , we notice that for and
and use the variation diminishing property (VDP) of the linear combinations of (see Gajek and Okolewski, 2003). Other special cases of we calculate separately in order to obtain the shapes of .
Faced with this knowledge, we conclude that satisfy conditions (A) with , for and if , for and or and if , as well as for and if . Moreover, we have
, . The value of in the local maximum point has to be positive, since the function starts and finishes with negative values and integrates to zero, which means that has to cross the -asis and changes the sign from negative to positive, finishing with negative value at . Therefore, we can use Proposition 1 in order to obtain the projection of onto and finally determine the desired bounds according to . Moreover, satisfy conditions () in case of the first record values () for , and we are entitled to use Proposition 2 then. Other cases can be dealt without the above results. These imply the particular shapes of the projections which can be one of the three possible kinds. The first one coincides with the original function (first values of the classic records for ), the second shape is the linear increasing function (classic record values for and or and ), and the last one is the projection coinciding with the function at the beginning and ultimately constant (first values of the th records for and or , ).
In order to simplify the notations, we will denote the projection of function onto with respect to by from now on.
3.1 Bounds for the classic records
In the proposition below we present the bounds on the classic record values (). This case does not require using the Proposition 1, since the shapes of the densities of records do not satisfy conditions (A), but possibly satisfy conditions ().
Assume that .
Let . If , then we have the following bound
with the equality attained for the exponential distribution function
If , then the set is nonempty and for we have
The equality in is attained for distribution functions IGFR that satisfy the following condition
Let now . We have the following bound
with the equality attained for the exponential distribution function .
Suppose , . Then we have the following bound
The equality in is attained for the following distribution function
Proof. Fix . Let us first consider case (i), i.e. . Here we have to add an additional restriction , which has been mentioned at the end of Section 2. If , then the function is increasing and concave, hence its projection onto is the same as . The bound can be determined via its norm, which square is given by
Taking into account that as well as are equal to one, formula implies .
Suppose now that . Note that in this case satisfy conditions (). Using Proposition 2, we have the following projection of onto the cone ,
An appropriate counterpart of function in our case is
with , since simple calculations show that
for , we conclude that takes the form
with . In consequence for and we have
where is given in . The square root of the expression above determines the optimal bound on .
Consider now case (iii) with and , which requires more explanation. With such parameters function is increasing and convex. This implies that its projection onto the cone of the nondecreasing and concave functions is linear increasing. The justification for this is similar as e.g. in Rychlik (2014, p.9). The only possible shape of the closest increasing and convex function to the function is the linear increasing one , say, which has at most two crossing points with . Since
(see e.g. Rychlik (2001)), we obtain
Next, in order to determine the optimal parameter , we need to minimize the distance between the function and its projection
For we have and . Therefore
we get the minimum of equal to . Since , we also obtain . Finally, the optimal bound can be determined by calculating the square root of
which equals .
Let finally consider the case (ii). Note that for function is increasing and concave, and the case is analogous to (i) with , when we get the bound equal to 1.
If , then is increasing and convex and its projection onto the cone of the nondecreasing and concave functions is linear, as in case (iii). Here the analogue to is . For , we have , , which gives us the distance function , which is minimized for . Hence , and we get the optimal bound equal to .
The distributions for which the equalities are attained in all the above cases can be determined using the condition with and .
3.2 Bounds for the th records,
As soon as we give some auxiliary calculations, we are ready to formulate the results on the upper bounds of the expected th record values, based on the the IGFR() family of distributions. For being the GPD distribution, we have the corresponding function of given by