My musings on a pioneering work of Erich Lehmann and its rediscoveries on some families of distributions

Abstract

The paper of Erich Lehmann (Lehmann 1953) is renowned for its ground-breaking contribution to rank tests within the area of nonparametric statistics. This work, in all likelihood, is known to every statistician. But, what is likely not known to many are some of the novel concepts and models that this paper succinctly introduced to the area of distribution theory. This, unfortunately, has led to some of these being rediscovered in the literature and then being referred to under different names. The purpose of this note is, therefore, two-fold: first to explain the key models that are contained in the mentioned work of Erich Lehmann, and second to point out how some of the known models discussed in the distribution theory and stochastic modeling literature are indeed present either explicitly or implicitly in the paper of Lehmann.

Keywords:

• Lehmann families
• exponentiated distributions
• transmuted distributions
• proportional hazards model
• proportional reversed hazards model

1. Introduction

Following the publication of two exhaustive volumes on Continuous Univariate Distributions by Johnson et al. (Johnson, Kotz, and Balakrishnan 1994, 1995), the area of Statistical Distribution Theory has exploded during the last 25 years, introducing many extensions, generalizations, and “new constructions” of some general families of distributions. These are clearly intended to provide greater flexibility while modeling statistical data. Unfortunately, in the ardor that ensued in this regard, some well-known work either got overlooked, or rediscovered! One such example is the classic paper of Lehmann (1953) in which several novel concepts and models pertaining to distribution theory were introduced, albeit for a study of nonparametric tests!

In this short note, I will first explain the key models that are present in Lehmann (1953) and then point out how some of the known models (that are currently studied and discussed in the distribution theory literature) under different names are contained either explicitly or implicitly in the above mentioned work of Lehmann.

2. Exponentiated distributions

By starting with the one-sided two-sample hypothesis testing problem, Lehmann points out that one could formulate the hypothesis test as $H_0:F(x)=G(x)$ vs. the alternative $H_1:G(x)\le F(x)$ for all x. He then states that one could take $G(x)=F(x-a)$ for some a > 0 (be reminded that it is a one-sided testing problem), analogous to parametric tests (like the Student-t test), and that the distribution of ranks would then depend not only on the location-shift a but also on the distributional form F(x) itself (see p. 23). Hence, he realizes the need for introducing a “natural nonparametric alternative” against which the distribution of ranks would be constant (see p. 24). With this intent, and keeping in mind that the alternative of interest is $G(x)\le F(x),$ he proposes the alternative $H{\prime }_1:G(x)={(F(x))}^k$ for some $k\ge 1$ (see p. 27). He then gives two intuitive reasonings for this particular choice: first is that when k is an integer, G is precisely the distribution function of the maximum of k iid random variables with distribution function F, and second is that with this choice Pr$(X<Y)=\frac{k}{k+1},$ where X is distributed as F and Y is distributed as G.

There are two key aspects to this choice: (i) as k increases, the distribution G would shift more and more to the right of F (recall here the notion of usual stochastic order $F\underset{\text{s}t}{\prec }G$), and (ii) so ranks of Y-values will tend to become larger in the combined sample when k increases. His Figure 1A–C on pages 26–28 visually demonstrates these two aspects for different choices of k.

The ingenuity in the proposal of the family $G(x)={(F(x))}^k$ is that it is a natural nonparametric family of distributions which is stochastically ordered with respect to F(x) for k > 1 as mentioned above (and naturally in the reverse ordering when $0<k<1$). Moreover, the “distance” between the two distributions is captured by the parameter k no matter what the “baseline” (in today’s jargon) distribution function F is!

It is evident that the use of any specific distributional form for F(x) will result in producing a corresponding family of parametric distributions. Let us take, for example, the two-parameter Weibull distribution with distribution function $F(x)=1-e^{-{(x/\sigma )}^{\alpha }},$ for $x,\sigma ,\alpha >0,$ where σ is the scale parameter and α is the shape parameter. Then, it simply results in the distribution function $G(x)={\{1-e^{-{(x/\sigma )}^{\alpha }}\}}^k,$ k > 0, which is what has been proposed as exponentiated-Weibull family by Mudholkar and Srivastava (1993); see also its subsequent use in the analysis of the well-known bus-motor-failure data by Mudholkar, Srivastava, and Freimer (1995).

Another choice for the baseline distribution is the two-parameter exponential distribution with distribution function $F(x)=1-e^{-(x-\mu )/\sigma },x>\mu ,\sigma >0,$ where μ is the location (threshold) parameter and σ is the scale parameter. This leads to the distribution function $G(x)={\{1-e^{-(x-\mu )/\sigma }\}}^k,$ k > 0, that has been introduced as generalized exponential distribution by Gupta and Kundu (1999).

In neither of these works, the original paper of Lehmann (1953) has been cited which, as mentioned earlier, is not surprising as many are not aware of the key distributional aspects in the paper of Lehmann! However, in the paper by Cordeiro, Ortega, and de Cunha (2013), the connection of Lehmann distributions of both types to exponentiated distributions has been noted. In fact, the author of this paper had a discussion with the first author (Gauss Cordeiro) of the mentioned manuscript about this specific issue!

During the last two decades, many more such “exponentiated distributions” have proliferated the distribution theory literature simply with different choices of F(x). It needs to be stated, of course, that these distributions, though are all part of the Lehmann family, could have their own intricacies in inferential aspects or when being used for modeling purposes.

3. PHR and PRHR distributions

In reliability theory, two useful reliability functions that have been studied extensively are hazard rate and reversed hazard rate functions. They are defined for lifetime distributions as (1) $$h_F(x)=\frac{f(x)}{1-F(x)}\hspace{1em}\mathrm{ }\text{and}\mathrm{ }\hspace{1em}{r}_F(x)=\frac{f(x)}{F(x)},\hspace{1em}x>0;$$(1) see, for example, Shaked and Shanthikumar (1994). While the hazard rate function seems to have been well studied in the literature, the reversed hazard rate function concept is more recent and possibly first introduced by Keilson and Sumita (1982). A detailed discussion on reversed hazard rate function and its properties was later provided by Block, Savits, and Singh (1998).

The Lehmann family of distributions G(x) is intrinsically connected with the reversed hazard rate function $r_F(x)$ though none of the mentioned papers made this connection. Interestingly, this was duly observed in a simultaneous independent work of Gupta, Gupta, and Gupta (1998), even explicitly mentioning Lehmann alternatives in the title of their paper. With F(x) as the baseline distribution function and $r_F(x)$ denoting the corresponding reversed hazard rate function, it is readily evident that (2) $$F(x)=\hspace{0.17em}\text{exp}\hspace{0.17em}\left\{-{\int }_x^{\infty }{r}_F(t)dt\right\}\hspace{1em}\mathrm{ }\text{and}\mathrm{ }\hspace{1em}f(x)=r_F(x)\hspace{0.17em}\text{exp}\hspace{0.17em}\left\{-{\int }_x^{\infty }{r}_F(t)dt\right\},$$(2) as stated in Finkelstein (2002). In order for the above integral to exist, with the condition that $F(0)=0$ over the support $[0,\infty ),$ we need to have ${\int }_0^{\infty }{r}_F(t)dt=\infty ,$ as specified by Block, Savits, and Singh (1998) and Finkelstein (2002). The connection between the function $r_F(x)$ and the Lehmann family becomes abundantly clear from the fact that (3) $$r_G(x)=\frac{g(x)}{G(x)}=\frac{k{(F(x))}^{k-1}f(x)}{{(F(x))}^k}=k{r}_F(x),$$(3) resulting in the so-called Proportional Reversed Hazard Rates (PRHR) family, as stated explicitly by Gupta, Gupta, and Gupta (1998). In (3)(3) $$r_G(x)=\frac{g(x)}{G(x)}=\frac{k{(F(x))}^{k-1}f(x)}{{(F(x))}^k}=k{r}_F(x),$$(3) , k will be interpreted as the proportionality parameter and will be linked to the covariates in the data, usually through a log-linear link function; one may see Kalbfleisch and Lawless (1991) wherein this is explicitly presented in their Eq. (20) (on p. 26), but referring to the function $r_G(x)$ as “reverse time hazards” and the distribution G(x) as a “power model”. The choices of Pareto distribution with $F(x)=1-{(1+x)}^{-\alpha },0<x,\alpha <\infty$ (see Gupta, Gupta, and Gupta 1998), and the Fréchet (or inverse Weibull) distribution with $F(x)=e^{-x^{-\gamma }},0<x,\gamma <\infty$ (see Di Crescenzo 2000) serve as good examples for the PRHR family. Unfortunately, some of the subsequently published works on the PRHR family of distributions still did not refer to the work of Lehmann in this regard; see, for example, Gupta, Gupta, and Sankaran (2004), Kundu and Gupta (2004), Badía and Berrade (2008), Sankaran and Sukumaran (2014), and Seo and Kim (2020).

The fundamental idea behind the construction of this family of distributions by Lehmann (1953) is that, in the two-sample case, we have two distributions (F, G), which may be represented, by a single function $\varphi ,$ as $(F,\varphi (F)),$ where $\varphi$ is a continuous non-decreasing function with $\varphi (0)=0$ and $\varphi =1;$ see this explicit comment on pages 25–26 in this regard. This is what ultimately leads to the distribution of ranks being constant, through the use of a probability integral transformation $U=F(X).$ Of course, he used $\varphi (F)=F^k$ as an example, stating its advantage of admitting a simple interpretation of it as an alternative and it being the distribution of the maxima of k iid random variables from F when k is a positive integer (see p. 27); also, on page 28, he goes on to say that a similar interpretation could be provided when k is any rational number. A heedful look at the distribution of ranks that Lehmann (1953) has derived shows its “symmetry” (meaning mirror-image) if the distribution $G=F^k$ is changed to $G=1-{(1-F)}^k;$ see Lehmann (1953, p. 29). This is quite intuitive in that the distribution of maxima of k iid random variables from F is being switched to that of the minima. One can only wonder whether Lehmann realized this property and so did not use it as another example, or he did not observe this property.

In any case, this has been duly noted by Shorack (1966, 1967) who, in fact, refers to the model $G(x)=1-{(1-F(x))}^k$ also as Lehmann alternative for the reasons explained above. Davies (1971) directly and explicitly refers to it as Lehmann alternative in addition to explaining its importance from a reliability testing perspective. Subsequently, in the nonparametric literature, both distributions, i.e., $G=F^k$ and $G=1-{(1-F)}^k,$ are commonly referred to as Lehmann families or Lehmann alternatives; for example, see Lin (1990), Lin and Sukhatme (1992), van der Laan and Chakraborti (1999), and Balakrishnan and Ng (2006). However, it is evident that this Lehmann family corresponds to the case wherein (4) $$h_G(x)=\frac{g(x)}{1-G(x)}=\frac{k{(1-F(x))}^{k-1}f(x)}{{(1-F(x))}^k}=k{h}_F(x),$$(4) which is the well-known Proportional Hazards Model. In (4)(4) $$h_G(x)=\frac{g(x)}{1-G(x)}=\frac{k{(1-F(x))}^{k-1}f(x)}{{(1-F(x))}^k}=k{h}_F(x),$$(4) , k is the proportionality parameter and is commonly linked to the covariates in the data through a log-linear link function. One of the earliest works in this regard, in fact, a prelude to the pioneering work of Cox (1972) on proportional hazards model under a semi-parametric setting, is the paper by Feigl and Zelen (1965) (see also, Cox 1964). Specifically, Feigl and Zelen used the model in (4)(4) $$h_G(x)=\frac{g(x)}{1-G(x)}=\frac{k{(1-F(x))}^{k-1}f(x)}{{(1-F(x))}^k}=k{h}_F(x),$$(4) , with baseline exponential distribution, to model the survival times of leukemia patients with white blood cell count as a covariate to estimate the parameter k. Strangely enough, all subsequent works on the model in (4)(4) $$h_G(x)=\frac{g(x)}{1-G(x)}=\frac{k{(1-F(x))}^{k-1}f(x)}{{(1-F(x))}^k}=k{h}_F(x),$$(4) in the areas of distribution theory and lifetime analysis, whether in a fully parametric or a semi-parametric form, refer to the model as PH distribution, while those in the area of nonparametric statistics refer to it as Lehmann family! Interestingly, in the book Analysis of Survival Data by Cox and Oakes (1984), the authors explicitly refer to this family as Lehmann family; see their Eq. (2.26) on page 23. In the paper by Hall and Wellner (2013), an explicit comment to the fact that the two Lehmann families correspond to proportional hazards and reversed proportional hazards distributions has been made.

One final comment that is worth making is that the two Lehmann families of distributions can be unified in a simple way to a single family form as $$G^{*}(x)={[1-{\{1-F(x)\}}^{\ell }]}^k,\hspace{1em}k,\ell >0,$$ with $\ell =1$ and k = 1 deducing to the two Lehmann families discussed so far.

4. Transmuted distributions

Shaw and Buckley (2007, 2009) used a transformation map to propose a quadratic rank transmuted distribution of the form (5) $$G(x)=(1+\lambda )F(x)-\lambda {(F(x))}^2,$$(5) where $\left|\lambda \right|\le 1$ and F(x) is the baseline distribution function. Since then, numerous authors have studied this transmuted family of distributions, by making different choices for the baseline distribution function F(x) (too many, in fact, to list here); see, for example, Aryal and Tsokos (2011) and Kozubowski and Podgorski (2016).

Turning our attention again to the paper of Lehmann (1953), in Section 6 of his paper, he introduces yet another family of distributions of the form (6) $$G(x)=qF(x)+p{(F(x))}^2,\hspace{1em}0\le p,q\le 1,$$(6) where $p+q=1;$ see Eq. (6.1) in Lehmann (1953, p. 34). It is evident that this family is another convenient nonparametric family of distributions from Lehmann’s viewpoint. Curiously, other than introducing this family of distributions and then using it in a subsequent power analysis of rank tests, Lehmann does not discuss any properties or intuitive interpretation for this family even though, in my view, it is as natural and appealing a distribution as his first family. Upon observing that the distribution in (6) is indeed a two-component mixture distribution and that $F(x)\ge {(F(x))}^2,$ we can reparametrize (6)(6) $$G(x)=qF(x)+p{(F(x))}^2,\hspace{1em}0\le p,q\le 1,$$(6) in the form of (5)(5) $$G(x)=(1+\lambda )F(x)-\lambda {(F(x))}^2,$$(5) with an expanded range for the parameter λ.

Another mixture formulation, using Lehmann family, can be provided as follows. Let us suppose Z, Y₁, and Y₂ are iid random variables from the baseline distribution F(x). Upon observing that the distribution in (6)(6) $$G(x)=qF(x)+p{(F(x))}^2,\hspace{1em}0\le p,q\le 1,$$(6) in indeed a two-component mixture distribution, we first define the random variable (7) $$\begin{array}{c}X\stackrel{d}{=}Z\hspace{1em}\text{with}\mathrm{ }\text{probability}\mathrm{ }q\\ \stackrel{d}{=}\max (Y_1,Y_2)\hspace{1em}\text{with}\mathrm{ }\text{probability}\mathrm{ }p,\end{array}$$(7) where $q=1-p,$ with its distribution function as in (6)(6) $$G(x)=qF(x)+p{(F(x))}^2,\hspace{1em}0\le p,q\le 1,$$(6) . Next, we use the well-known “triangle rule” for order statistics (see, for example, Arnold, Balakrishnan, and Nagaraja 1992) which states that (8) $$\begin{array}{c}Z\stackrel{d}{=}\min (Y_1,Y_2)\hspace{1em}\mathrm{ }\text{with}\mathrm{ }\text{probability}\mathrm{ }\frac{1}{2}\\ \stackrel{d}{=}\max (Y_1,Y_2)\hspace{1em}\mathrm{ }\text{with}\mathrm{ }\text{probability}\mathrm{ }\frac{1}{2}.\end{array}$$(8) By combining the two mixture forms in (7)(7) $$\begin{array}{c}X\stackrel{d}{=}Z\hspace{1em}\text{with}\mathrm{ }\text{probability}\mathrm{ }q\\ \stackrel{d}{=}\max (Y_1,Y_2)\hspace{1em}\text{with}\mathrm{ }\text{probability}\mathrm{ }p,\end{array}$$(7) and (8)(8) $$\begin{array}{c}Z\stackrel{d}{=}\min (Y_1,Y_2)\hspace{1em}\mathrm{ }\text{with}\mathrm{ }\text{probability}\mathrm{ }\frac{1}{2}\\ \stackrel{d}{=}\max (Y_1,Y_2)\hspace{1em}\mathrm{ }\text{with}\mathrm{ }\text{probability}\mathrm{ }\frac{1}{2}.\end{array}$$(8) , we arrive at (9) $$\begin{array}{c}X\stackrel{d}{=}\min (Y_1,Y_2)\hspace{1em}\mathrm{ }\text{with}\mathrm{ }\text{probability}\mathrm{ }\frac{q}{2}\\ \stackrel{d}{=}\max (Y_1,Y_2)\hspace{1em}\mathrm{ }\text{with}\mathrm{ }\text{probability}\mathrm{ }p+\frac{q}{2}.\end{array}$$(9)

The distribution of X obtained readily from (9)(9) $$\begin{array}{c}X\stackrel{d}{=}\min (Y_1,Y_2)\hspace{1em}\mathrm{ }\text{with}\mathrm{ }\text{probability}\mathrm{ }\frac{q}{2}\\ \stackrel{d}{=}\max (Y_1,Y_2)\hspace{1em}\mathrm{ }\text{with}\mathrm{ }\text{probability}\mathrm{ }p+\frac{q}{2}.\end{array}$$(9) , after a reparametrization, turns out to be identical to Shaw and Buckley’s transmuted distribution in (5)(5) $$G(x)=(1+\lambda )F(x)-\lambda {(F(x))}^2,$$(5) . The above mixture representation of the transmuted distribution in (5)(5) $$G(x)=(1+\lambda )F(x)-\lambda {(F(x))}^2,$$(5) has been offered by Granzotto, Louzada, and Balakrishnan (2017). They then proposed a cubic rank transmuted distribution by considering the random variable X with distribution function G as a mixture of the three order statistics obtained from iid random variables $Y_1,Y_2,Y_3$ from the baseline distribution F(x). Though this differs from the Lehmann family in (6)(6) $$G(x)=qF(x)+p{(F(x))}^2,\hspace{1em}0\le p,q\le 1,$$(6) , it just turns out to be an extension with one more mixture component with distribution function ${(F(x))}^3$!

5. Concluding remarks

In this paper, we have highlighted the novel concepts and statistical models that have been introduced in the pioneering paper of Lehmann (1953). Even though this paper is well known to researchers in the area of nonparametric statistics, it seems that many in the area of distribution theory are not aware of this important aspect of the paper. This, unfortunately, has led to some of these distributions being rediscovered in the literature under different names. For this reason, we have explained the key distributions that are present in Lehmann (1953) and as to how some of the known models in the distribution theory literature are present either explicitly or implicitly in the ground-breaking work of Lehmann!

Acknowledgments

The author thanks the Natural Sciences and Engineering Research Council of Canada for funding this research through an Individual Discovery Grant 2020-06733, and the Associate Editor and the Reviewer for some useful comments and suggestions on an earlier version of this manuscript.