# The Ara OB 1a Association

Scott J. Wolk  Fernando Comerón  Tyler Bourke

###### Abstract

The Ara OB1a association is one of the closest sites where triggered star
formation is visible for multiple generations of massive stars. At about 1.3
kpc distance, it contains complex environments including cleared young
clusters, embedded infrared clusters, CO clouds with no evidence of star
formation, and clouds with evidence of ongoing star formation. In this review
we discuss the research on this region spanning the last half-century. It has
been proposed that the current configuration is the result of an expanding
wave of neutral gas set in motion between 10–40 million years ago in
combination with photoionization from the current epoch.

Harvard–Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA
02138; USA

European Southern Observatory,Karl-Schwarzschild-Strasse 2, 85748 Garching,
Germany

Harvard–Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA
02138; USA

##  1\. Historical Summary (1960–1985)

Ara OB1a, while somewhat enigmatic, may be one of the best examples of
triggered star formation in the local Galaxy. The triggering in the most
active portion is easily imagined from images of the $<$ 3 Myr NGC 6193/RCW
108-IR complex such as Figure 1. However the full association may be as much
as 50 Myr in age and cover several degrees on the sky. Ara OB1 was first
studied in detail by Whiteoak (1963). He used photographic photometry and
objective prism spectroscopy to identify about 35 O and B star members.
Whiteoak’s work was a follow up to an H$\alpha$ survey by Rodgers, Campbell &
Whiteoak (RCW; 1960), which cataloged 181 H$\alpha$ emission regions in the
southern sky, including RCW 108. NGC 6193 is an open cluster, discovered by
James Dunlop in 1826, that is dominated by a pair of O stars, HD 150135 and HD
150136. These are the brightest optically revealed O stars in the association
and are thought to be responsible for ionizing the bright rim of emission to
the west (NGC 6188, discovered by John Herschel in 1836) which separates NGC
6193 from RCW 108–IR. The youth of the region is clear in sky survey plates
and Figure 1 which show concurrent regions of ionized gas and dust lanes.
Whiteoak noted two additional clusters in the region, NGC 6204 – about 2
degrees to the northeast – and NGC 6167 – about 1 degree to the southwest.

Ara OB1a is a compact association covering about 1 sq. degree around a central
cluster – NGC 6193. It is generally thought to be about 1.3 kpc away and can
be equated with RCW 108. There is some confusion in the literature as to what
is actually meant by RCW 108. The original definition of Rodgers, Campbell &
Whiteoak (1960) refers to all the region where H$\alpha$ nebulosity is
detected, for which they give a size of 210′x 120′centered at (l,b = 336.49,
–1.48). Straw et al. (1987) used RCW 108–IR to refer to the embedded IR
cluster about 15′ to the west of the O stars in NGC 6193 and which is
identified with IRAS 16362–4845. The confusion arises when RCW 108-IR is
abbreviated by dropping the ”–IR”. For the remainder of this paper we will
refer to the embedded cluster as RCW 108–IR or IRAS 16362–4845.

Moffat & Vogt (1973) measured photometry for 13 stars within 4′ of the center
of NGC 6193 and found EB-V= 0.4 and a distance of 1360 pc. Herbst & Havlen
(1977) describe Ara OB1 as having a “diamond ring” appearance, with RCW 108–IR
as the “diamond” and a thin circular dust ring making up almost half of the
“ring”. They performed photoelectric and photographic photometry on 702 stars
in the region. Herbst (1974, 1975) identified parts of this region as an “R
association” as there were three early type stars associated with reflection
nebulosity. These stars may mark a separate site of star formation within Ara
OB1a. We will discuss this further in Section 6.

Mid–infrared MSX & IRAS maps of the region show several condensations. The
brightest is coincident with RCW 108–IR and two apparently related peaks IRAS
16379–4856 and IRAS 16348–4849. One of the earliest radio studies of the
region was in the survey by Shaver & Goss (1970). They made a 5 GHz and 408
MHz survey of over 250 Galactic radio source including RCW 108. RCW 108 was
remarkable for having one of the smallest emission regions – unresolved at 3′
and having a relatively high density and temperature of the electron
population.

In addition to Ara OB1a, Whiteoak (1963) identified a background O star
cluster coincident with NGC 6193 but with a different distance modulus. While
the foreground Ara OB1a has a distance modulus of 10.5$\pm 0.6$ the second
group of about 13 O and B stars (Ara OB1b) has a distance modulus of 12.7 $\pm
0.5$ or about 3500 pc. The central O stars of the two associations are offset
by about 2o along the Galactic plane (Humphreys 1978), so this is a concern
for membership determination.

Figure 1.: Salient features of the most active portion of Ara OB1a. This image
is a B,V H$\alpha$ (blue, green and red respectively) from the wide field
imager on the MPI-ESO 2.2m telescope. The field of view is 32′x 32′ or about
12 pc on a side.

##  2\. NGC 6193

NGC 6193 is the open cluster to the east of the bright emission rim NGC 6188.
In this region, Herbst & Havlen (1977) measured over 700 stars
photometrically. Of these, 59 had photometric distances and reddening
consistent with cluster membership. This appears to be complete to earlier
than A0. This survey covered an area about 40′ on a side. They found a
slightly steeper than usual reddening law. Their best-fit photometric distance
was 1320$\pm 120$ pc. Several other groups have estimated the distance to the
cluster using photometric parallax techniques and obtain results ranging from
$\approx$1100 to $\approx$ 1400 pc (Moffat & Vogt 1973, Fitzgerald 1987,
Kalcheva & Georgiev 1992). Vazquez & Feinstein (1992) obtained an age of about
3 Myr by fitting the upper main sequence, they also determined a distance of
about 1410 $\pm 120$ pc using an R = 3.1 reddening law. They noted that the
steep reddening found by Herbst & Havlen (1977) was due to a binarity-induced
color shift. Extinction is low (A${}_{V}\sim 0.5$).

It has been difficult to obtain a good catalog of stellar members. The 59
stars in Herbst & Havlen (1977) are only identified on a star chart. Similar
practices were used by Moffat & Vogt (1973), Arnal et al. (1988) and Vazquez &
Feinstein (1992). The deepest published study of NGC 6193 is the $Chandra$
zero order image of the $Chandra$ gratings observation of HD 150135. HD 150136
is a remarkable O+O spectroscopic binary with a short 2.66 day orbital period
(Niemela & Gamen 2005). HD 150136 also has unusual radio properties, being a
non-thermal radio emitter (Benaglia, Cappa, & Koribalski 2001) which made it
an interesting $Chandra$ gratings target in its own right (Skinner et al.
2005). This deep spectrum produces a useful zero-order image. The zero-order
data include the central 2′$\times$ 2′ region of the cluster. This
observation, which should be fairly complete to 1 solar mass, reveals 43 X-ray
sources within 4 square arcminutes (Skinner et al 2005). Only 11 of these had
previous optical identifications, however all 43 were detected at H band and
are likely cluster members. In addition to those sources cataloged by Skinner
et al. there are about 30 X-ray point sources visible outside of the central
2′$\times$ 2′ region in this data set. Most of these have I-band magnitudes
consistent with cluster membership and are between 2′ and 6.8′ from HD 150136.
Wolk et al. (2008) obtained a 100 ks $Chandra$ observation of RCW 108–IR. Due
to the orientation of the telescope pointing, about 64 square arcminutes of
the NGC 6193 field was covered, mostly to the south and west of HD 150136.
These data were a little less than 10 times deeper than the observation of
Skinner et al. An additional 99 X-ray sources associated with NGC 6193 were
detected. Most of these have infrared counterparts, which supports membership
in Ara OB1a. This brings the total identified membership of NGC 6193 up to
about 200\. But we emphasize the non-uniformity of the data. This is
especially true of the Chandra pointings which cover perhaps 20% of the area
of NGC 6193. A recent shallow $Spitzer/IRAC$ map of the region detected about
185 Class II sources, associated with both NGC 6193 and RCW 108–IR, and 13
Class I sources associated with RCW 108–IR alone (Wolk et al. 2008). But this
observation was centered on RCW 108–IR and so again does not provide a uniform
sample of NGC 6193.

##  3\. IRAS 16362–4845/RCW 108–IR

In addition to the NGC 6193 cluster, the nearby source IRAS 16362–4845 harbors
the extremely young stellar cluster RCW 108–IR, which has been recently
studied (Comerón et al. 2005, Comerón & Schneider 2007, Wolk et al. 2008). The
bright rim NGC 6188 marks the border between the HII region and a dense dark
nebula containing RCW 108–IR. The bright rim is produced by the ionization of
the molecular cloud hosting IRAS 16362–4845 by the two central stars of NGC
6193. The dark cloud contains a high emission measure knot (Wilson et al.
1970), which appears to be a compact HII region (Frogel & Persson 1974). The
comprehensive UBVRI study by Herbst & Havlen (1977) proposed this was a site
of very recent star formation.

RCW 108–IR is a young, compact cluster partially embedded in its parent
molecular cloud. Straw et al. (1987) used photometry at IR wavelengths
(1.2-100 $\mu$m) to perform the first spatially complete survey of the
cluster. They report on 55 objects; $<$ 20 of these have optical counterparts.
In addition to the point sources, there is diffuse IR emission. The total
luminosity of the cluster is about 1.8$\times 10^{5}$ L⊙, dominated by at
least two early O stars. The full aggregate of O and B stars appears
responsible for the ionization of the diffuse emission (Comerón et al. 2005).
The primary exciting source star is identified as IRS 29 (Straw et al. 1987) –
probably an O8 star. Straw et al. also identify at least one protostar. They
put forward a physical model of the region in which photoionization of the
older stars in the Ara OB1a association ionized the bright rim (NGC 6188)
which lies at the edge of a finger of molecular material which thickens from
east to west (Figure 2).

Figure 2.:  A simple model of the interface between RCW 108 and NGC 6193 from
Straw et al. (1987). It now appears that the older stars lie somewhat behind
the molecular cloud since none are seen superimposed over it.

Several studies at radio wavelengths have been completed to understand the
physical properties of the dark cloud. Goss & Shaver (1970) detected continuum
emission at 5 GHz, Whiteoak & Gardner (1974) detected H2CO absorption at 4830
MHz, and CO emission at 115 GHz was detected by Whiteoak & Otrupcek (1982).
The hydrogen recombination lines H109$\alpha$ and H166$\alpha$ were also
measured (Wilson et al. 1970; Cersosimo 1982). RCW 108 IRS 29 is the second
strongest water source seen by SWAS (second to the BN object in Orion; Gary
Melnick-private communication) with a peak flux of over 7000 Jy in the 557 GHz
line. A follow up study suggests that gaseous H2O is largely restricted to a
thin layer of gas near the cloud surface.

There have been two recent near-IR through mm studies of RCW 108–IR. Urquhart
et al. (2004) compared the 2MASS and MSX observations of the cloud with radio
recombination lines and radio continuum. They find that RCW 108–IR is an
ultracompact HII region, less than 0.1 pc in scale with a core–halo
morphology. They discuss 8 new sources within the UCHII region and report 3
sources detected at 25 $\mu$m and set back about 2.5′ from the bright rim. A
more detailed study has been recently published by Comerón et al. (2005). From
their high resolution CO map they estimate a total mass of the cloud at 8000
M⊙ with $<$ 200 M⊙ in the molecular concentration harboring the compact HII
region. They produced a very high resolution JHKs map of the region (Figure
3). They identify 25 stars whose luminosities suggest spectral types earlier
than A0 under the assumption that there is no significant circumstellar
contribution to the K-band flux and conclude that the ionization of the UCHII
region is provided by this aggregate. They suggest that low–mass star
formation has yet to commence here. IRS 29 is found to be an O9 star from the
visible spectrum of the compact HII nebula, which is in agreement with its
infrared photometry. Out of 4365 stars brighter than Ks=14.5 in the whole
13′$\times$ 13′ field, 87 are found to have strong disk signatures, most of
them located in the molecular cloud that contains RCW108–IR.

Figure 3.: A JHKS (blue, green and red respectively) image of RCW 108–IR from
the NTT. The field of view is about 13′ on a side, corresponding to about 6 pc
on a side. Five-sigma detection limits in the least-exposed areas of the image
are typically J=18.6, H=17.7, K=17.2 mag. The cloud is very opaque immediately
north and south of the emission nebula. Most of the stars with optically thick
disks are either within this emission or along the eastern edge of the cloud.
Inset: A J (blue), H (green), and K (red) color composite of RCW 108-IR
obtained using adaptive optics at the VLT. The frames are centered on the
brightest source in the near-infrared, at RA(2000) = 16:40:00.2, Dec(2000)
=-48:51:40. The field is 52′′ $\times$ 57′′.

In their 100 ks $Chandra$ pointing centered on RCW 108–IR, Wolk et al. (2008)
detect 32 point sources in the central arcminute, and 65 within the central
2′(Figure 4). They find a sharp rise in the absorption column of log NH $>$
21.2 (A${}_{V}>10$) for these stars. By contrast X-ray sources associated with
NGC 6193 have inferred extinctions between 0 and 5 AV. About 10 of the 65
sources in the core region do not have counterparts in the deep survey of
Comerón et al.

Figure 4.: $Chandra$/ACIS-I X-ray map of a somewhat wider region than shown in
Figure 3. The colors indicate X-ray energy – red for soft X-rays ($<1$ keV),
blue for hard X-rays ($>$ 2.4 keV) and green in between. Hard X-rays penetrate
gas better than soft X-rays leading to X-ray ”blueing” equivalent to
optical/infrared reddening. Thus the stars associated with RCW 108–IR appear
blue and the stars to the east and south associated with NGC 6193 appear
yellow or red. The region is about 20′ across. From Wolk et al. (2008).

Recent JHKsL′ observations of the stellar content of RCW 108-IR have been
carried out by Comerón & Schneider (2007) using adaptive optics-assisted
imaging at the VLT. The superior resolution attained with adaptive optics
provides a much more detailed and deep view than those available in previous
studies. The deep imaging reveals faint members of the cluster, probably
including massive brown dwarfs. This proves that, unlike suggested by Comerón
et al.(2005), the cluster does contain low-mass stars. However, such members
are present in numbers far below those expected when the bright end of the
K-band luminosity function is extrapolated to fainter magnitudes (see e.g.
Muench et al. 2002). This is interpreted as evidence for a top-heavy initial
mass function, perhaps related to special circumstances that triggered star
formation in the cluster (Sect. 6). The revised mass of the cluster based on
these new observations is estimated to be $\sim 370$ M⊙. Comerón & Schneider
(2007) also report the existence of a point-like Class I source detected only
at 3.8 $\mu$m, as well as by Spitzer, projected on the border between the HII
region and its surrounding molecular cloud.

##  4\.  NGC 6204 and NGC 6167

NGC 6204 is an open cluster about 2 degrees northeast of NGC 6193. Whiteoak
(1963) identified 78 possible members. He noted a wide variation in
extinction, which prevented a good distance estimate. Hogg (1965) identified
this clump as two distinct clusters. One cluster is still known as NGC 6204
and is estimated to be at 1180 $\pm 50$ pc. The stars in the other cluster,
Hogg 22, are about twice as distant and hence not part of Ara OB1a or Ara
OB1b. Forbes & Short (1996) performed photoelectric photometry on 36 member
stars of NGC 6204 and find an age of about 125 Myr. This is much older than
the other clusters in this region. Thus, these stars are probably not
associated unless winds or shocks from extinct stars in this cluster triggered
all that followed. The age difference between NGC 6204 and NGC 6193/RCW108–IR
seems to safely exclude any role of the cluster in triggering star formation.
Even if the velocity difference between the cluster and NGC 6193 were just a
fraction of the typical velocity dispersion in a star forming complex, that
would already imply that NGC 6204 formed far from its current location and
that it was benign (with no stars with strong winds or able to explode as
supernovae) by the time it got close to NGC 6193 or RCW 108–IR. Nonetheless,
Waldhausen et al. (1999) argue that NGC 6204 is associated with NGC 6193
because their polarization angles are close (about 15o).

NGC 6167 about 1 degree southwest of NGC 6193 was also poorly defined in the
original work by Whiteoak (1963) due to highly variable absorption. Brück &
Smyth (1967) identified this as a real cluster at 1200 $\pm 200$pc and 40 Myr
old. Moffat & Vogt (1975) determined a distance of 600 pc, however their
argument against the 1.2 kpc distance rests on a single, perhaps foreground,
star. Rizzo & Bajaja (1994) assert the age to be closer to 10 Myr. Waldhausen
et al. (1999) suggest that this is two overlain clusters since there appear to
be two polarizations (offset by 15o).

##  5\. IRAS 16379–4856 and IRAS 16348–4849

Figure 5.: $Spitzer$/IRAC image of RCW 108–IR (IRAC bands 1, 2, 3 and 4 are
blue, yellow, orange and red respectively.) Data are from a $Spitzer$ early
release observation – Melnick-PI. The field of view is about 22′ on a side,
corresponding to about 10 pc on a side. The red is dominated by PAH emission
that fills much of the space and shows a deficit of stars in Figure 3. There
are slight enhancements in the PAH emission on the southeastern and western
portions of the field associated with IRAS 16379–4856 & IRAS 16348–4849. Most
of the stars with optically thick disks are either within the core PAH
emission or along the eastern edge.

The two IRAS/CO peaks closest to RCW 108–IR are briefly discussed here. To the
east is a small diffuse optical nebula near IRAS 16379–4856. Wolk et al.
(2008) detect a complex of bright unresolved X–ray emission coincident with
this region. A search of the 2MASS catalog for K-band excess sources in this
region reveals the strongest concentration of such sources to be associated
with this emission, at least 25 sources with K-band excesses in a 2′ $\times$
1′ region. These sources all lie on the northwestern portion of a circular
ring of 8 $\mu$m emission about 10′ in diameter. There are two 21 $\mu$m MSX
point sources in the region. One centered among the 25 K-band excess sources,
the other located near the inner edge of the ring coincident with the centroid
of the IRAS source. This cloud is coincident with region H of the CO survey by
Arnal et al. (2003). It is one of three CO clumps coincident with an IRAS
source (RCW 108–IR and IRAS 16348–4849 - see below – are the other two). They
estimate the mass of the cloud to be 5800 M⊙, second only to the core region
of RCW 108–IR.

MSX data indicate a mid-IR cloud about 10′ west of RCW 108–IR and coincident
with IRAS 16348–4849 with a peak 8 $\mu$m flux of about 20% that of RCW 108.
The overall extent of the cloud is about 10′ on a side. Tendrils of warm dust
connecting the cloud and RCW 108 are evident in the MSX and $Spitzer$ images.
Two sources from the recent 12CO observations by Arnal et al. (2005) lie
within this dark cloud. Five B and A stars have been identified in this cloud;
the most massive is the B2IV star HD 149658. 2MASS data show about 20 stars in
the cloud with K${}_{S}<$ 15 and K-band excesses indicative of disks. Most of
these stars lie within a circular region of 20 $\mu$m emission at about 1% the
level of RCW 108–IR (Figure 5).

These two regions, like RCW 108–IR, lie between NGC 6193 and the putative
center of expansion. IRAS 16379–4856 is almost the same projected distance as
RCW 108 from HD 150135/6 at the center of NGC 6193. So, the nominal cluster
near IRAS 16379–4856 appears to be a sibling of RCW 108. Likewise formed by
photoionization from the OB stars in NGC 6193 striking a cloud which appears
to have been swept out by an event related to NGC 6167. The stars near IRAS
16357–4832 appear to compose a sibling cluster as well. The 8 $\mu$m
morphology suggests that it was torn off of RCW 108 itself. This cluster is
intermediate in mass between the former two. Additional bright CO and 8 $\mu$m
emission is visible for a degree north of RCW 108. While the details as
described here are speculative at this point, this is clearly a dynamic and
active association.

##  6\. Triggered Star Formation

It is the morphology of the region that makes Ara OB1a fascinating. NGC 6193
appears in relation to NGC 6188/IRAS 16362–4845 in much the same way that the
$\sigma$ Orionis cluster (Walter et al. 1997, see also chapter by Walter et
al.) relates to the Horsehead nebula. The emission in the open cluster is
radiation bounded by the molecular cloud containing the embedded cluster RCW
108–IR, and is spectacularly revealed in the figures as a clear ridge between
them (NGC 6188; Figure 1). This appears as one of the clearest examples of
triggered star formation in the sky. But which mechanism and how it works is
still an open issue.

Herbst & Havlen (1977) originally put forward the hypothesis that Ara OB1a was
an example of triggered star formation. Their favored model was that the
trigger was a supernova event. The $prima-facie$ evidence for this was the
circular shape of the dust ring upon which RCW 108–IR sits as a “diamond.”
Straw et al. (1987) countered with the fact that RCW 108–IR is surrounded by
some more evolved objects and point to the bright rim NGC 6188 as evidence
that photoionization is very strong in the region. Thus, they favor the model
originally detailed by Elmegreen & Lada (1977) in which successive sites of
star formation are triggered by photoionization from the previous generation.

Radio observations have much to contribute to this portion of the discussion
since the radio data trace the gas and dust before the onset of protostellar
collapse. Phillips et al. (1986) mapped the region in CO and find a CO cloud
centered 2–3′ southeast of the bright stars of NGC 6193. They hypothesize that
the cloud is behind NGC 6193 and is being compressed by the winds of the OB
stars. However, there is no evidence for embedded sources within this cloud.
The fragmented structure of the cloud supports its interpretation as the
heavily eroded remnant of a now dispersed, larger cloud, possibly the
progenitor of NGC 6193 (Comerón et al. 2005).

Arnal et al. (1987) find a connection between the stellar population of Ara
OB1a and an expanding neutral hydrogen ring centered near NGC 6167. The
connection is that they derive the same distance to the gas as has been
independently determined for the stars of $\approx 1400pc$. At that distance
the HI ring is about 30 pc in radius and 20 pc in width expanding at 10 km
s-1. Thus, Arnal et al. derive an upper age of this HI ring of 2$\times
10^{6}$ yr and an age closer to 1 Myr assuming a wind blown bubble (Weaver et
al. 1977). While this is somewhat younger than current age estimates of NGC
6193, the age of NGC 6193 is based on very few stars. If the formation of NGC
6193 were the result of a supernova in NGC 6167 the problem of the age is more
severe and in the opposite sense as NGC 6193 should be quite young, $\sim 0.1$
Myr, in this case (Straw et al. 1987).

Figure 6.: A wide view of Ara OB1a. The base image is a DSS2 red plate about 2
degrees on a side. The top left quadrant is the region shown in Figure 1. The
clusters NGC 6193 and NGC 6167 are indicated as well as RCW 108 and several
IRAS point sources. The three IRAS sources nearest NGC 6193 all have 8 $\mu$m
counterparts in MSX data. IRAS 60 $\mu$m contours of 300, 450, 600, 1200 and
2400 Jy/Sr are indicated. The large circle is used to connect the various
features. There is a chain of CO clouds following along the eastern side of
the circle.

Rizzo & Bajaja (1994) follow up on the Arnal et al. finding with a more
detailed HI study. They confirm the initial result. In their model, the 10 Myr
stars of NGC 6167 formed the HI shell. This triggered the 3 Myr stars in NGC
6193. In their model the HI shell continues on to trigger RCW 108–IR. However,
RCW 108–IR is misplaced in their Figure 6. It seems more likely that
photoionization from the stars in NGC 6193 is triggering RCW 108 in the
reverse direction relative to the outward propagating HI ring. Supporting
evidence for external triggering in the clouds that host RCW 108–IR is given
by Comerón et al. (2005) who found the cluster members they could identify are
not symmetrically distributed throughout the cloud, but clumped along the
ionization front as well as the core.

At a smaller scale, Comerón & Schneider (2007) consider that their findings on
the stellar population of RCW 108-IR provide evidence that its embedded
cluster formed as a result of external triggering. In particular, a top-heavy
initial mass function like that derived by those authors was suggested by
Elmegreen & Lada (1977) as a possible byproduct of sequential star formation.
This may be a consequence of the warm temperature of the gas in the shocked
layer located ahead of the ionization front driven by the hot stars in NGC
6193, as well as of turbulence induced by instabilities in that layer and of
coalescence of unstable fragments. Moreover, Comerón & Schneider (2007) also
point out that the spatial distribution of massive stars in RCW 108-IR differs
from that commonly found in other clusters containing high-mass members, which
tend to concentrate towards the cluster center as a result of the formation
conditions and early dynamical evolution (Bonnell et al. 2007). This is not
expected for externally triggered star formation, and indeed is not observed
in RCW 108-IR.

##  7\. The Future of Ara OB 1a

Will more star formation take place in Ara OB1a? We believe this is the case
and that it is already underway. A recent CO survey of the region finds over a
dozen CO clouds in the region covering an area almost 3 degrees on a side
(Arnal et al. 2003). While several groups of clouds are kinematically
coherent, there is a spread in velocities of almost 20 km s-1. Three of the
concentrations are adjacent to the bright rim separating RCW 108 and NGC 6193.
IRAS data (see Figure 6) show three concentrations of cool dust in the
vicinity of RCW 108–IR spaced about 5 pc apart. These concentrations are
bright from 8 $\mu$m to 100 $\mu$m. The brightest of these is RCW 108–IR
itself. The IRAS peaks are coincident with 13CO peaks found by Yamaguchi et
al. (1999). The 13CO peaks then make a chain to the south. Arnal et al. (2003)
identify five separate 12CO features in this cloud. The chain extends almost
1.5o south of RCW 108–IR. In fact, the CO peaks taken in concert with the 5
IRAS peaks (including IRAS 16379–4856 and IRAS 16348–4849 adjacent to RCW
108–IR) cover almost exactly 180 degrees of arc in a near perfectly circular
alignment 1.03o from $l$=335:29:12 $b$=$-$1:42:23, about 8 pc from the center
of NGC 6167. More specifically, Comerón et al. (2005) estimate a current star
formation efficiency in the main cloud hosting IRAS 16362–4845/RCW 108–IR to
be well below 10%, suggesting that most of star formation in the cloud still
has to take place. They also speculate that its future evolution may lead to a
complex similar to the Orion Nebula Cluster, with the aggregate of IRAS
16362–4845 being the equivalent to the Trapezium cluster.

Acknowledgments. We acknowledge many useful comments from the anonymous
referee, one of which was worked directly into the text. We gratefully
acknowledge the financial support of NASA grant GO4-5013X (Chandra) and from
NASA contract NAS8-39073 (CXC). Archival data was obtained from the Two Micron
All Sky Survey (2MASS), a joint project of the University of Massachusetts and
the Infrared Processing and Analysis Center/California Institute of
Technology, funded by the National Aeronautics and Space Administration and
the National Science Foundation and additional data were obtained with the
Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory,
California Institute of Technology under a contract with NASA. This research
has made use of the SIMBAD database, operated at CDS, Strasbourg, France, and
of NASA’s Astrophysics Data System Bibliographic Services.

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# Estimating the scale parameters of several exponential distributions under
order restriction

Suchandan Kayala, Lakshmi Kanta Patrab111<EMAIL_ADDRESS>
aDepartment of Mathematics, National Institute of Technology Rourkela
bDepartment of Mathematics, Indian Institute of Technology Bhilai

###### Abstract

In the present work, we have investigated the problem of estimating parameters
of several exponential distributions with ordered scale parameters under the
linex loss function. We have considered estimating ordered scale parameters
when the location parameters are known and unknown. For every case, we
consider a class of equivariant estimators, and sufficient condition is
obtained under which this class of estimators improves upon the usual
estimator. Using this result, we have shown that the restricted maximum
likelihood estimator is inadmissible. Finally, for every case, we conduct a
simulation study to compare the risk performance of the proposed estimators.

Keywords: Decision theory, Best affine equivatiant estimator; Linex loss;
Inadmissibility; Restricted maximum likelihood estimator.

##  1 Introduction

Researchers are often confronted with the problem of estimating parameters
when they are ordered in many experimental situations, such as industrial,
agriculture, clinical trials, and economic experiments. For example, it is
natural to assume that the salary of a highly-skilled employee is greater than
that of a low-skilled employee in a company. The average yield of a crop after
using a certain type of fertilizer is expected to be more than that when no
fertilizer is used. Because of various application during the last sixty
years, the problem of estimating restricted parameters have drawn considerable
attention from various researchers. For a good review of the problem of
estimation under order restriction on the parameters, we refer to Barlow et
al. (1972), Silvapulle and Sen (2004) and van Eeden (2006).

Let $(X_{i1},\ldots,X_{in_{i}})$ be a random sample drawn from the $i$-th
population $\Pi_{i},$ $i=1,\ldots,k~{}(k\geq 2)$. The probability density
function of the $i$-th population $\Pi_{i}$ is given by

$\displaystyle
f_{i}(x;\mu_{i},\sigma_{i})=\left\\{\begin{array}[]{ll}\frac{1}{\sigma_{i}}~{}\exp{\Big{(}-\frac{x-\mu_{i}}{\sigma_{i}}\Big{)}},&\textrm{if
$x>\mu_{i},$}\\\ 0,&\textrm{otherwise,}\end{array}\right.$ (1.3)

where $-\infty<\mu_{i}<\infty$ and $\sigma_{i}>0.$ This paper is concerned
with the problem of estimating the parameters of $k~{}(k\geq 2)$ exponential
populations with ordered scale parameters. To increase system reliability, it
is required to make some changes in different stages to correct the design
deficiencies. Suppose there are $k$ stages in which the changes are made.
Therefore, it is expected that reliability of the system at each stage
increases. Let $\lambda_{i}$ be a measure of the reliability at the $i$-th
stage, $i=1,\ldots,k.$ Therefore, it is worthwhile to assume that
$\lambda_{1}\leq\ldots\leq\lambda_{2}.$ Most of the work on ordered parameter
estimation for exponential populations has been done with respect to the
squared error loss function. For some reference, in this direction we refer to
Kaur and Singh (1991), Vijayasree and Singh (1991, 1993), Kushary et al.
(1989), Jin and Pal (1991), Pal and Kushary (1992) Vijayasree et al. (1995),
Jana et al. (2016) and Petropoulos (2017). We will mainly focus on point
estimation, and we will particularly emphasize on finding estimators which
dominate classical estimators such as maximum likelihood estimator, best
affine equivariant estimator, and best scale equivariant estimator in the
unrestricted problem with respect to linex loss function. The linex loss
function is given by

$\displaystyle
L(\triangle)=p^{\prime}[\exp\\{p\triangle\\}-p\triangle-1],~{}\triangle=\delta-\theta,~{}p\neq
0,~{}p^{\prime}>0.$ (1.4)

Here $p$ and $p^{\prime}$ are shape and scale parameters, respectively.
Throughout the paper, we assume $p^{\prime}=1.$ Note that when $|p|$ tends to
$0,$ then $(1.2)$ reduces to the squared error loss function. For a detailed
discussion on linex loss function, we refer to Zellner (1986).

The rest of the paper is organized as follows. In Section 2 we have considered
the estimation of ordered scale parameters when location parameters are known.
We proved the inadmissibility of the best scale equivariant estimator and
restricted maximum likelihood estimator. Improved estimation of ordered scale
parameters with unknown location parameters is studied in Section 3. Finally,
for every case, we have compared the relative risk performance of the proposed
improved estimators.

##  2 Estimating ordered scale parameters when location parameters are known

In this section, we consider the estimation of ordered scale parameters of
$k\geq 2$ exponential distributions when the location parameters are known. So
without loss of generality, we assume $\mu_{i}=0,~{}i=1,2,\ldots,k.$ In this
section, we consider estimation of $\sigma_{i}$’s subject to the order
restriction $\sigma_{1}\leq\sigma_{2}\leq\ldots\leq\sigma_{k}.$ The loss
function is taken to be

$\displaystyle
L(\sigma_{i},\delta_{i})=\exp\left\\{p\left(\frac{\delta_{i}-\sigma_{i}}{\sigma_{i}}\right)\right\\}-p\left(\frac{\delta_{i}-\sigma_{i}}{\sigma_{i}}\right)-1,~{}p\neq
0.$ (2.1)

Denote $S_{i}=\sum_{j=1}^{n_{i}}X_{ij}.$ Then
$\underline{S}=(S_{1},S_{2},\ldots,S_{k})$ is a complete and sufficient
statistic for $\underline{\sigma}=(\sigma_{1},\sigma_{2},\ldots,\sigma_{k}).$
Further, $S_{1},S_{2},\ldots,S_{k}$ are independently distributed, where
$S_{i}$ follows gamma distribution with shape parameter $n_{i}$ and scale
parameter $\sigma_{i}.$ For convenience, we denote $S_{i}\sim
Gamma(n_{i},\sigma_{i}),$ $i=1,2,\ldots,k.$ The probability density function
of $S_{i}$ is given by

$\displaystyle
f(x;n_{i},\sigma_{i})=\frac{1}{\sigma_{i}^{n_{i}}\Gamma(n_{i})}e^{-\frac{x}{\sigma_{i}}}x^{n_{i}-1},~{}x>0,$
(2.2)

where $\Gamma(.)$ denotes usual gamma function. Now consider the scale group
of transformations
$G_{a_{i}}=\\{g_{a_{i}}:g_{a_{i}}(x_{ij})=a_{i}x_{ij},~{}a_{i}>0,~{}j=1,2,\ldots,n_{i},i=1,2,\ldots,k\\}.$
Under this transformation, $S_{i}\rightarrow a_{i}S_{i}$ and
$\sigma_{i}\rightarrow a_{i}\sigma_{i}.$ A standard argument shows that any
scale equivariant estimator is of the form

$\displaystyle\delta_{i_{c}}(\underline{S})=cS_{i}$ (2.3)

for any constant $c.$ The following theorem provides the best scale
equivariant estimator of $\sigma_{i},$ when there is no restriction on
$\sigma_{i},$ $i=1,2,\ldots,k.$ The proof of this theorem is omitted for the
sake of brevity.

###### Theorem 2.1

Under the linex loss function $(2.1),$ the (unrestricted) best scale
equivariant estimator of $\sigma_{i}$ is
$\delta_{i_{c_{0i}}}(\underline{S})=c_{0i}S_{i},$ where
$c_{0i}=p^{-1}(1-e^{-p/(n_{i}+1)}),$ $i=1,2,\ldots,k.$

###  2.1 Improving upon the (unrestricted) best scale equivariant estimator

In this subsection, we take a subgroup of the scale group of transformations
$G_{a_{i}}$. Consider
$G_{a}=\\{g_{a}:g_{a}(x_{ij})=ax_{ij},~{}a>0,~{}j=1,2,\ldots,n_{i},~{}i=1,2,\ldots,k\\}.$
Then the form of the scale equivariant estimator is

$\displaystyle{\delta_{\psi}}_{i}(\underline{S})$ $\displaystyle=$
$\displaystyle
S_{i}\psi_{i}\left(\frac{S_{1}}{S_{i}},\frac{S_{2}}{S_{i}},\ldots,\frac{S_{i-1}}{S_{i}},\frac{S_{i+1}}{S_{i}},\ldots,\frac{S_{k}}{S_{i}}\right)$
(2.4) $\displaystyle=$ $\displaystyle
S_{i}\psi_{i}\left(\underline{U}\right),$

where $\underline{U}=(U_{1},U_{2},\ldots,U_{i-1},U_{i+1},\ldots,U_{k}),$
$U_{j}=S_{j}/S_{i}$ for $j\neq i,~{}j=1,2,\ldots,k$ and $U_{i}=S_{i}.$ The
risk function of the estimators of the form $(2.4)$ is

$\displaystyle R(\sigma_{i},{\delta_{\psi}}_{i})$ $\displaystyle=$
$\displaystyle
E\left[\exp\left\\{\displaystyle{p\left(\frac{S_{i}\psi_{i}(\underline{U})}{\sigma_{i}}-1\right)}\right\\}-\displaystyle
p\left(\frac{S_{i}\psi_{i}(\underline{U})}{\sigma_{i}}-1\right)-1\right]$
(2.5) $\displaystyle=$ $\displaystyle
E^{\underline{U}}R_{1}\left(\sigma_{i},\underline{U},\delta_{\psi_{i}}\right),$

where $R_{1}\left(\sigma_{i},\underline{U},{\delta_{\psi}}_{i}\right)$
represents the conditional risk function of ${\delta_{\psi}}_{i}$ given
$\underline{U}=\underline{u},$ where
$\underline{u}=(u_{1},u_{2},\ldots,u_{k})$. It is given by

$\displaystyle
R_{1}(\sigma_{i},\underline{U},{\delta_{\psi}}_{i})=E\left[\left(\exp\left\\{\displaystyle{p\left(\frac{S_{i}\psi_{i}(\underline{U})}{\sigma_{i}}-1\right)}\right\\}-\displaystyle
p\left(\frac{S_{i}\psi_{i}(\underline{U})}{\sigma_{i}}-1\right)-1\right)\Big{|}\underline{U}=\underline{u}\right].$
(2.6)

Now, given $\underline{U}=\underline{u}$, the value of
$\psi_{i}(\underline{u})$ minimizing $(2.6)$ can be obtained from the
following equation:

$\displaystyle
E\left[U_{i}\exp\left\\{\displaystyle\frac{pU_{i}\psi_{i}(\underline{U})}{\sigma_{i}}\right\\}\Big{|}\underline{U}=\underline{u}\right]=\exp\\{p\\}E\left(U_{i}|\underline{U}=\underline{u}\right).$
(2.7)

To solve the above equation, we need to derive conditional distribution of
$U_{i}$ given $\underline{U}=\underline{u}.$ After some calculations, it can
be shown that the distribution of $U_{i}|\underline{U}=\underline{u}$ is gamma
with shape parameter $\sum_{j=1}^{k}n_{j}$ and scale parameter
$\left(\sigma_{i}^{-1}+\sum_{j(\neq i)=1}^{k}(u_{j}/\sigma_{j})\right)^{-1}.$
Thus, after simplifications, from $(2.7)$ we obtain

$\displaystyle\psi_{i}(\underline{u},\underline{\sigma})=\frac{1}{p}\left(1-\exp\left\\{-\frac{p}{n+1}\right\\}\right)\left(1+\sum_{j(\neq
i)=1}^{k}\frac{u_{j}\sigma_{i}}{\sigma_{j}}\right),$ (2.8)

where $n=\sum_{j=1}^{k}n_{j}$. To apply Brewster and Zidek (1974) technique,
it is required to find out supremum and infimum of
$\psi_{i}(\underline{u},\underline{\sigma})$ given by $(2.8)$. For $p\neq 0,$
under the constraint $\sigma_{1}\leq\sigma_{2}\leq\ldots\leq\sigma_{k}$, we
obtain

$\displaystyle\sup_{\underline{\sigma}}\psi_{i}(\underline{u},\underline{\sigma})$
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\displaystyle\frac{1}{p}\left(1-\exp\left\\{-\displaystyle\frac{p}{n+1}\right\\}\right)\left(1+\sum_{j=2}^{k}u_{j}\right),&\textrm{if
$i=1$}\\\ +\infty,&\textrm{if $i=2,\ldots,k.$}\end{array}\right.$ (2.11)
$\displaystyle=$ $\displaystyle{\psi_{i}}^{*}(\underline{u}),~{}\mbox{(say)}$
(2.12)

and

$\displaystyle\inf_{\underline{\sigma}}\psi_{i}(\underline{u},\underline{\sigma})$
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\displaystyle\frac{1}{p}\left(1-\exp\left\\{-\displaystyle\frac{p}{n+1}\right\\}\right),&\textrm{if
$i=1$}\\\
\displaystyle\frac{1}{p}\left(1-\exp\left\\{-\displaystyle\frac{p}{n+1}\right\\}\right)\left(1+\sum_{j=1}^{i-1}u_{j}\right),&\textrm{if
$i=2,\ldots,k$}\end{array}\right.$ (2.15) $\displaystyle=$
$\displaystyle{\psi_{i}}_{*}(\underline{u}),~{}\mbox{(say)}.$ (2.16)

Now, we define a function $\psi_{i0}(\underline{u})$ as follows:

$\displaystyle\psi_{i0}(\underline{u})=\left\\{\begin{array}[]{ll}{\psi_{i}}_{*}(\underline{u}),&\textrm{if
$\psi_{i}(\underline{u})\leq{\psi_{i}}_{*}(\underline{u})$}\\\
{\psi_{i}}(\underline{u}),&\textrm{if
${\psi_{i}}_{*}(\underline{u})<\psi_{i}(\underline{u})<{\psi_{i}}^{*}(\underline{u})$}\\\
{\psi_{i}}^{*}(\underline{u}),&\textrm{if
$\psi_{i}(\underline{u})\geq{\psi_{i}}^{*}(\underline{u})$}.\end{array}\right.$
(2.20)

Thus, we have the following theorem which provides sufficient conditions for
improving the estimators of the form $(2.4)$.

###### Theorem 2.2

Let $\delta_{\psi_{i}}(\underline{U})=S_{i}\psi_{i}\left(\underline{U}\right)$
be a scale equivariant estimator given by $(2.4)$ and $\psi_{i0}$ be defined
in $(2.11).$ If there exists some $\underline{\sigma}$ such that
$P_{\underline{\sigma}}\left(\psi_{i0}(\underline{U})\neq\psi_{i}(\underline{U})\right)>0,$
then under the order restriction
$\sigma_{1}\leq\sigma_{2}\leq\ldots\leq\sigma_{k}$, with respect to linex loss
function $(2.1),$ the estimator $\delta_{\psi_{i0}}(\underline{U})$ improves
upon $\delta_{\psi_{i}}(\underline{U}).$

The following corollary is immediate from the Theorem 2.2. We denote
$d_{1}=p^{-1}(1-\exp\\{-p/(n_{1}+n_{2}+1)\\})$ and
$d_{2}=p^{-1}(1-\exp\\{-p/(n+1)\\})$.

###### Corollary 2.3

Suppose it is known in prior that
$\sigma_{1}\leq\sigma_{2}\leq\ldots\leq\sigma_{k}$. Then with respect to linex
loss function given by $(2.1),$
$(i)$ the unrestricted best scale equivariant estimator of $\sigma_{1}$ is
inadmissible and dominated by the estimator given by

$\displaystyle\delta_{\psi_{10}}(\underline{U})=\left\\{\begin{array}[]{ll}d_{1}S_{1},&\textrm{if
$c_{01}<d_{1}$}\\\ c_{01}S_{1},&\textrm{if $d_{1}\leq c_{01}\leq
d_{1}(1+U_{2})$}\\\ d_{1}(1+U_{2})S_{1},&\textrm{if
$c_{01}>d_{1}(1+U_{2})$};\end{array}\right.$ (2.24)

$(ii)$ the unrestricted best scale equivariant estimator of $\sigma_{i},$
$i=2,\dots,k$ is inadmissible and dominated by the estimator given by

$\displaystyle\delta_{\psi_{i0}}(\underline{U})=\left\\{\begin{array}[]{ll}d_{2}\left(1+\sum_{j=1}^{i-1}U_{j}\right)S_{i},&\textrm{if
$c_{0i}<d_{2}\left(1+\sum_{j=1}^{i-1}U_{j}\right)$}\\\
c_{0i}S_{i},&\textrm{otherwise.}\end{array}\right.$ (2.27)

Next, we consider the maximum likelihood estimator of $\sigma_{i},$
$i=1,2,\ldots,k$ under the restriction
$\sigma_{1}\leq\sigma_{2}\dots\leq\sigma_{k},$ which is given by

$\displaystyle\delta_{iRML}=\max_{s\leq i}\min_{t\geq
i}\left\\{\frac{\sum_{j=s}^{t}S_{j}}{\sum_{j=s}^{t}n_{j}}\right\\}.$ (2.28)

In particular, consider $k=2.$ Then the maximum likelihood estimators of
$\sigma_{1}$ and $\sigma_{2}$ under the restriction $\sigma_{1}\leq\sigma_{2}$
reduces to

$\displaystyle\delta_{1RML}=\min\left\\{\frac{Y_{1}}{n_{1}},\frac{Y_{1}+Y_{2}}{n_{1}+n_{2}}\right\\}$
(2.29)

and

$\displaystyle\delta_{2RML}=\max\left\\{\frac{Y_{2}}{n_{2}},\frac{Y_{1}+Y_{2}}{n_{1}+n_{2}}\right\\},$
(2.30)

respectively. Note that the above estimators are scale equivariant of the form
$(2.4)$. Thus as an application of the Theorem $2.2$, we get the following
corollary.

###### Corollary 2.4

Let $\sigma_{1}\leq\sigma_{2}.$ Then for $k=2,$ under the linex loss function
$(2.1)$, the estimators

$\displaystyle\delta_{1IRML}(\underline{U})=\left\\{\begin{array}[]{ll}d_{1}S_{1},&\textrm{if
$\phi_{01}<d_{1}$}\\\ \phi_{01}S_{1},&\textrm{if $d_{1}\leq\phi_{01}\leq
d_{1}(1+U_{1})$}\\\ d_{2}S_{1},&\textrm{if
$d_{2}(1+U_{2})<\phi_{01}$};\end{array}\right.$ (2.34)

and

$\displaystyle\delta_{2IRML}(\underline{U})=\left\\{\begin{array}[]{ll}d_{1}(1+U_{1})S_{2},&\textrm{if
$\phi_{02}<d_{1}(1+U_{1})$}\\\
\phi_{02}S_{2},&\textrm{otherwise,}\end{array}\right.$ (2.37)

where
$\phi_{01}=\min\left\\{\frac{1}{n_{1}},\frac{1+U_{2}}{n_{1}+n_{2}}\right\\}$
and
$\phi_{02}=\max\left\\{\frac{1}{n_{2}},\frac{1+U_{1}}{n_{1}+n_{2}}\right\\}$
have uniformly smaller risk than the restricted maximum likelihood estimators
of $\sigma_{1}$ and $\sigma_{2}$, respectively.

###  2.2 A numerical study

In this subsection, we carry out a simulation study to compare the risk
performance of the proposed estimators numerically. We have generated $50,000$
random samples from two independent exponential distributions using $R$
software for the simulation study. The percentage relative risk improvement
(PRRI) of an estimator $\delta_{1}$ over another estimator $\delta_{2}$ is
defined as

$\displaystyle
PRRI(\delta_{1},\delta_{2})=\left(\frac{Risk(\delta_{2})-Risk(\delta_{1})}{Risk(\delta_{2})}\right)\times
100\%$ (2.38)

In the following tables, we present PRRI of various estimators for different
sample sizes and different values of $p$. We have considered sample sizes
$(n_{1},n_{2})$ as $(3,5),$ $(10,15),$ $(8,5)$ and $(15,10).$ The values of
$p$ are taken to be $0.5$ and $-0.5$. The following points can be observed
from Table $1$ and Table $2$.

  1. 1.

With respect to the BSEE, the PRRI of the improved BSEE decreases when sample
sizes $(n_{1},n_{2})$ increases with the constraint $n_{1}<n_{2}$. Similar
observation is noticed for $n_{1}>n_{2}$ case.

  2. 2.

For fixed $(n_{1},n_{2})$, the PRRI of the improved BSEE (with respect to the
BSEE) increases if $(\sigma_{1},\sigma_{2})$ increases. Further, if we
increase the values of $\sigma_{2}$ by keeping $\sigma_{1}$ fixed, then the
PRRI of the improved BSEE (with respect to the BSEE) decreases. This
observation ensures that after some distance, the risk function of the
improved BSEE coincides with that of the BSEE.

Table 1: The PRRI of various estimators for $k=2$ and $p=0.5.$ The 3rd, 4th
and 5th columns respectively present the PRRI of improved BSEE, restricted MLE
and improved restricted MLE over the BSEE whereas the 6th, 7th and 8th columns
present the PRRI of improved BSEE, BSEE and improved restricted MLE over the
restricted MLE.

$(n_{1},n_{2})$ | $(\sigma_{1},\sigma_{2})$ | $(\delta_{\psi_{10}},\delta_{\psi_{20}})$ | $(\delta_{1RML},\delta_{2RML})$ | $(\delta_{1IRML},\delta_{2IRML})$ | $(\delta_{\psi_{10}},\delta_{\psi_{20}})$ | $(\delta_{1c_{01}},\delta_{2c_{02}})$ | $(\delta_{1IRML},\delta_{2IRML})$
---|---|---|---|---|---|---|---
(3,5) | (0.2,0.5) | (3.89,6.21) | (18.26,-28.32) | (17.27,-27.01) | (-14.39,29.13) | (-18.26,24.86) | (1.26,0.89)
| (0.2,1.0) | (0.37,0.58) | (-32.19,-33.29) | (-32.43,-33.22) | (24.63,25.41) | (24.35,24.97) | (0.23,0.05)
| (0.2,1.5) | (0.13,0.1) | (-51.64,-33.30) | (-51.84,-33.28) | (34.14,25.05) | (34.05,24.97) | (0.13,0.01)
| (0.5,1.0) | (6.19,9.78) | (25.57,-32.73) | (24.89,-30.47) | (-26.03,32.03) | (-34.35,24.66) | (0.80,1.70)
| (0.5,1.5) | (2.24,3.39) | (4.87,-33.21) | (3.27,-32.59) | (-2.76,27.47) | (-5.12,24.92) | (1.68,0.46)
| (0.5,2.0) | (0.71,1.33) | (-16.48,-33.27) | (-17.39,-33.09) | (14.76,25.97) | (14.15,24.96) | (0.78,0.14)
| (1.0,1.5) | (9.64,16.83) | (25.53,-31.53) | (29.74,-27.38) | (-21.35,36.77) | (-34.30,23.97) | (5.65,3.15)
| (1.0,2.0) | (6.19,9.78) | (25.57,-32.73) | (24.98,-30.47) | (-26.03,32.03) | (-34.35,24.66) | (0.78,1.70)
| (1.0,2.5) | (3.74,5.67) | (16.38,-33.08) | (15.33,-31.91) | (-15.11,29.12) | (-19.59,24.86) | (1.25,0.88)
(10,15) | (0.2,0.5) | (0.65,0.76) | (14.7,-10.80) | (13.62,-10.61) | (-16.48,10.44) | (-17.24,9.75) | (1.26,0.17)
| (0.2,1.0) | (0,0.002) | (-14.97,-10.80) | (-14.87,-10.81) | (13.03,9.76) | (13.03,9.75) | (0,0)
| (0.2,1.5) | (0,0) | (-15.99,-10.81) | (-15.99,-10.80) | (13.79,9.75) | (13.8,9.75) | (0,0)
| (0.5,1.0) | (2.53,2.90) | (27.96,-10.81) | (26.28,-9.83) | (-35.29,12.37) | (-38.81,9.57) | (2.32,0.88)
| (0.5,1.5) | (0.13,0.19) | (0.86,-10.80) | (0.47,-10.77) | (-0.73,9.92) | (-0.87,9.75) | (0.38,0.034)
| (0.5,2.0) | (0.02,0.015) | (-11.58,-10.81) | (-11.64,-10.80) | (10.40,9.76) | (10.38,9.75) | (0.051,0.003)
| (1.0,1.5) | (8.66,10.72) | (13.76,-10.77) | (17.78,-6.16) | (-5.92,19.40) | (-15.96,9.72) | (4.56,4.16)
| (1.0,2.0) | (2.53,2.91) | (27.96,-10.80) | (26.29,-9.83) | (-35.29,12.37) | (-38.80,9.75) | (2.32,0.88)
| (1.0,2.5) | (0.65,0.76) | (14.71,-10.80) | (13.62,-10.61) | (-16.48,10.44) | (-17.24,9.75) | (1.26,0.17)
(8,5) | (0.2,0.5) | (2.36,6.41) | (-13.92,-33.35) | (9.52,-32.10) | (14.29,29.87) | (12.22,25.06) | (20.58,1.01)
| (0.2,1.0) | (0.167,0.42) | (-0.23,-34.23) | (2.01,-34.16) | (0.40,25.82) | (0.23,25.5) | (2.22,0.05)
| (0.2,1.5) | (0.03,0.05) | (-11.55,-34.25) | (-11.14,-34.24) | (10.38,25.55) | (10.35,25.51) | (0.36,0.01)
| (0.5,1.0) | (4.16,12.74) | (-47.64,-32.24) | (-4.29,-29.51) | (35.08,34.02) | (32.27,24.38) | (29.36,2.06)
| (0.5,1.5) | (1.37,3.38) | (0.30,-33.92) | (13.65,-33.21) | (1.07,27.85) | (-0.30,25.33) | (13.38,0.52)
| (0.5,2.0) | (0.51,1.10) | (4.32,-34.18) | (9.39,-33.97) | (-3.98,26.30) | (-4.51,25.47) | (5.30,0.16)
| (1.0,1.5) | (7.07,25.77) | (-116.0,-28.55) | (-31.36,-23.05) | (56.98,42.26) | (53.70,22.21) | (39.18,4.28)
| (1.0,2.0) | (4.16,12.74) | (-47.64,-32.24) | (-4.29,-29.51) | (35.08,34.02) | (32.27,24.38) | (29.36,2.06)
| (1.0,2.5) | (2.36,6.41) | (-13.92,-33.45) | (9.52,-32.10) | (14.29,29.87) | (12.22,25.06) | (20.58,1.01)
(15,10) | (0.2,0.5) | (0.66,1.47) | (-5.75,-15.33) | (5.73,-14.91) | (6.01,14.57) | (5.44,13.29) | (10.86,0.36)
| (0.2,1.0) | (0,0.001) | (-2.63,-15.33) | (-2.56,-15.33) | (2.56,13.29) | (2.56,13.29) | (0.065,0)
| (0.2,1.5) | (0,0) | (-9.51,-15.33) | (-9.51,-15.3) | (9.05,13.29) | (9.05,13.29) | (0,0)
| (0.5,1.0) | (2.13,5.05) | (-61.69,-15.23) | (-28.66,-13.47) | (39.47,17.60) | (38.15,13.22) | (20.42,1.53)
| (0.5,1.5) | (0.17,0.43) | (10.06,-15.33) | (14.22,-15.25) | (-11.0,13.67) | (-11.18,13.29) | (4.62,0.072)
| (0.5,2.0) | (0.004,0.03) | (5.62,-15.34) | (6.18,-15.33) | (-5.95,13.32) | (-5.96,13.29) | (0.59,0.002)
| (1.0,1.5) | (6.38,17.11) | (-195.63,-14.12) | (-99.18,-7.55) | (68.33,27.37) | (66.17,12.37) | (32.62,5.75)
| (1.0,2.0) | (2.13,5.05) | (-61.69,-15.23) | (-28.66,-13.47) | (39.47,17.61) | (38.15,13.22) | (20.42,1.52)
| (1.0,2.5) | (0.66,1.47) | (-5.75,-15.32) | (5.73,-14.91) | (6.06,14.57) | (5.44,13.29) | (10.86,0.36)

Table 2: The PRRI of various estimators for $k=2$ and $p=-0.5.$ The 3rd, 4th
and 5th columns respectively present the PRRI of improved BSEE, restricted MLE
and improved restricted MLE over the BSEE whereas the 6th, 7th and 8th columns
present the PRRI of improved BSEE, BSEE and improved restricted MLE over the
restricted MLE.

$(n_{1},n_{2})$ | $(\sigma_{1},\sigma_{2})$ | $(\delta_{\psi_{10}},\delta_{\psi_{20}})$ | $(\delta_{1RML},\delta_{2RML})$ | $(\delta_{1IRML},\delta_{2IRML})$ | $(\delta_{\psi_{10}},\delta_{\psi_{20}})$ | $(\delta_{1c_{01}},\delta_{2c_{02}})$ | $(\delta_{1IRML},\delta_{2IRML})$
---|---|---|---|---|---|---|---
(3,5) | (0.2,0.5) | (4.49,5.92) | (20.66,-10.19) | (18.62,-7.99) | (-20.38,14.63) | (-26.04,9.25) | (2.56,2.00)
| (0.2,1.0) | (0.46,0.61) | (-4.43,-10.47) | (-4.76,-10.31) | (4.68,10.04) | (4.24,9.48) | (0.31,0.15)
| (0.2,1.5) | (0.13,0.11) | (-13.15,-10.48) | (-13.35,-10.45) | (11.74,9.58) | (11.62,9.48) | (0.17,0.02)
| (0.5,1.0) | (7.28,10.13) | (24.31,-9.73) | (24.13,-5.78) | (-22.49,18.10) | (-32.12,8.87) | (0.24,3.59)
| (0.5,1.5) | (2.79,3.57) | (14,66,-10.35) | (12.77,-9.11) | (-13.9,12.62) | (-17.18,9.38) | (2.21,1.13)
| (0.5,2.0) | (1.07,1.42) | (3.39,-10.44) | (2.02,-10.03) | (-2.40,10.74) | (-3.51,9.46) | (1.42,0.37)
| (1.0,1.5) | (11.19,17.19) | (18.68,-8.24) | (25.19,1.50) | (-9.21,23.39) | (-22.97,7.61) | (8.0,6.22)
| (1.0,2.0) | (7.28,10.12) | (24.31,-9.73) | (24.13,-5.78) | (-22.49,18.10) | (-32.12,8.17) | (0.23,3.59)
| (1.0,2.5) | (4.49,5.92) | (20.66,-10.19) | (18.62,-7.99) | (-20.38,14.63) | (-26.40,9.25) | (2.56,2.00)
(10,15) | (0.2,0.5) | (0.75,0.77) | (16.79,-3.68) | (15.28,-3.35) | (-19.28,4.29) | (-20.18,3.55) | (1.81,0.32)
| (0.2,1.0) | (0,0.002) | (-4.67,-3.68) | (-4.67,-3.68) | (4.63,3.55) | (4.63,3.55) | (0.0)
| (0.2,1.5) | (0,0) | (-5.32,-3.68) | (-5.32,-3.68) | (5.05,3.55) | (5.05,3.55) | (0,0)
| (0.5,1.0) | (2.8,2.92) | (25.65,-3.68) | (24.12,-2.17) | (-30.74,6.37) | (-34.51,3.55) | (2.06,1.46)
| (0.5,1.5) | (0.18,0.19) | (6.65,-3.68) | (5.99,-3.61) | (-6.39,3.73) | (-7.12,3.55) | (0.70,0.072)
| (0.5,2.0) | (0.02,0.02) | (-2.33,-3.69) | (-2.38,-3.68) | (2.30,3.57) | (2.28,3.55) | (0.05,0.005)
| (1.0,1.5) | (9.67,10.71) | (5.51,-3.64) | (12.33,2.77) | (4.41,13.85) | (-5.83,3.51) | (7.22,6.19)
| (1.0,2.0) | (2.79,2.92) | (25.65,-3.68) | (24.12,-2.17) | (-30.75,6.37) | (-34.51,3.55) | (2.07,1.46)
| (1.0,2.5) | (0.75,0.77) | (16.79,-3.68) | (15.27,-3.35) | (-19.28,4.29) | (-20.18,3.55) | (1.81,0.32)
(8,5) | (0.2,0.5) | (2.66,6.57) | (-26.18,-9.71) | (4.18,-7.42) | (22.85,14.84) | (20.75,8.84) | (24.53,2.07)
| (0.2,1.0) | (0.22,0.44) | (4.84,-10.74) | (7.76,-10.61) | (-4.85,10.09) | (-5.08,9.69) | (3.07,0.12)
| (0.2,1.5) | (0.04,0.05) | (-1.30,-10.77) | (-0.77,-10.75) | (1.32,9.77) | (1.29,9.72) | (0.52,0.02)
| (0.5,1.0) | (4.66,12.96) | (-69.06,-8.16) | (-11.62,-3.79) | (43.60,19.53) | (40.85,7.54) | (33.97,0.04)
| (0.5,1.5) | (1.55,3.53) | (-6.13,-10.32) | (11.44,-9.08) | (7.25,12.55) | (5.78,9.35) | (16.56,1.12)
| (0.5,2.0) | (0.59,1.16) | (5.17,-10.67) | (11.82,-10.27) | (-4.83,10.69) | (-5.44,9.64) | (7.01,0.67)
| (1.0,1.5) | (7.87,25.78) | (-105.43,-3.67) | (-41.70,4.48) | (63.79,28.42) | (60.69,4.54) | (44.30,7.86)
| (1.0,2.0) | (4.65,12.96) | (-69.07,-8.17) | (-11.62,-3.79) | (43.61,19.53) | (40.85,7.54) | (33.97,4.04)
| (1.0,2.5) | (2.65,6.57) | (-26.18,-9.71) | (4.78,-7.43) | (22.86,14.84) | (20.74,8.84) | (24.54,2.07)
(15,10) | (0.2,0.5) | (0.73,1.47) | (-15.12,-4.97) | (-0.25,-4.29) | (13.77,6.14) | (13.13,4.74) | (12.91,0.05)
| (0.2,1.0) | (0,0) | (2.41,-4.99) | (2.49,-4.99) | (-2.47,4.75) | (-2.47,4.75) | (0.08,0)
| (0.2,1.5) | (0,0) | (-3.02,-4.99) | (-3.02,-4.99) | (2.94,4.75) | (2.94,4.75) | (0,0)
| (0.5,1.0) | (2.31,5.00) | (-82.77,-4.85) | (-39.81,-2.28) | (46.54,9.41) | (45.28,4.63) | (23.50,2.45)
| (0.5,1.5) | (0.02,0.40) | (6.50,-4.99) | (11.87,-4.84) | (-6.74,5.13) | (-6.94,4.75) | (5.74,0.14)
| (0.5,2.0) | (0.007,0.03) | (7.99,-4.99) | (8.72,-4.98) | (-8.68,4.77) | (-8.69,4.75) | (0.79,0.01)
| (1.0,1.5) | (6.83,16.87) | (-224.95,-3.46) | (-118.74,5.39) | (72.99,19.66) | (71.01,3.35) | (36.58,8.56)
| (1.0,2.0) | (2.30,5.00) | (-82.77,-4.85) | (-39.81,-2.82) | (46.54,9.41) | (45.28,4.63) | (23.50,2.45)
| (1.0,2.5) | (0.73,1.47) | (-15.12,-4.97) | (-0.25,-4.29) | (13.77,6.14) | (13.13,4.74) | (12.91,0.64)

##  3 Estimation of ordered scale parameters when location parameters are
unknown

Throughout the present section, we assume that $\sigma_{i}$’s and $\mu_{i}$’s,
$i=1,\ldots,k$ are unknown. Further, we assume an additional condition on the
scale parameters as $\sigma_{1}\leq\ldots\leq\sigma_{k}$. Let us denote

$X_{i(1)}=\min_{1\leq j\leq
n_{i}}X_{ij},~{}~{}T_{i}=\sum_{j=1}^{n_{i}}\left(X_{ij}-X_{i(1)}\right),$

where $i=1,\ldots,k.$ It is known that
$\left(\underline{X_{(1)}},\underline{T}\right)$ is a complete and sufficient
statistic, where $\underline{X_{(1)}}=(X_{1(1)},\ldots,X_{k(1)})$ and
$\underline{T}=(T_{1},\ldots,T_{k})$. Moreover,
$(X_{1(1)},\ldots,X_{k(1)},T_{1},\ldots,T_{k})$ are independently distributed
with $X_{i(1)}\sim E\left(\mu_{i},\frac{\sigma_{i}}{n_{i}}\right)$ and
$T_{i}\sim Gamma(n_{i}-1,\sigma_{i})$, where $i=1,\ldots,k.$

We consider the problem of estimation of the scale parameters $\sigma_{i}$
associated with the order restriction
$\sigma_{1}\leq\sigma_{2}\leq\ldots\leq\sigma_{k}$ when the location
parameters $\mu_{i}$’s are unknown. In this direction, we take linex loss
function given by $(2.1)$. Consider the affine group of transformations
$G_{a_{i},b_{i}}=\\{g_{a_{i},b_{i}}(x)=a_{i}x_{ij}+b_{i},~{}j=1,\ldots,n_{i},~{}i=1,\ldots,k\\}$.
Now under this group, the form of the affine equivariant estimators can be
obtained as

$\displaystyle\delta_{id}(\underline{X_{(1)}},\underline{T})=dT_{i},$ (3.1)

where $d$ is a constant. The following theorem provides the best affine
equivariant estimator of $\sigma_{i}$ when there is no restriction on
$\sigma_{i}$’s.

###### Theorem 3.1

Under linex loss function given by $(2.1),$ the (unrestricted) best affine
equivariant estimator of $\sigma_{i}$ is
$\delta_{id_{0i}}(\underline{X_{(1)}},\underline{T})=d_{0i}T_{i}$, where
$d_{0i}=\frac{1}{p}(1-e^{-p/n_{i}})~{}i=1,\ldots,k.$

Proof. The proof is simple and hence omitted for the sake of brevity. $\Box$

###  3.1 Improving upon the (unrestricted) best affine equivariant estimator

Consider a subgroup of $G_{a_{i},b_{i}}$ as $G_{a,b_{i}}$. Under the
transformation
$g_{a,b_{i}}=ax_{ij}+b_{i},~{}i=1,\ldots,k,~{}j=1,\ldots,n_{i},$ the form of
the estimators is

$\displaystyle\delta_{\phi_{i}}\left(\underline{X_{(1)}},\underline{T}\right)$
$\displaystyle=$ $\displaystyle
T_{i}\phi_{i}\left(\frac{T_{1}}{T_{i}},\ldots,\frac{T_{i-1}}{T_{i}},\frac{T_{i+1}}{T_{i}},\ldots,\frac{T_{k}}{T_{i}}\right)$
(3.2) $\displaystyle=$ $\displaystyle
T_{i}\phi_{i}\left(\underline{V}\right),$

where $\underline{V}=(V_{1},\ldots,V_{i-1},V_{i+1},\ldots,V_{k}),$
$V_{j}=T_{j}/T_{i}$ for $j\neq i$ and $V_{i}=T_{i}.$ The risk of the
estimators of the form $(3.2)$ is

$\displaystyle R\left(\sigma_{i},{\delta_{\phi}}_{i}\right)$ $\displaystyle=$
$\displaystyle
E\left[\exp\Big{\\{}p\Big{(}\frac{T_{i}\phi_{i}\left(\underline{V}\right)}{\sigma_{i}}-1\Big{)}\Big{\\}}-\displaystyle
p\Big{(}\frac{T_{i}\phi_{i}(\underline{V})}{\sigma_{i}}-1\Big{)}-1\right]$
(3.3) $\displaystyle=$ $\displaystyle
E^{\underline{V}}R_{1}\left(\sigma_{i},\underline{V},{\delta_{\phi_{i}}}\right),$

where $R_{1}\left(\sigma_{i},\underline{V},{\delta_{\phi}}_{i}\right)$
represents the conditional risk of ${\delta_{\phi}}_{i}$ given
$\underline{V}=\underline{v}$. It is given by

$\displaystyle
R_{1}\left(\sigma_{i},\underline{V},{\delta_{\phi}}_{i}\right)=E\left[\left(\exp\left\\{p\left(\frac{T_{i}\phi_{i}\left(\underline{V}\right)}{\sigma_{i}}-1\right)\right\\}-\displaystyle
p\left(\frac{T_{i}\phi_{i}(\underline{V})}{\sigma_{i}}-1\right)-1\right)\Big{|}\underline{V}=\underline{v}\right].$
(3.4)

Note that $R_{1}(.,.,.)$ is a convex function in $\phi_{i}(.)$. Therefore, the
choice of $\phi_{i}(.)$ which minimizes $R_{1}(.,.,.)$ given by $(3.4)$ can be
obtained from the following relation:

$\displaystyle
E\left[T_{i}\exp\left\\{\left(\frac{pT_{i}\phi_{i}\left(\underline{V}\right)}{\sigma_{i}}\right)\right\\}\Big{|}\underline{V}=\underline{v}\right]=\exp\\{p\\}E\left(T_{i}|\underline{V}=\underline{v}\right).$
(3.5)

To get $\phi_{i}\left(\underline{v}\right)$ from $(3.5)$, it is required to
derive the distribution of
$T_{i}|\underline{V}=\underline{v},~{}i=1,\ldots,k.$ It can be shown that
$T_{i}|\underline{V}=\underline{v}\sim
Gamma\left(n-k,(\frac{1}{\sigma_{i}}+\sum_{j(\neq
i)=1}^{k}\frac{v_{j}}{\sigma_{j}})^{-1}\right)$. Thus after some
simplification, from $(3.5)$ we obtain

$\displaystyle\phi_{i}\left(\underline{v},\underline{\sigma}\right)=\frac{1}{p}\left(1-\exp\left\\{-\displaystyle\frac{p}{n-k+1}\right\\}\right)\left(1+\sum_{j(\neq
i)=1}^{k}\frac{v_{j}\sigma_{i}}{\sigma_{j}}\right).$ (3.6)

Next, we obtain the infimum and supremum of
$\phi_{i}\left(\underline{v},\underline{\sigma}\right)$ given by $(3.6)$ with
respect to $\underline{\sigma}$. Under the order restriction
$\sigma_{1}\leq\ldots\leq\sigma_{k}$ and $p\neq 0,$ the infimum and supremum
of $\phi_{i}\left(\underline{v},\underline{\sigma}\right)$ can be obtained as

$\displaystyle\inf_{\underline{\sigma}}\phi_{i}\left(\underline{v},\underline{\sigma}\right)$
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\displaystyle\frac{1}{p}\left(1-\exp\left\\{-\displaystyle\frac{p}{n-k+1}\right\\}\right),&\textrm{if
$i=1$}\\\
\displaystyle\frac{1}{p}\left(1-\exp\left\\{-\displaystyle\frac{p}{n-k+1}\right\\}\right)\left(1+\sum_{j=1}^{i-1}v_{j}\right),&\textrm{if
$i=2,\ldots,k$}\end{array}\right.$ (3.9) $\displaystyle=$
$\displaystyle{\phi_{i}}_{*}\left(\underline{v}\right),~{}\mbox{(say)}$ (3.10)

and

$\displaystyle\sup_{\underline{\sigma}}\phi_{i}\left(\underline{v},\underline{\sigma}\right)$
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}\displaystyle\frac{1}{p}\left(1-\exp\left\\{-\displaystyle\frac{p}{n-k+1}\right\\}\right)\left(1+\sum_{j=2}^{k}v_{j}\right),&\textrm{if
$i=1$}\\\ \infty,&\textrm{if $i=2,\ldots,k$}\end{array}\right.$ (3.13)
$\displaystyle=$
$\displaystyle{\phi_{i}}^{*}\left(\underline{v}\right),~{}\mbox{(say)}.$
(3.14)

Now, we define the following function

$\displaystyle\phi_{i0}\left(\underline{v}\right)=\left\\{\begin{array}[]{ll}{\phi_{i}}_{*}\left(\underline{v}\right),&\textrm{if
$\phi_{i}\left(\underline{v}\right)\leq{\phi_{i}}_{*}\left(\underline{v}\right)$}\\\
{\phi_{i}}\left(\underline{v}\right),&\textrm{if
${\phi_{i}}_{*}\left(\underline{v}\right)<\phi_{i}\left(\underline{v}\right)<{\phi_{i}}^{*}\left(\underline{v}\right)$}\\\
{\phi_{i}}^{*}\left(\underline{v}\right),&\textrm{if
$\phi_{i}\left(\underline{v}\right)\geq{\phi_{i}}^{*}\left(\underline{v}\right)$}.\end{array}\right.$
(3.18)

Thus we have the following result which provides the sufficient conditions for
the improvement of the estimators of the form given by $(3.2)$.

###### Theorem 3.2

Consider the estimator of the form
$\delta_{\phi_{i}}\left(\underline{X_{(1)}},\underline{T}\right)=T_{i}\phi_{i}(\underline{V})$.
Let $\phi_{i0}(\underline{v})$ be a function as defined in $(3.9)$. If there
exists some $\underline{\sigma}$ such that
$P_{\underline{\sigma}}\left(\phi_{i0}\left(\underline{V}\right)\neq\phi_{i}\left(\underline{V}\right)\right)>0,$
then under the order restriction
$\sigma_{1}\leq\sigma_{2}\leq\ldots\leq\sigma_{k}$, the estimator
$\delta_{\phi_{i0}}\left(\underline{X_{(1)}},\underline{T}\right)$ dominates
$\delta_{\phi_{i}}\left(\underline{X_{(1)}},\underline{T}\right)$ with respect
to the linex loss function $(2.1).$

In the following we present a corollary which is immediately follows from the
Theorem $3.2.$ For convenience of the presentation, we denote
$e_{1}=p^{-1}\left(1-\exp\\{-p/(n_{1}+n_{2}-1)\\}\right)$ and
$e_{2}=p^{-1}\left(1-\exp\\{-p/(n-k+1)\\}\right)$.

###### Corollary 3.3

Let $\sigma_{1}\leq\sigma_{2}\leq\ldots\leq\sigma_{k}$. Then under the loss
function given by $(2.1)$
$(i)$ the unrestricted best affine equivariant estimator of $\sigma_{1}$ is
inadmissible and dominated by the estimator given by

$\displaystyle\delta_{\phi_{10}}\left(\underline{X_{(1)}},\underline{T}\right)=\left\\{\begin{array}[]{ll}e_{1}T_{1},&\textrm{if
$d_{01}<e_{1}$}\\\ d_{01}T_{1},&\textrm{if $e_{1}\leq d_{01}\leq
e_{1}(1+V_{2})$}\\\ e_{1}(1+V_{2})T_{1},&\textrm{if
$d_{01}>e_{1}(1+V_{2})$};\end{array}\right.$ (3.22)

$(ii)$ the unrestricted best affine equivariant estimator of $\sigma_{i},$
$i=2,\dots,k$ is inadmissible and dominated by the estimator given by

$\displaystyle\delta_{\phi_{i0}}\left(\underline{X_{(1)}},\underline{T}\right)=\left\\{\begin{array}[]{ll}e_{2}\left(1+\sum_{j=1}^{i-1}V_{j}\right)T_{i},&\textrm{if
$d_{0i}<e_{2}\left(1+\sum_{j=1}^{i-1}V_{j}\right)$}\\\
d_{0i}T_{i},&\textrm{otherwise.}\end{array}\right.$ (3.25)

Now we consider the maximum likelihood estimation of
$\sigma_{i},~{}i=1,\ldots,k$ when there is an order restriction on
$\sigma_{i}$’s. Under the restriction
$\sigma\leq\sigma_{2}\leq\ldots\leq\sigma_{k}$, the restricted maximum
likelihood estimator is given by

$\displaystyle\delta^{2}_{iRML}=\max_{s\leq i}\min_{t\geq
i}\left\\{\frac{\sum_{j=s}^{t}T_{j}}{\sum_{j=s}^{t}n_{j}}\right\\},~{}i=1,\ldots,k.$
(3.26)

Let $k=2.$ Then the the restricted maximum likelihood estimators of
$\sigma_{1}$ and $\sigma_{2}$, while $\sigma_{1}\leq\sigma_{2}$ are
respectively given by

$\displaystyle\delta^{2}_{1RML}=\min\left\\{\frac{T_{1}}{n_{1}},\frac{T_{1}+T_{2}}{n_{1}+n_{2}}\right\\}$
(3.27)

and

$\displaystyle\delta^{2}_{2RML}=\max\left\\{\frac{T_{2}}{n_{2}},\frac{T_{1}+T_{2}}{n_{1}+n_{2}}\right\\}.$
(3.28)

Note that the estimators given by $(3.13)$ and $(3.14)$ belongs to the class
of affine equivariant estimators $(3.1)$. Thus as an application of the
Theorem $3.2,$ one can easily obtain corresponding improved estimators in
terms of the risk function as follows.

###### Corollary 3.4

Assume that $\sigma_{1}\leq\sigma_{2}.$ Then for $k=2,$ under the linex loss
function $(2.1),$ the estimators

$\displaystyle\delta^{2}_{1IRML}\left(\underline{X_{(1)}},\underline{T}\right)=\left\\{\begin{array}[]{ll}e_{1}T_{1},&\textrm{if
$\xi_{01}(\underline{V})<e_{1}$}\\\ \xi_{01}(\underline{V})T_{1},&\textrm{if
$e_{1}\leq\xi_{01}(\underline{V})\leq e_{1}(1+V_{2})$}\\\
e_{1}(1+V_{2})T_{1},&\textrm{if
$e_{1}(1+V_{2})<\xi_{01}(\underline{V})$};\end{array}\right.$ (3.32)

and

$\displaystyle\delta^{2}_{2IRML}\left(\underline{X_{(1)}},\underline{T}\right)=\left\\{\begin{array}[]{ll}e_{1}(1+V_{1})T_{2},&\textrm{if
$\xi_{02}(\underline{V})<e_{1}(1+V_{1})$}\\\
\xi_{02}(\underline{V})T_{2},&\textrm{otherwise,}\end{array}\right.$ (3.35)

where
$\xi_{01}(\underline{V})=\min\left\\{\frac{1}{n_{1}},\frac{T_{2}}{T_{1}}\frac{1}{n_{1}+n_{2}}\right\\}$
and
$\xi_{02}=\max\left\\{\frac{1}{n_{2}},\frac{T_{1}}{T_{2}}\frac{1}{n_{1}+n_{2}}\right\\}$
have uniformly smaller risk than the restricted maximum likelihood estimators
of $\sigma_{1}$ and $\sigma_{2}$, respectively.

###  3.2 A numerical study

Here we carry out a simulation study to compare the risk functions of the
proposed estimators numerically. We have generated $50,000$ random samples
from two independent exponential distributions using $R$ software. The
percentage relative risk improvement (PRRI) of an estimator $\delta_{1}$ over
another estimator $\delta_{2}$ is defined as

$\displaystyle
PRRI(\delta_{1},\delta_{2})=\left(\frac{Risk(\delta_{2})-Risk(\delta_{1})}{Risk(\delta_{2})}\right)\times
100\%$ (3.36)

In the following tables, we present PRRI of various estimators for different
sample sizes and different values of $p$. We have considered sample sizes
$(n_{1},n_{2})$ as $(3,5),$ $(10,15),$ $(8,5)$ and $(15,10).$ The values of
$p$ are taken to be $0.5$ and $-0.5$. The following points can be observed
from Table $1$ and Table $2$.

  1. 1.

With respect to the BSEE, the PRRI of the improved estimators decreases when
sample sizes $(n_{1},n_{2})$ increases with the constraint $n_{1}<n_{2}$.
Similar observation is noticed for $n_{1}>n_{2}$ case.

  2. 2.

For fixed $(n_{1},n_{2})$, the PRRI of the improved estimators (with respect
to the BSEE) increases if $(\sigma_{1},\sigma_{2})$ increases. Further, if we
increase the values of $\sigma_{2}$ by keeping $\sigma_{1}$ fixed, then the
PRRI of the improved estimators (with respect to the BSEE) decreases. This
observation ensures that after some distance, the risk function of the
improved BSEE coincides with that of the BSEE.

  3. 3.

It can seen that the risk performance of restricted MLE and improved
restricted MLE increases when the sample sizes $(n_{1},n_{2})$ increases with
the constraint $n_{1}<n_{2}$.

Table 3: The PRRI of various estimators for $k=2$ and $p=0.5.$ The 3rd, 4th
and 5th columns respectively present the PRRI of improved BAEE, restricted MLE
and improved restricted MLE over the BAEE whereas the 6th, 7th and 8th columns
present the PRRI of improved BAEE, BAEE and improved restricted MLE over the
restricted MLE.

$(n_{1},n_{2})$ | $(\sigma_{1},\sigma_{2})$ | $(\delta_{\phi_{10}},\delta_{\phi_{20}})$ | $(\delta^{2}_{1RML},\delta^{2}_{2RML})$ | $(\delta^{2}_{1IRML},\delta^{2}_{2IRML})$ | $(\delta_{\phi_{10}},\delta_{\phi_{20}})$ | $(\delta_{1d_{01}},\delta_{2d_{02}})$ | $(\delta^{2}_{1IRML},\delta^{2}_{2IRML})$
---|---|---|---|---|---|---|---
(3,5) | (0.2,0.5) | (4.35,6.54) | (12.26,-0.71) | (13.50,3.94) | (-9.00,7.20) | (-13.09,0.71) | (1.41,4.62)
| (0.2,1.0) | (0.88,1.06) | (7.26,-1.01) | (7.44,-0.23) | (-6.88,2.05) | (-7.87,1.00) | (0.19,0.68)
| (0.2,1.5) | (0.42,0.28) | (2.04,-1.03) | (2.44,-0.85) | (-2.03,1.29) | (-2.46,1.02) | (0.04,0.18)
| (0.5,1.0) | (6.06,10.21) | (9.89,-0.37) | (12.22,7.01) | (-4.25,10.55) | (-10.98,0.37) | (2.58,7.35)
| (0.5,1.5) | (3.01,4.30) | (12.04,-0.86) | (12.87,2.15) | (-10.26,5.12) | (-13.69,0.85) | (0.84,2.98)
| (0.5,2.0) | (1.41,2.04) | (9.87,-0.97) | (10.12,0.39) | (-9.27,2.98) | (-10.87,0.96) | (0.38,1.35)
| (1.0,1.5) | (8.26,15.98) | (1.69,0.65) | (6.53,12.11) | (6.68,15.42) | (-1.71,-0.66) | (4.92,11.53)
| (1.0,2.0) | (6.06,10.21) | (9.89,-0.37) | (12.22,7.01) | (-4.25,10.55) | (-10.98,0.36) | (2.58,7.35)
| (1.0,2.5) | (4.35,6.54) | (12.25,-0.71) | (13.50,3.94) | (-9.00,7.20) | (-13.97,0.70) | (1.41,4.62)
(10,15) | (0.2,0.5) | (0.83,0.90) | (15.79,-0.44) | (15.82,-0.24) | (-17.76,1.35) | (-18.76,0.44) | (0.03,0.69)
| (0.2,1.0) | (0,0.002) | (0.18,-0.44) | (0.18,-0.45) | (-0.18,0.44) | (-0.18,0.44) | (0,0.001)
| (0.2,1.5) | (0,0) | (-0.53,-0.44) | (-0.53,-0.45) | (0.52,0.45) | (0.53,0.44) | (0,0)
| (0.5,1.0) | (2.94,3.16) | (17.64,-0.44) | (17.85,2.11) | (-17.84,3.59) | (-21.42,0.44) | (0.25,2.54)
| (0.5,1.5) | (0.15,0.26) | (9.59,-0.44) | (9.59,-0.25) | (-10.43,0.70) | (-10.61,0.44) | (0.01,0.19)
| (0.5,2.0) | (0.01,0.02) | (2.42,-0.44) | (-2.42,-0.43) | (-2.47,0.46) | (-2.49,0.44) | (0,0.01)
| (1.0,1.5) | (9.05,10.87) | (-6.79,-0.42) | (-5.15,8.79) | (14.84,11.24) | (6.36,0.42) | (1.54,9.16)
| (1.0,2.0) | (2.94,3.16) | (17.64,-0.45) | (17.85,2.11) | (-17.84,3.59) | (-21.42,0.44) | (0.25,2.54)
| (1.0,2.5) | (0.83,0.90) | (15.79,-0.45) | (15.82,0.25) | (-17.76,1.34) | (-18.76,0.44) | (0.03,0.69)
(8,5) | (0.2,0.5) | (2.73,8.67) | (-54.73,0.39) | (-7.14,5.99) | (37.14,8.31) | (35.37,-0.39) | (30.75,5.62)
| (0.2,1.0) | (0.28,0.79) | (1.23,-1.07) | (8.78,-0.54) | (-0.96,1.84) | (-1.25,1.06) | (7.64,0.52)
| (0.2,1.5) | (0.10,0.14) | (3.54,-1.13) | (5.49,-1.04) | (-3.59,1.26) | (-3.67,1.12) | (2.02,0.08)
| (0.5,1.0) | (4.38,15.67) | (-95.99,2.40) | (-20.71,12.05) | (51.21,13.59) | (48.97,-2.46) | (38.41,9.88)
| (0.5,1.5) | (1.68,5.04) | (-29.96,-0.42) | (1.26,2.88) | (24.35,5.40) | (23.06,0.42) | (24.03,3.29)
| (0.5,2.0) | (0.62,1.87) | (-6.67,-0.94) | (8.00,0.31) | (6.81,2.79) | (6.22,0.93) | (13.37,1.24)
| (1.0,1.5) | (6.96,28.68) | (-162.75,7.87) | (-39.43,23.68) | (64.59,22.58) | (61.94,-8.55) | (46.93,17.15)
| (1.0,2.0) | (4.37,15.67) | (-95.99,2.40) | (-20.71,12.05) | (51.21,13.59) | (48.97,-2.46) | (38.41,9.88)
| (1.0,2.5) | (2.73,8.67) | (-54.73,0.39) | (-7.14,5.99) | (36.13,8.31) | (35.37,-0.39) | (30.75,5.62)
(15,10) | (0.2,0.5) | (0.79,1.79) | (-35.54,-0.44) | (-13.89,0.95) | (26.81,2.23) | (26.22,0.44) | (15.97,1.39)
| (0.2,1.0) | (0.005,0.01) | (5.55,-0.47) | (5.88,-0.47) | (-5.87,0.48) | (-5.88,0.47) | (0.35,0.01)
| (0.2,1.5) | (0,0) | (0.76,-0.47) | (0.79,-0.48) | (-0.77,0.47) | (-0.77,0.47) | (0.30,0)
| (0.5,1.0) | (2.25,5.80) | (-105.05,-0.24) | (-15.86,4.30) | (52.32,6.03) | (51.23,0.25) | (25.94,4.53)
| (0.5,1.5) | (0.27,0.59) | (-5.78,-0.46) | (3.08,-0.03) | (5.72,1.05) | (5.47,0.56) | (8.38,0.44)
| (0.5,2.0) | (0.03,0.10) | (7.78,-0.48) | (9.49,-0.44) | (-8.40,0.54) | (-8.43,0.47) | (1.86,0.04)
| (1.0,1.5) | (6.56,18.19) | (-242.13,1.25) | (-110.63,15.21) | (72.69,17.15) | (17.77,-1.27) | (38.43,14.13)
| (1.0,2.0) | (2.25,5.80) | (-105.05,-0.25) | (-51.86,4.30) | (52.33,6.03) | (51.23,0.24) | (24.94,4.53)
| (1.0,2.5) | (0.79,1.79) | (-35.54,-0.44) | (-13.89,0.95) | (26.81,2.23) | (26.22,0.44) | (15.97,1.39)

Table 4: The PRRI of various estimators for $k=2$ and $p=-0.5.$ The 3rd, 4th
and 5th columns respectively present the PRRI of improved BAEE, restricted MLE
and improved restricted MLE over the BAEE whereas the 6th, 7th and 8th columns
present the PRRI of improved BAEE, BAEE and improved restricted MLE over the
restricted MLE.

$(n_{1},n_{2})$ | $(\sigma_{1},\sigma_{2})$ | $(\delta_{\phi_{10}},\delta_{\phi_{20}})$ | $(\delta^{2}_{1RML},\delta^{2}_{2RML})$ | $(\delta^{2}_{1IRML},\delta^{2}_{2IRML})$ | $(\delta_{\phi_{10}},\delta_{\phi_{20}})$ | $(\delta_{1d_{01}},\delta_{2d_{02}})$ | $(\delta^{2}_{1IRML},\delta^{2}_{2IRML})$
---|---|---|---|---|---|---|---
(3,5) | (0.2,0.5) | (5.18,6.98) | (3.01,-0.57) | (4.75,8.16) | (2.25,7.50) | (-3.09,0.56) | (1.81,8.67)
| (0.2,1.0) | (1.09,1.16) | (2.65,-0.98) | (2.89,0.71) | (-1.61,2.11) | (-2.72,0.96) | (0.25,1.67)
| (0.2,1.5) | (0.44,0.31) | (0.45,-1.01) | (0.52,-0.51) | (-0.02,1.31) | (-0.46,0.99) | (0.06,0.48)
| (0.5,1.0) | (7.32,10.77) | (-0.84,-0.11) | (2.44,12.65) | (8.09,10.86) | (0.84,0.11) | (3.25,12.74)
| (0.5,1.5) | (3.69,4.64) | (4.07,-0.77) | (5.12,5.27) | (-0.39,5.37) | (-4.24,0.76) | (1.07,5.98)
| (0.5,2.0) | (1.92,2.22) | (3.72,-0.92) | (4.19,2.15) | (-1.87,3.12) | (-3.87,0.91) | (0.48,3.05)
| (1.0,1.5) | (10.14,16.63) | (-11.23,1.21) | (-4.45,19.45) | (19.22,15.61) | (10.10,-1.22) | (6.09,18.25)
| (1.0,2.0) | (7.31,10.77) | (-0.84,-0.11) | (2.43,12.65) | (8.09,10.87) | (0.84,0.11) | (3.26,12.75)
| (1.0,2.5) | (5.18,6.98) | (3.00,-0.57) | (4.75,8.16) | (2.25,7.50) | (-3.09,0.56) | (1.81,8.68)
(10,15) | (0.2,0.5) | (0.94,0.91) | (11.1,-0.35) | (11.1,0.83) | (-17.76,1.34) | (-18.76,0.44) | (0.03,0.69)
| (0.2,1.0) | (0,0.002) | (-0.08,-0.08) | (-0.08,-0.34) | (-0.18,0.45) | (0.53,0.44) | (0,0)
| (0.2,1.5) | (0,0) | (-0.56,-0.56) | (-0.56,-0.35) | (0.52,0.44) | (0.53,0.44) | (0,0)
| (0.5,1.0) | (3.23,3.17) | (10.45,-0.35) | (10.79,3.58) | (-17.85,3.59) | (-21.42,0.45) | (0.25,2.54)
| (0.5,1.5) | (0.23,0.26) | (6.71,-0.35) | (6.72,0.005) | (-10.43,0.71) | (-10.61,0.45) | (0.006,0.19)
| (0.5,2.0) | (0.02,0.02) | (1.47,-0.35) | (1.47,-0.32) | (-2.47,0.46) | (-2.48,0.45) | (0,0.013)
| (1.0,1.5) | (10.02,10.83) | (-19.06,-0.31) | (-16.81,12.4) | (14.84,11.25) | (6.36,0.41) | (1.54,9.17)
| (1.0,2.0) | (3.22,3.17) | (10.49,-0.35) | (10.79,3.58) | (-17.84,3.59) | (-21.42,0.44) | (0.25,2.54)
| (1.0,2.5) | (0.93,0.91) | (11.12,-0.35) | (11.14,0.83) | (-17.76,1.35) | (-18.76,0.45) | (0.03,0.69)
(8,5) | (0.2,0.5) | (3.09,8.9) | (-79.04,1.18) | (-15.82,9.99) | (37.14,8.31) | (35.37,-0.39) | (30.75,5.62)
| (0.2,1.0) | (0.35,0.81) | (-6.05,-0.81) | (3.94,0.16) | (-0.96,1.84) | (-1.25,1.06) | (7.64,0.52)
| (0.2,1.5) | (0.08,0.13) | (0.51,-0.89) | (3.09,-0.71) | (-3.59,1.26) | (-3.67,1.12) | (2.03,0.08)
| (0.5,1.0) | (4.68,16.02) | (-130.67,3.78) | (-30.31,18.17) | (51.21,13.59) | (48.97,-2.46) | (38.41,9.88)
| (0.5,1.5) | (1.92,5.18) | (-47.95,0.09) | (-6.5,5.53) | (24.35,5.40) | (23.06,0.41) | (24.03,3.29)
| (0.5,2.0) | (0.76,1.93) | (-17.64,-0.62) | (1.71,1.58) | (6.81,2.79) | (6.23,0.93) | (13.37,1.24)
| (1.0,1.5) | (7.71,28.81) | (-215.37,10.58) | (-50.04,32.48) | (64.59,22.58) | (61.94,-8.55) | (46.93,17.15)
| (1.0,2.0) | (4.88,16.02) | (-130.68,3.78) | (-30.32,18.17) | (51.21,13.59) | (48.97,-2.46) | (38.41,9.88)
| (1.0,2.5) | (3.09,8.95) | (-79.04,1.17) | (-15.82,9.98) | (36.14,8.32) | (35.37,-0.39) | (30.75,5.62)
(15,10) | (0.2,0.5) | (0.85,1.78) | (-51.33,-0.62) | (-23.48,1.56) | (26.81,2.23) | (26.22,0.44) | (15.97,1.39)
| (0.2,1.0) | (0.008,0.01) | (3.73,-0.67) | (4.17,-0.66) | (-5.87,0.49) | (-5.88,0.47) | (0.35,0.01)
| (0.2,1.5) | (0,0) | (0.53,-0.67) | (0.56,-0.67) | (-0.77,0.47) | (-0.77,0.47) | (0.30,0)
| (0.5,1.0) | (2.45,5.72) | (-135.07,-0.36) | (-65.83,6.27) | (52.32,6.03) | (51.23,0.25) | (25.94,4.54)
| (0.5,1.5) | (0.29,0.57) | (-15.18,-0.65) | (-3.66,0.08) | (5.72,1.05) | (5.47,0.56) | (8.39,0.44)
| (0.5,2.0) | (0.03,0.06) | (3.75,-0.67) | (5.97,-0.59) | (-8.41,0.54) | (-8.43,0.48) | (1.86,0.04)
| (1.0,1.5) | (7.02,17.92) | (-303.68,1.55) | (-131.17,20.14) | (72.69,17.15) | (17.77,-1.27) | (38.43,14.14)
| (1.0,2.0) | (2.45,5.72) | (-135.07,-0.36) | (-65.82,6.27) | (52.33,6.03) | (51.23,0.24) | (24.94,4.53)
| (1.0,2.5) | (0.86,1.79) | (-51.54,-0.62) | (-23.48,1.56) | (26.81,2.23) | (26.22,0.44) | (15.97,1.39)

##  4 Conclusion

In the present communication, we have studied the component wise estimation of
scale parameters of several exponential distributions when the scale
parameters are ordered with respect to the linex loss function. In Section 2,
we have studied the estimation of ordered scale parameters when the location
parameters are known. The inadmissibility of the usual estimator is proved by
deriving a class of improved estimators. Using this result, we have obtained
an estimator which improves upon the restricted MLE. When location parameters
are unknown, the estimation of ordered scale parameters is investigated in
section 3. It is proved that the BAEE is inadmissible by deriving a class of
improved estimators. We have also established the inadmissibility of
restricted MLE. Finally, for both cases, we have compared the risk
performances of the proposed estimators through simulations.

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