# The Truncated Lindley Distribution with Applications in Astrophysics

*Reviewer 1:*Anonymous

*Reviewer 2:*Anonymous

**Round 1**

*Reviewer 1 Report*

The author study the Lindsley distribution (LD) and some of its variants and applies them to different areas of astrophysics: the initial mass function of stars, the luminosity function of SDSS galaxies, the photometric maximum of 2MRS galaxies, and the Malmquist bias of these galaxies.

While I have not gone through the (numerous) equations in details, the results seem correct, but of limited interest to the community. In addition, several points should be clarified before the paper can be considered for publication in Galaxies

Major comments:

- The paper has a lot of equations and calculations, and is repetitive (Sections 2,3, and 4 are the same calculations applied to three different cases: the Lindley distribution, the LD with scale, and the truncated LD with scale). It would be more clear to summarize the key points (definition, mean, variance, and a few other key quantities) into one section, and defer the other calculations to the Appendix. While the LD itself certainly deserves its own paragraph, the other two could be introduced together in a second subsection of the LD section as they are very similar.

- Most of the times, the results are stated with no comment or discussion. It would add to the paper to interpret the results and explain why the LD is interesting rather than just stating best fit results.

- The LD does not seem to do as well as the Schechter function, and has no theoretical backing, therefore it is not clear whether it is useful to show it here, although, admittedly a negative result is still a result.

- How is it that the LD does not fit the galaxy luminosity distribution as well as the Schechter one, while for the photometric maxium case, it works better?

- what is the justification for eq. 61?

Minor comments :

- Fig. 4 takes the whole page, and its purpose is not clear.

- Eqs. (37–39) have a strange (black square character), is it meant to be Psi*?

- Malmquist bias usually referts the the preferential detection of brighter sources at high redshift. Here, it seems to simply mean the mean brightness as a function of redshift. I think it would be good to clarify, and if necessary, remove the term "Malmquist bias"

*Author Response*

Here you will find the answer organized by points.

Point of the referee 1)

The paper has a lot of equations and calculations, and is repetitive

(Sections 2,3, and 4 are the same calculations applied to three different

cases: the Lindley distribution, the LD with scale, and the truncated LD

with scale). It would be more clear to summarize the key points

(definition, mean, variance, and a few other key quantities) into one

section, and defer the other calculations to the Appendix. While the LD

itself certainly deserves its own paragraph, the other two could be

introduced together in a second subsection of the LD section as they are

very similar.

Answer of the author to point 1)

The Lindley family is now concentrated in Section 2 as suggested.

Some results are reported in two appendices as suggested.

Point of the referee 2)

- Most of the times, the results are stated with no comment or discussion.

Answer of the author to point 2)

*) An example of the utility for the truncated IMF has been

reported before Section 4.

*) A comment has been inserted after equation (31)

on M^*

Point of the referee 3)

- It would add to the paper to interpret the results and explain why the LD

- is interesting rather than just stating best fit results.

Answer of the author to point 3)

As an example now the problem of the IMF is better introduced

, see beginning of Section 3.

The truncated Lindley distribution has been introduced

because there is maximum and a minimum in the IMF for stars,

see text in Section 3.

Point of the referee 4)

- The LD does not seem to do as well as the Schechter function, and has no

- theoretical backing, therefore it is not clear whether it is useful to

- show it here, although, admittedly a negative result is still a result.

- How is it that the LD does not fit the galaxy luminosity distribution as

- well as the Schechter one, while for the photometric maxium case, it works

- better?

Answer of the author to point 4)

a) the target for a new LF for galaxies has been summarized at the

end of section 4.1.

b) the parameters for the two LFs here used in the

case of the photometric maximum are chosen minimizig

the chisquare, see new text after formula (54)

Point of the referee 5)

- what is the justification for eq. 61?

Answer of the author to point 5)

Old formula (61) now (56) allows to match observational

and theoretical data. This comment has been inserted

after equation (56).

Point of the referee 6)

- Fig. 4 takes the whole page, and its purpose is not clear.

Answer of the author to point 6)

a) a new figure has been inserted

b) an evaluation of the reduction in the number of stars

after a given time is given ,

see new text before section 4.

Point of the referee 7)

- Eqs. (3739) have a strange (black square character), is it meant to be

- Psi*?

Answer of the author to point 7)

The strange black square character in Eqs. (3739)

now equations (3437) has been introduced

by the Galaxies version of the pdf.

I will see how to cook this fact in the future with Galaxies.

Point of the referee 8)

- Malmquist bias usually referts the the preferential detection of brighter

- sources at high redshift. Here, it seems to simply mean the mean

- brightness as a function of redshift. I think it would be good to clarify,

- and if necessary, remove the term "Malmquist bias"

Answer of the author to point 8)

I have adopted the point of view of the referee and I am now

speaking of averaged absolute magnitude.

The term Malmquist bias has been removed.

*Reviewer 2 Report*

Introduction

The introduction part should be improved in order to introduce more about the background of related astrophysics and explain why the Lindley distribution is valuable/necessary to be applied to the astrophysics. This is a paper on Astrophysics & astronomy, but after I read the introduction part I find the necessary scientific background for astrophysics is absent.

Section 2 & 3.

The author listed many basic equations, which can be deduced easily. Are all these equations necessary to be listed in the paper? I suggest the author just list the important and necessary equations.

Section 4.

Considering that the truncated Lindley distribution is the key point in this paper, I suggest that the author give some common on the truncated Lindley distribution and discuss its advantages.

Section 5.

The author should give a brief introduction about the background of the IMF of stars and explain why the truncated Lindley distribution is a preferable choice.

Section 6.

Equation (37) : What does the solid square mean? Is it a free parameter? Maybe this is not a good choice to use such symbol to represent a parameter.

Overall, I think the paper put too many mathematical formulas, which make the scientific concentration seems to be flooded. I think the paper should be improved greatly in order to highlight the scientific theme.

*Author Response*

Point of the referee 1)

Introduction

The introduction part should be improved in order to introduce more about

the background of related astrophysics and explain why the Lindley

distribution is valuable/necessary to be applied to the astrophysics. This

is a paper on Astrophysics & astronomy, but after I read the introduction

part I find the necessary scientific background for astrophysics is absent.

Answer of the author to point 1)

Now the introduction contains a detailed discussion on

the IMF for stars,

the LF for galaxies,

the photometric maximum for galaxies and

the range in absolute magnitude for galaxies

versus the redshift.

Point of the referee 2)

Section 2 & 3.

The author listed many basic equations, which can be deduced easily. Are all

these equations necessary to be listed in the paper? I suggest the author

just list the important and necessary equations.

Answer of the author to point 2)

The non necessary equations are now inserted in appendix A and B

Point of the referee 3)

Section 4.

Considering that the truncated Lindley distribution is the key point in this

paper, I suggest that the author give some common on the truncated Lindley

distribution and discuss its advantages.

Answer of the author to point 3)

The introduction now contains a more clear discussion on the

need to substitute the zero and infinity in the PDFs

with finite values.

The truncated Lindley PDF was not yet analyzed in astrophysics.

Point of the referee 4)

Section 5.

The author should give a brief introduction about the background of the IMF

of stars and explain why the truncated Lindley distribution is a preferable

choice.

Answer of the author to point 4)

a) The problem of the IMF is now better introduced

in two places : the introduction

and the beginning of Section 3.

The truncated Lindley distribution has been introduced

because there is maximum and a minimum in the IMF for stars,

see text between (28-29) in Section 3.

Point of the referee 5)

Section 6.

Equation (37) : What does the solid square mean? Is it a free parameter?

Maybe this is not a good choice to use such symbol to represent a

parameter.

Answer of the author to point 5)

The strange black square character in old equation (37)

now equation (32) has been introduced

in the Galaxies version of the pdf.

I will see how to cook this fact in the future with the Galaxies

redaction.

**Round 2**

*Reviewer 1 Report*

Most of the points have been satisfactorily answered, and I find the manuscript more clear, although the motivations and the usefulness of this family of distributions are still not completely convincing to me.

I only have one last comment: Fig. 5 compares the Schechter and Lindley LFs, and clearly, the Lindley LF is not a good fit. Then in section 4.3, the author calculates the truncated Lindley LF (TLLF). If this LF is meant to be a better fit than Schechter, why is it not plotted on Fig. 5, and the best-fit values should be reported (compared with table 3).

If the fit is not better, then I don't see the point of the TLLF in this section, since the rest of section 4 uses the non-truncated Lindley LF.

*Author Response*

Point of the referee 1

I only have one last comment:

Fig. 5 compares the Schechter and Lindley LFs,

and clearly, the Lindley LF is not a good fit.

Then in section 4.3, the

author calculates the truncated Lindley LF (TLLF).

If this LF is meant to be

a better fit than Schechter,

why is it not plotted on Fig. 5, and the

best-fit values should be reported (compared with table 3).

If the fit is not better, then I don't see the point of

the TLLF in this

section, since the rest of section 4 uses the non-truncated Lindley LF.

Answer to point 1 of the author

Now table 3 contains the parameters of the truncated Lindley LF.

The differences between truncated and not truncated Lindley

LF are now outlined , see red text before Section 3.3.

*Reviewer 2 Report*

Section 3: The author shoud give at least a brief discussion on his result. E.g., Discuss the physical meaning, compare the Lindley method with traditional ones and thus highlight the strengths of the Lindley method.

Section 4: This section has the same problem. The author should give a more detailed discussion on the result. The author have a good mathematical background and present much mathematical description, however the discussion on aspect of astronomy is deficient.

Section 5: This setion should be adjusted once Section 3 and 4 are revised.

*Author Response*

Points of the referee

Section 3: The author shoud give at least a

brief discussion on his result.

E.g., Discuss the physical meaning,

compare the Lindley method with

traditional ones and thus highlight the strengths of

the Lindley method.

Section 4: This section has the same problem.

The author should give a more

detailed discussion on the result.

The author have a good mathematical

background and present much mathematical description,

however the discussion

on aspect of astronomy is deficient.

Section 5: This setion should be adjusted

once Section 3 and 4 are revised.

Answer of the author

I have introduced five modifications to the text as suggested by

the referee. They are marked in red.