It is widely believed that under some fairly general conditions, MLEs are consistent, asymptotically normal, and efficient. Stephen Stigler has elegantly documented some of Fisher’s troubles when he wanted a proof. You want proof? Of course you can pile on assumptions so that the proof is easy. If checking your assumptions in any particular case is harder than checking the conclusion in that case, you will have joined a great tradition.
I used to think that efficiency was a thing for the theorists (I can live with inefficiency), that normality was a thing of the past (we can simulate), but that—in spite of Ralph Waldo Emerson—consistency is a thing we should demand of any statistical procedure. Not any more. These days we can simulate in and around the conditions of our data, and learn whether a novel procedure behaves as it should in that context. If it does, we might just believe the results of its application to our data. Other people’s data? That’s their simulation, their part of the parameter space, their problem. Maybe some theorist will take up the challenge, and study the procedure, and produce something useful. But if we’re still waiting for that with MLEs in general (canonical exponential families are in good shape), I wouldn’t hold my breath for this novel procedure. By the time a few people have tried the new procedure, each time checking its suitability by simulation in their context, we will have built up a proof by simulation. Shocking? Of course.
Some time into my career as a statistician, I noticed that I don’t check the conditions of a theorem before I use some model or method with a set of data. I think in statistics we need derivations, not proofs. That is, lines of reasoning from some assumptions to a formula, or a procedure, which may or may not have certain properties in a given context, but which, all going well, might provide some insight. The evidence that this might be the case can be mathematical, not necessarily with epsilon-delta rigour, simulation, or just verbal. Call this “a statistician’s proof ”. This is what I do these days. Should I be kicked out of the IMS?

After reading many astronomy literature, I develop a notion that astronomers like to use the maximum likelihood as a robust alternative to the chi-square minimization for fitting astrophysical models with parameters. I’m not sure it is truly robust because not many astronomy paper list assumptions and conditions for their MLEs.

Often I got confused with their target parameters. They are not parameters in statistical models. They are not necessarily satisfy the properties of probability theory. I often fail to find statistical properties of these parameters for the estimation. It is rare checking statistical modeling procedures with assumptions described by Prof. Speed. Even derivation is a bit short to be called “rigorous statistical analysis.” (At least I wish to see a sentence that “It is trivial to derive the estimator with this and that properties”).

Common phrases I confronted from astronomical literature is that authors’ strategy is statistically rigorous, superior, or powerful without showing why and how it is rigorous, superior, or powerful. I tried to convey these pitfalls and general restrictions in their employed statistical methods. Their strategy is not “statistically robust” nor “statistically powerful” nor “statistically rigorous.” Statisticians have own measures of “superiority” to discuss the improvement in their statistics, analysis strategies, and methodology.

It has not been easy since I never intend to case specific fault picking every time I see these statements. A method believed to be robust can be proven as not a robust method with your data and models. By simulations and derivations with the sufficient description of conditions, your excellent method can be presented with statistical rigors.

Within similar circumstances for statistical modeling and data analysis, there’s a trade off between robustness and conditions among statistical methodologies. Before stating a particular method adopted is robust or rigid, powerful or insensitive, efficient or inefficient, and so on; derivation, proof, or simulation studies are anticipated to be named the analysis and procedure is statistically excellent.

Before it gets too long, I’d like say that statistics have traditions for declaring working methods via proofs, simulations, or derivations. Each has their foundations: assumptions and conditions to be stated as “robust”, “efficient”, “powerful”, or “consistent.” When new statistics are introduced in astronomical literature, I hope to see some additional effort of matching statistical conditions to the properties of target data and some statistical rigor (derivations or simulations) prior to saying they are “robust”, “powerful”, or “superior.”

• [2006MNRAS.369.1334S]: semiparametric technique based on a maximum likelihood (ML) approach and Voronoi tessellation (VT). Besides, I wonder if Section 3.3, the cluster detection algorithm works similarly to a source detection algorithm in high energy astrophysics if tight photon clusters indicate sources. By the way, what is the definition of sources? Depending on the definitions, determining the right thresholds for detections would change; however, it seems like (brute) Monte Carlo simulations i.e. empirical approaches are employed for setting thresholds. Please, note that my questionnaire is irrelevant to this paper, which I enjoyed reading very much.
• [2004MNRAS.347.1241S]: similar to the above because of the same methodology, ML, VT, and color slide/filter for cluster detection
• [2002AJ....123.1807G]: cut and enhance (CE) cluster detection method. From the abstract: The method is semiparametric, since it uses minimal assumptions about cluster properties in order to minimize possible biases. No assumptions are made about the shape of clusters, their radial profile, or their luminosity function. On the contrary, I wish they used nonparametric which seems more proper in a statistical sense instead of semiparametric judging from their methodology description.
• [2002A%26A...383.1100N]: statistics related keywords: time series; discrete Fourier transform; long range dependence; log-periodogram regression; ordinary least squares; generalized least squares. The semiparametric method section seems too short. Detail accounts are replaced by reference papers from Annals of Statistics. Among 31 references, 15 were from statistics journals and without reading them, average readers will not have a chance to understand the semiparametric approach.

You might want to check out wiki:Semiparametric about semiparametric (model) from the statistics standpoint.

The following books that I checked from libraries some years back related to semiparametric methods, from which you could get more information about semeparametric statistics. Unfortunately, applications and examples in these books are heavily rely on subjects such as public health (epidemiology), bioinformatics, and econometrics.

• Rupert, Wand, and Carroll (2003) Semiparametric Regression, Cambridge University Press
• Härdle, Müller, Sperlich, and Werwatz (2004) Nonparametric and Semiparametric Models, Spinger
• Horowitz (1998) Semiparametric Methods in Econometrics (Lecture Notes in Statistics) , Springer

There seem more recent publications from 2007 and 2008 about semiparametric methods, targeting diverse but focused readers but no opportunities for me to have a look on them. I just want to point out that many occasions we confront that full parametrization of a model is not necessary but those nuisance parameters determines the shape of a sampling distribution for accurate statistical inference. Semiparametric methods described in above papers are very limited from statistics viewpoints. Astronomers can take a way more advantages from various semiparametrical strategies. There are plenty of rooms for developing semiparametric approaches to various astronomical data analysis and inference about the parameters of interest. It is almost unexplored.

• [astro-ph:0805.1290]R. Barnard, L. Shaw Greening, U. Kolb
  A multi-coloured survey of NGC 253 with XMM-Newton: testing the methods used for creating luminosity functions from low-count data

• [astro-ph:0805.1469] Philip J. Marshall et al.
  Automated detection of galaxy-scale gravitational lenses in high resolution imaging data

• [astro-ph:0805.1470] E. P. Kontar, E. Dickson, J. Kasparova
  Low-energy cutoffs and in electron spectra of solar flares: statistical survey (It is not statistically rigorous but the topic can be connected to dip tests or gap tests in statistics)

• [astro-ph:0805.1936] J. Yee & B. Gaudi
  Characterizing Long-Period Transiting Planets Observed by Kepler (discusses uncertainty in light curves and Fisher matrix)

• [astro-ph:0805.1994] the QUad collaboration: C. Pryke et al.
  Second and third season QUaD CMB temperature and polarization power spectra (What is jackknife tests? A brief scan of the paper does not register with my understanding of jackknifing. It looks more close to cross validation. Another slog topic shall come: bootstrap, cross validation, jackknife, and resampling.)

• [astro-ph:0805.2121] N. Cole et al.
  Maximum Likelihood Fitting of Tidal Streams With Application to the Sagittarius Dwarf Tidal Tails

• [astro-ph:0805.2155] J Yoo & M Zaldarriaga
  Improved estimation of cluster mass profiles from the cosmic microwave background

• [astro-ph:0805.2207] A.Vikhlinin et al.
  Chandra Cluster Cosmology Project II: Samples and X-ray Data Reduction (it mentions calibration uncertainty and background, can it be a reference to stacking, coadding, source detection, etc?)

• [astro-ph:0805.2325] J.M. Loh
  A valid and fast spatial bootstrap for correlation functions

• [astro-ph:0805.2326] T. Wickramasinghe, M. Struble, J. Nieusma
  Observed Bimodality of the Einstein Crossing Times of Galactic Microlensing Events

With the 5th year WMAP data release, there were many WMAP related papers and among them, most statistical papers are listed. WMAP specific/related:

• [astro-ph:0803.0586] J. Dunkley et. al.
    Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Likelihoods and Parameters from the WMAP data (likelihoods)

• [astro-ph:0803.0715] B. Gold et. al.
    Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Galactic Foreground Emission (MCMC)

• [astro-ph:0803.0889] Ichikawa, Sekiguchi, & Takahashi
    Probing the Effective Number of Neutrino Species with Cosmic Microwave Background

And others:

• [astro-ph:0802.4464] M. Sahlén et.al.
    The XMM Cluster Survey: Forecasting cosmological and cluster scaling-relation parameter constraints

• [astro-ph:0803.0918] J.M. Colberg et.al.
    The Aspen–Amsterdam Void Finder Comparison Project (TFE, tessellation field estimator)

• [astro-ph:0803.0885] J.Ballot et.al.
    On deriving p-mode parameters for inclined solar-like stars (MLE, maximum likelihood estimator)

By the way, I noticed [astro-ph:0802.4464] used Monte Carlo Markov Chain, whereas [astro-ph:0803.0715] used Markov chain Monte Carlo. Do they mean different? Or the former is a typo?