Not really because inferential statistics are are based on probability samples. Without a probability model summary statisics are fine but inference doesn't really make sense.

BTW you might look at Devore and Berk. Mathematical Statisics in the z-library google instructions for it's use. BEST WISHES, David Booth

I might add a question What is the design of the study?

Mike Lane The study is of perceived home environment among senior citizens. A municipality is the study area. Here, we have total number of households but we do not have exact information about the presence/absence of senior citizen (age > 70 years). So, could not do random sampling and did purposive sampling got collection of information from households. I have socio-economic variables and want to relate with their higher and lower perception of home environment. Is it possible? Which socio-economic factor contribute most to their perception? Thank you.

I would be hesitant in this case to make inferences about socio-economic effects with a purposive sampling design. You would likely introduce some kind of bias.

Thank you for the response. Much appreciated. @Mike Lane

Harish Bahadur Chand  - 

Yes, you can, BUT you need auxiliary data on the entire population; you can use a prediction (i.e. a regression model-based approach); and you need to chose your sample accordingly, without leaving too much room for bias in your selection.  (Balanced sampling is noted below, and also for highly skewed establishment surveys, cutoff or quasi-cutoff (multiple attrubute) sampling is noted.) 

Actually it really depends on your approach. 

Consider this:

Brewer(2014), "Three controversies in the history of survey sampling," Survey Methodology, Dec 2013 -  article Ken Brewer wrote as a consequence of receiving the Waksberg Award: 

https://www150.statcan.gc.ca/n1/pub/12-001-x/2013002/article/11883-eng.htm

There is the probability-of-selection-based (randomized design-based) approach, the prediction-based (model-based) approach, and a combined or a model-assisted design-based approach.  For the strictly model-based approach you can use purposive sampling (and you can trade some added bias for greatly reduced variance in the case of establishment surveys in a presentation linked below).   And you need auxiliary data on the entire population. 

The model-based approach mimics and even beats the randomized approach for "representativeness" when using balanced sampling.  A simple example of balanced sampling is when you have one auxiliary variable, x, related to y, in a ratio regression model, y = bx + e (where e is heteroscedastic), and you choose yi members such that their corresponding xi values have a mean about equal to the population mean of the xi.  

Here is a paper which picks the sample size needed for a single population, or subpopulation, using a frequently used ratio model, and compares it to the methodology used for simple random sampling when determining sample size requirements.  Results for highly skewed data are compared using balanced sampling and cutoff sampling.

Note that a key here for cutoff/quasi-cutoff sampling, is that as has been well established (see Cochran(1953), Sampling Techniques, 1st ed, page 205), when looking for the best measure of size (which makes the best predictor), the same data item in a previous census is best for an item on a current sample.  This works very well for highly skewed establishment surveys. 

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Examples can be found through the following where this has been used with high success for many cases over many years:

Invited presentation for mathematical statisticians at the US Energy Information Administration:   

https://www.researchgate.net/publication/319914742_Quasi-Cutoff_Sampling_and_the_Classical_Ratio_Estimator_-_Application_to_Establishment_Surveys_for_Official_Statistics_at_the_US_Energy_Information_Administration_-_Historical_Development, 2017, using prediction. 

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Here are a couple of books using the model-based approach: 

Chambers, R, and Clark, R(2012), An Introduction to Model-Based Survey Sampling with Applications, Oxford Statistical Science Series

Valliant, R, Dorfman, A.H., and Royall, R.M.(2000), Finite Population Sampling and Inference: A Prediction Approach, Wiley Series in Probability and Statistics

Many other textbooks at least have some information on this approach.

Also consider this:

Royall, R.M.(1992), "The model based (prediction) approach to finite population sampling theory," Institute of Mathematical Statistics Lecture Notes - Monograph Series, Volume 17, pp. 225-240. 

The paper is available under Project Euclid, open access:

 https://projecteuclid.org/euclid.lnms/1215458849. 

Also please note the following on using covariates:

https://www.researchgate.net/publication/331481770_Comparing_Alternatives_for_Estimation_from_Nonprobability_Samples

So yes, inference can be done with a nonrandom sample, if you use prediction (regression), have good regressor/predictor/auxiliary data on the entire population, and choose your data so that bias and variance together are not too high.  Since variance is easier to measure, I suggest minimizing bias first. 

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Please note the following:

Inference based on randomization means using probability in a way that should "average out" over infinitely many repeated applications - though the current application could be highly misleading.  However, the prediction-based approach means considering the sample actually collected. 

However, as in regression applications in general, you can have a sample problem.  That is the reason for cross-validation rather than only looking at model fit to a particular sample, using say, graphical residual analyses. 

The model-based approach can be used with balanced sampling, or you can use a random sample - if you are being less rigorous.  :-) 

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But when people say that they have a nonrandom sample, and it is just a convenience sample with no auxiliary data, that is quite problematic and not recommended for inference. 

Here a "purposive sample" is noted.  That could fit with a prediction-based approach, but since auxiliary data were not mentioned, I doubt it.  But it depends upon circumstances and what auxiliary data are available, and how the purposive sample was selected. 

Cheers - Jim

Just to add some more weight to Jim's arguments, please have a look at "Comparison of Purposive and Random Sampling Schemes for Estimating Capital Expenditure." Author(s): T. S. Karmel and Malti Jain. Source: Journal of the American Statistical Association, Vol. 82, No. 397 (Mar., 1987), pp. 52-57. You may then want to re-assess the value of a model-based approach.