Why do we not care about completeness, sufficiency of an estimator as much anymore? When we begin to learn Statistics, we learn about seemingly important class of estimators that satisfy the properties sufficiency and completeness. However, when I read recent articles in Statistics I can hardly find any papers that address complete sufficient statistics. Why would we not care about completeness, sufficiency of an estimator as much anymore? 
 A: We still care. However, a large part of statistics is now based on a data-driven approach where these concepts may not be essential or there are many other important concepts. 
With computation power and lots of data, a large body of statistics is devoted to provide models that solve specific problems (such as forecasting or classification) that can be tested using the given data and cross-validation strategies. So, in these applications, the most important characteristics of  models are that they have a good fit to the data and claimed ability to forecast out of sample.
Furthermore, some years ago, we were very interested in unbiased estimators. We still are. However, in that time, in rare situations one could consider to use an estimator that is not unbiased. In situations where we are interested in out of sample forecasts, we may accept  an estimator that is clearly biased (such as Ridge Regression, LASSO and Elastic Net) if they are able to reduce the out of sample forecast error. Using these estimators actually we “pay” with bias to reduce the variance of the error or the possibility of overfitting.
This new focus of the literature has also brought new concepts such as sparsistency. In statistical learning theory we study lots of bounds to understand the ability of the generalization of a model (this is crucial). See for instance the beautiful book  "Learning From Data" by Abu-Mostafa et al.
Related fields such as econometrics have also been suffering the impact of these changes. Since  this field is strongly based on statistical inference and it is fundamental to work with unbiased estimators associated with models that come from the theory, the changes are slower. However, several attempts have been introduced and machine learning (statistical learning) is becoming essential to deal for instance high dimensional databases. 
Why is that? 
Because economists, in several situations, are interested in the coefficients and not in the predictable variable. For instance, imagine a work that tries to explain corruption-level using a regression model such as: $$\text{corruptionLevel} = \beta_0 + \beta_1 \text{yearsInPrison} + \beta_2 \text{numberConvicted} + \cdots$$
Note that the coefficients $\beta_1$ and $\beta_2$ provide information to guide the public policy. Depending on the values of the coefficients, different public policies will be carried out. So, they cannot be biased. 
If the idea is that we should trust in the coefficients of the econometric regression model and we are working with high dimensional databases, maybe we may accept to pay with some bias to receive in return lower variance: “Bias-variance tradeoff holds not only for forecasts (which in the case of a linear model are simply linear combinatons of the estimated coefficients) but also for individual coefficients. One can estimate individual coefficients more accurately (in terms of expected squared error) by introducing bias so as to cut variance. So in that sense biased estimators can be desirable. Remember: we aim at finding the true value. Unbiasedness does not help if variance is large and our estimates lie far away from the true value on average across repeated samples.” - @Richard_Hardy
This idea has motivated researchers to look for solutions that sound good for economists as well. Recent literature has approached this problem by choosing focus variables that are not penalized. These focus variables are the ones that are important to guide public policy. In order to avoid the omitted variables bias, they also run a regression of this focus variables on all the other independent variables using a shrinking procedure (such as Lasso). The ones with coefficients different from zero are also included in the regression model as well. They ensure that asymptotics of this procedure is good. See here a paper of one of the leader of the field. See for instance this overview by leaders of the field.
A: We do care but usually either the issue is taken care of, or we're not making a specific distributional assumption with which we could apply those considerations.


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*Many of the usual estimators for  commonly used parametric models are either fully efficient under the usual distributional assumptions for that model or asymptotically efficient under those model assumptions. Unless we're dealing with fairly small sample sizes, there's nothing to do.
Consider generalized linear models as an obvious example.

*We often don't have a fully explicit parametric distributional model. We might use a robust procedure, or we might be looking at some convenient estimator along with a bootstrap for dealing with bias and estimating standard error.
Without an explicit distribution to even start looking at sufficiency or completeness for, there's nothing to do.
(Consider that there may be little point in finding an efficient estimator for a model you're sure will be wrong... what might make more sense would be finding one that does reasonably well in some kind of neighborhood of an approximate model. A good part of the theory for robustness takes a particular sense of the word "neighborhood" when considering a question like this.)

In the comments below Nick Cox points out that "deviations from the ideal -- are often perfectly tolerable"; this is certainly the case. Box wrote "Remember that all models are wrong; the practical question is how wrong do they have to be to not be useful." To me this is a pretty central issue, but I'd add "and in what particular ways" after "how wrong". 
It's important to understand the behavior of the tools we use away from the situation they're best at; when do they perform quite well, when do they perform badly (and hopefully what else might do at least as well in a similar range of circumstances). 
We need to keep in mind that statistical tools like tests, estimates and intervals all have several senses in which we expect them to 'perform' (e.g. significance level and power, bias and variance, interval width and coverage); for example, there's often a tendency to focus very hard on significance level on tests without paying attention to power.
These issues are less clean-cut than looking at completeness or sufficiency, and we don't have a nice array of "neat" theorems to use. In many cases we may need to use coarser but simpler tools - like simulation - to get much of a sense of what may happen. [In some situations it helps to understand something of the tools of robustness to have clues about what things it might make sense to simulate. It's good to have a sense of what it takes to make something go completely off the rails. I've seen people report that a test has "good robustness to skewness" while simulating nothing more extreme than an exponential distribution, for example, and only examining type I error rate.]
