Model misspecification effects for biased samples

Abstract

The model misspecification effects on the maximum likelihood estimator are studied when a biased sample is treated as a random one as well as when a random sample is treated as a biased one. The relation between the existence of a consistent estimator under model misspecification and the completeness of the distribution is also considered. The cases of the weight invariant distribution and the scale parameter distribution are examined and finally an example is presented to illustrate the results.

Appendix

Proof of Lemma 1

For x sufficiently large we have

$$\begin{aligned} \int _{x/2}^{x}f(t)dt> f(x)\int _{x/2}^{x}dt=\frac{1}{2}xf(x), \end{aligned}$$

and since $\int _0^{\infty }f(t)dt\,{<}\,\infty$ from the Cauchy criterion for proper integrals (Sharma 2004) we obtain $\lim \nolimits _{x \rightarrow \infty } xf(x)<2\lim \nolimits _{x \rightarrow \infty } \int \nolimits _{x/2}^{x}f(t)dt =0$.

The same argument holds when $x \,{\rightarrow }\, 0+$. If f is locally increasing and thus bounded in a neighborhood of $x_0\,{=}\,0$, the result is trivial. If f is locally decreasing then for x close to 0 we obtain

$$\begin{aligned} \int _{x/2}^{x}f(t)dt> f(x)\int _{x/2}^{x}dt=\frac{x}{2}f(x) \end{aligned}$$

from which the result follows trivially for $x \,{\rightarrow }\, 0+$. $\square$