I am studying so-called alternating renewal process, and I have a question concerning estimation of on-time/off-time distributions.
Settings
Let $\{(Z_0,Y_0),(Z_1,Y_1),(Z_2,Y_2),...\}$ be a sequence of pairs of positive random variables in $\mathbb{Z}_{>0}=\{1,2,3,\cdots\}$, and assume that $Y_i$ and $Z_i$ are independent and identically distributed random variables whose associated distributions are $f_Y(m)$ and $f_Z(m)$, respectively. Also, let $t\in \mathbb{Z}_{>0}$. Then, one can introduce $S_{Z;n},S_{Y;n}$ and $N_t$ by $$S_{Y;n}=\sum_{i=0}^{n}(Z_i+Y_i),\quad S_{Z;n}=S_{Y;n-1}+Z_n,\ where\ S_{Y;-1}:=0;$$$$N_t=\mathrm{card}(\{n|S_{Y;n}\in[0,t)\}).$$ Now, define two positive constant integers $t_0$ and $T$, and observe the process for $t\in[t_0,t_0+T]$. The observation procedure can be given as follows. (1) If $S_{Z;N_{t_0}}\le t_0$, define $Y_*$ by $Y_*=S_{Y;N_{t_0}}-t_0$, or otherwise, define $Z_*$ by $Z_*=S_{Z;N_{t_0}}-t_0$. (2) Similarly, if $S_{Z;N_{{t_0}+T}}< t_0+T$, define $Y_{**}$ by $Y_{**}=t_0+T-S_{Z;N_{{t_0}+T}}$, or otherwise define $Z_{**}$ by $Z_{**}=t_0+T-S_{Y;N_{t_0+T}-1}$. (3) We obtain a new sequence (hereafter "subsequence") $O=\{[(Z_*,Y_{N_{t_0}})\ or\ (0,Y_*)],(Z_{N_{t_0}+1},Y_{N_{t_0}+1}),(Z_{N_{t_0}+2},Y_{N_{t_0}+2}),\cdots,(Z_{N_{t_0+T}-1},Y_{N_{t_0+T}-1}),$$[(Z_{**},0)\ or\ (Z_{N_{t_0+T}},Y_{**})]\}$. Repeating this observation for independent $\{(Z_i,Y_i)\}$ sequences gives us ensemble of subsequences.
Question
Can we construct estimators of $f_Z(m)$ and $f_Y(m)$ (for $1\le m\le T$) from the subsequence $O$, or from the ensemble of subsequences? If yes, how can we do that? If no, what extra assumptions are needed (for example, what if we know $f_Z(m)\propto\exp(-m/\lambda)$, or how rapidly should $f_Z(m)$ and $f_Y(m)$ decay)?
I appreciate that one can naively make a "histogram" of $Z_i$ and $Y_i$ by defining $p_Z(m)$ and $p_Y(m)$ as $$p_X(m):=\frac{\mathrm{card}\{n\in[N_{t_0},N_{t_0+T}-1]|X_n=m\}}{N_{t_0+T}-N_{t_0}}\ where\ X=Z,Y,$$ and that one can expect that $p_Z(m)$ and $p_Y(m)$ converge to $f_Z(m)$ and $f_Y(m)$ respectively in a limit of $T\rightarrow\infty$. However, if $T$ is finite, I suspect that the problem gets complicated for two reasons: (1) The process is censored by a window of finite length, and thus we have to compensate the missing intervals, (2) The denominator $N_{t_0+T}-N_t$ of $p_X(m)$ $(X=Z,Y)$ is also a random number and is sensitive to rare events of large $m$. Thus, I have no idea how I can appropriately estimate $f_Z(m),f_Y(m)$ so far.
I would be grateful if you share your idea to tackle this problem, or you recommend me some literatures (if any) concerning this problem.
Note
My numerical experience with the case $f_Z(m)\propto m^{-6/5}\exp(m/\xi_{Z})$ and $f_Y(m)\propto \exp(m/\xi_{Y})$ with $\xi_{Y}\ll T$ tells me that the naive histogram over many realization, that is, estimating $p_X(m)$ from set of independent subsequences $\{O_i\}_{i=1}^M$ by $$p_X(m):=\frac{\sum_{i=1}^M\mathrm{card}\{n\in[N_{t_0;i},N_{t_0+T;i}-1]|X_n=m\}}{\sum_{i=1}^M(N_{t_0+T;i}-N_{t_0;i})}\ where\ X=Z,Y$$ underestimates $f_Z(t)$ (in particular $\xi_{Z}$) for $\xi_{Z}\sim T$. Meanwhile I have found that the two estimators shown below give us rather reasonable agreement with $f_Y(t)$:
Calculating the sample probability for each realization $O_i(i=1,2,\cdots,M)$ and averaging over the ensemble; that is, $$q_X(m):=\sum_{i=1}^{M}\frac{\mathrm{card}\{n\in[N_{t_0;i},N_{t_0+T;i}-1]|X_n=m\}}{M(N_{t_0+T;i}-N_{t_0;i})}.$$ It should be noted that $q_X(m)$ itself does not satisfy normalization in general, that is, $\sum_{i=1}^{T-1}q_X(m)\ne 1$.
Giving a correction to the forementioned histogram by multiplying $1/(1-m/T)$ for each $m$, considering that the interval of length $l$ is censored by the window (and therefore cannot be observed) with probability of approximately $m/T$, even if either of the edge of the interval is inside the window: $$r(m):=p_X(m):=\frac{\sum_{i=1}^M(\mathrm{card}\{n\in[N_{t_0;i},N_{t_0+T;i}-1]|X_n=m\}/(1-m/T))}{\sum_{i=1}^M\sum_{m=1}^T(\mathrm{card}\{n\in[N_{t_0;i},N_{t_0+T;i}-1]|X_n=m\}/(1-m/T))}.$$
Also, it seems to me that giving a rigorous estimation of the estimation is rather complicated task (e.g. E. E. Alvarez, Journal of Statistical Planning and Inference 131, 209-229 (2005)), and thus I am wondering how we can evaluate the "goodness" of the heuristically constructed (approximate) estimators. Any suggestions or comments are gratefully appreciated.