# Why are estimation results sharply different from the actual population (exponential distr.)

For my work, I analyze the field failure data sometimes and make decisions accordingly. (spare parts quantity, optimum preventive replacement point, ...)

I made an experiment in Excel. Using inverse cumulative function, I created 10^3 x (operational duration at failure) data.

$F(x) = 1 − \exp(−λ x)$, and $\text{INV}F(u) = −\ln(1−u) / λ$

I set $λ$ as 250. I checked mean, standard deviation and $λ$ to control my process. Everything seems fine. ( mean = standard deviation = $1/λ$ )

Since defense industry is not lucky enough to have lots of data from its sources, I select 24 of that 10^3 data which obey this rule: 164 < selected data < 331

164 and 331 have no special aim, just for experiment.

Then I analyzed my 24 data and those 24 were best explained by 2 parameters Weibull distribution. Parameter values were: (6.43, 264.73)

6.43 is the shape parameter and since it's>1, this means that my item is wearing out. however my 10^3 population was a perfect exponential and exponential means that failure rate is constant (no wear out)

I want to ask: Why was I so far away from exponential distribution with my 24 chosen data? Did putting boundaries for those 24 data create biased data and destroyed my experiment?

• "data which obey this rule: 164" ... what does this mean? "164 and 331 have no special aim, just for experiment"? I really haven't the faintest concept of what you might mean by these sentences. – Glen_b -Reinstate Monica Dec 30 '14 at 9:32
• sorry, I edited. I selected 24 data that are between 164 & 331. As I've said, no special meaning, I just tried to increase level of reality. In my 10^3 set, data was between 6 & 3076. – Andre Chenier Dec 30 '14 at 10:10
• The distribution you are sampling from is not exponential if you set those limits on the range. The distribution would look like just the part of pdf of exponential that lies with in that range. It won't actually look like an exponential – Kamster Dec 30 '14 at 10:25

By selecting data within a certain range, it no longer has the original distribution. You have changed it so it is no longer exponential

(It actually becomes a doubly-truncated exponential; equivalently an exponential that's shifted and truncated. No matter how you describe it, it's not exponential.)

$\hspace{2cm}^\text{Histogram of large random sample, and theoretical density for exponential }$
$\hspace{2cm}^\text{on the left, and doubly truncated exponential on the right.}$

So it's hardly surprising that a different distribution fits the data better.

Did putting boundaries for those 24 data create biased data and destroyed my experiment?

Well, I wouldn't put it quite that way, but you have the gist of it.