Introduction to Sufficient Estimator and Sufficient Statistics
An estimator $\hat{\theta}$ is sufficient if it makes so much use of the information in the sample that no other estimator could extract from the sample, additional information about the population parameter being estimated.
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The sample mean $\overline{X}$ utilizes all the values included in the sample so it is a sufficient estimatorof the population mean $\mu$.
Sufficient estimators are often used to develop the estimator that has minimum variance among all unbiased estimators (MVUE).
If a sufficient estimator exists, no other estimator from the sample can provide additional information about the population being estimated.
If there is a sufficient estimator, then there is no need to consider any of the non-sufficient estimators. A good estimator is a function of sufficient statistics.
Let $X_1, X_2,\cdots, X_n$ be a random sample from a probability distribution with unknown parameter $\theta$, then this statistic (estimator) $U=g(X_1, X_,\cdots, X_n)$ observation gives $U=g(X_1, X_2,\cdots, X_n)$ does not depend upon population parameter $\Theta$.
Sufficient Statistics Example
The sample mean$\overline{X}$ is sufficient for the population mean $\mu$ of a normal distribution with known variance. Once the sample mean is known, no further information about the population mean $\mu$ can be obtained from the sample itself, while the median is not sufficient for the mean; even if the median of the sample is known, knowing the sample itself would provide further information about the population mean $\mu$.
Mathematical Definition of Sufficiency
Suppose that $X_1,X_2,\cdots,X_n \sim p(x;\theta)$. $T$ is sufficient for $\theta$ if the conditional distribution of $X_1,X_2,\cdots, X_n|T$ does not depend upon $\theta$. Thus \[p(x_1,x_2,\cdots,x_n|t;\theta)=p(x_1,x_2,\cdots,x_n|t)\] This means that we can replace $X_1,X_2,\cdots,X_n$ with $T(X_1,X_2,\cdots,X_n)$ without losing information.
Using Descriptive statistics we can organize the data to get the general pattern of the data and check where data values tend to concentrate and try to expose extreme or unusual data values. Let us start learning about the Frequency Distribution Table and its construction.
Frequency and Frequency Distribution
A frequency distribution is a compact form of data in a table that displays the categories of observations according to their magnitudes and frequencies, such that similar or identical numerical values are grouped. The categories are also known as groups, class intervals, or simply classes. The classes must be mutually exclusive, showing the number of observations in each class. The number of values falling in a particular category is called the frequency of that category, denoted by $f$.
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A Frequency Distribution Table shows us a summarized grouping of data divided into mutually exclusive classesand the number of occurrences in a class. Frequency distribution is a way of showing raw (ungrouped or unorganized) data into grouped or organized data to show results of sales, production, income, loan, death rates, height, weight, temperature, etc.
Relative Frequency
The relative frequency of a category is the proportion of observed frequency to the total frequency, obtained by dividing the observed frequency by the total frequency and denoted by r.f. The sum of the RF column should be one, except for rounding errors. Multiplying each relative frequencyof a class by 100, we can get the percentage occurrence of a class. A relative frequency captures the relationship between a class total and the total number of observations.
Decide on the number of classes. The number of classes is usually between 5 and 20. Too many classes or too few classes might not reveal the basic shape of the data set, also it will be difficult to interpret such a frequency distribution. The maximum number of classes may be determined by the formula: \[\text{Number of Classes} = C = 1 + 3.3 log (n)\] \[\text{or} \quad C = \sqrt{n} \quad {approximately}\]where $n$ is the total number of observations in the data.
Calculate the range of the data ($Range = Max – Min$) by finding the minimum and maximum data values. The range will be used to determine the class interval or class width.
Decide about the width of the class denoted by h and obtained by \[h = \frac{\text{Range}}{\text{Number of Classes}}= \frac{R}{C} \] Generally, the class interval or class width is the same for all classes. The classes all taken together must cover at least the distance from the lowest value (minimum) in the data set to the highest (maximum) value. Also note that equal class intervals are preferred in frequency distribution, while unequal class intervals may be necessary in certain situations to avoid a large number of empty or almost empty classes.
Decide the individual class limits and select a suitable starting point for the first class, which is arbitrary; it may be less than or equal to the minimum value. Usually, it is started before the minimum value in such a way that the midpoint (the average of the lower and upper-class limits of the first class) is properly placed.
Take an observation and mark a vertical bar (|) for the class it belongs to. A running tally is kept till the last observation. The tally counts indicate five.
Find the frequencies, relative frequency, cumulative frequency, etc., as required.
Frequency Distribution Table
A frequency distribution is said to be skewed when its mean and median are different. The kurtosis of a frequency distribution is the concentration of scores at the mean, or how peaked the distribution appears if depicted graphically, for example, in a histogram. If the distribution is more peaked than the normal distribution, it is said to be leptokurtic; if less peaked, it is said to be platykurtic.
There are many objectives of time series analysis. The one of major Objectives of Time Series is to identify the underlying structure of the Time Series represented by a sequence of observations by breaking it down into its components (Secular Trend, Seasonal Variation, Cyclical Trend, Irregular Variation).
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Objectives of Time Series Analysis
The objectives of Time Series Analysis are classified as follows:
Description
Explanation
Prediction
Control
The description of the objectives of time series analysis is as follows:
Description of Time Series Analysis
The first step in the analysis is to plot the data and obtain simple descriptive measures (such as plotting data, looking for trends, seasonal fluctuations, and so on) of the main properties of the series. In the above figure, there is a regular seasonal pattern of price change although this price pattern is not consistent. The Graph enables us to look for “wild” observations or outliers (not appear to be consistent with the rest of the data). Graphing the time series makes possible the presence of turning points where the upward trend suddenly changed to a downward trend. If there is a turning point, different models may have to be fitted to the two parts of the series.
Explanation
Observations were taken on two or more variables, making it possible to use the variation in a one-time series to explain the variation in another series. This may lead to a deeper understanding. A multiple regression model may be helpful in this case.
Prediction
Given an observed time series, one may want to predict the future values of the series. It is an important task in sales forecasting and is the analysis of economic and industrial time series. Prediction and forecasting are used interchangeably.
Control
When time series is generated to measure the quality of a manufacturing process (the aim may be) to control the process. Control procedures are of several different kinds. In quality control, the observations are plotted on a control chart and the controller takes action as a result of studying the charts. A stochastic model is fitted to the series. Future values of the series are predicted and then the input process variables are adjusted to keep the process on target.
Image taken from: http://archive.stats.govt.nz
The figure shows that there is a regular seasonal pattern of price change although this price pattern is not consistent.
In quality control, the observations are plotted on the control chart and the controller takes action as a result of studying the charts.
A stochastic model is fitted to the series. Future values of the series are predicted and then the input process variables are adjusted to keep the process on target.
